Let's Do Mathematics 6 – Worktext A

n ca t Wo r k t e xt x

6A

for learner learners 11 - 12 yea r s o l d

Aligned to the US Common Core State Standards

n ca t Wo r k t e xt x

6A

for le arne arners 11 - 12 yea r s o l d

Let’s Do Mathematics

Re ga le du ca tio n

Let’s Do Mathematics is a series covering levels K-6 and is fully aligned to o the United States Common Core State Standards (USCCSS). Each level consists of two books boo (Book A and Book B) and combines textbook-style presentation of concepts oncepts pts as well as workbook practice.

Central to the USCCSS is the promotion of problem-solving skills ls and reasoning.

Let’s Do Mathematics achieves this by teaching and presenting ng g concepts through throu a problem-solving based pedagogy and using the concrete-pictorial-abstract pictorial-abstract torial-abstract ((CPA) approach. Learners acquire knowledge and understanding a ng off concepts through thr th guided progression beginning with concrete examples and experiences which then w flow into pictorial representations and finally mastery at the and symbolic level. e abstract an a This approach ensures that learners develop a fundamental understanding of concepts damental ental understa underst rather than answering questions by learned procedures edures es and algorithms. algorit

Key features of the series include:

Anchor Task

1

3

Integers Inte

Anchorr Task

Open-ended activities serve as the starting point for understanding new vities concepts. Learners engage in activities e and discussions to form concrete experiences before the conceptt is formalized.

Fractions

Anchor Task

o

ChicFinae g

Cheesecake Recipe

–8ºCghh: 4º4

Serves 10 people

Prep Time Cook Time Cooling Time

2º Hig 2º 1 12 LLow: ––1

Cheesecake

–2

12

y Tuesda

sd Wedne

0

8

ay

–2

–14

0

–7

14 Graham Crackers

1

1 cup pecans 2

4 tbsp butter

1 cup sugar 4 1 tsp cinnamon 2

3 1 tbsp flour 2

5 eggs

2 egg yolks

Friday

2

Let’s Learn n

Crust

2 1 lbs cream cheese 2 3 cup sour cream 4 1 tsp salt 2 3 1 cups sugar 4

–12

4

y Monda

ay Thursd

Concepts are presented sented in a clear and a colorful manner.. Worked rked problems problem provide learners step-by-step ers with ith guided step-by ste progression through Series gh examples. Ser mascots provide rovide de guidance through throug throu helpful comments observations omments ments and observa when new concepts oncepts are introduced. intro

30 minutes 1 hour 15 minutes

Ingredients

98 9 8

Let’s ’ L Learn

When findi ng the area of triangles, height. we first

need to find

Lets find the

the base and

base and heig

(a)

A

hts of som

e more trian

gles.

N

The height must be perpendic ular to the base.

In this triangle, the height is not a side length.

height

B

M

We can choo se any side of the trian let's take the gle to be the base to be base. For the BC. triangle ABC, The height of the trian gle is chosen base . This is a right given by the perpendi cular heig height.

-angled trian ht to our gle, so AB is the perpendi cular BC is the base and AB is the height.

P

C

(b)

base

O

If we choose the base to be MO, the is perpendi cular to the height is given base. by the line NP

which

T

A

height

height

S

base

B

base

C

For a base SU, the perp endicular heig triangle at line TV. ht is

U

located outs ide

V

the

26

27

ii

Let’s Practice

D

(d)

Let’s Practice 1.

Identify the base and

height of the triangles

base =

.

height =

B

(a)

Learners demonstrate their understanding of concepts through a range of exercises and problems to be completed in a classroom environment.

Questions provide a varying degree of guidance and scaffolding as learners progress to mastery of the concepts.

base = height = F

G

E

C

A

F

Re ga le du ca tio n

(e)

Q

(b)

base =

base =

ight = height

height =

I

H

G

R

S

P

W

(f)

M

(c)

base =

height =

base =

height =

Z

X

Y

33

P

O

At Home

1.

Complete the followin g. Show your working in its simplest form. and write your answer

(a)

4 x 7 5

At Home

Further practice designed to be completed without the guidance of a teacher. Exercises and problems in this section follow on from those completed under Let’s Practice.

(c)

2.

Multiply the mixed numbers. Show your yo working and answer in its simples write your t form. orm.

(a) 3 2 x 5 5

(b) 6 x 2 3

5 x 8 12

(d) 10 x 5 6

(b) 2 x 3 5 8

(c)

(e) 6 x 3 7

(f)

4x22 3

(d) 7 x 3 3 5

7 x 9 5

(e) 12 x 4 1 8

(f)

8x53 12

1 12

1 13

Hands On

Learners are encouraged to ‘learn by doing’ through the use of group activities and the use of mathematical manipulatives.

ngle with the Hands On area of a recta is half the of a triangle ao the area Show that height. and he 12 cm. same base a height of 16 cm and of h widt has a gle below The rectangle of 192 cm2 . height of 12 area cm and a It has an ar base of 16 a has te page the opposite tri gle on The trian th so the dotted lines the m. g cm. up alon to neatly fill . Then cut grid below m the page triangle from nge pieces in the Cut out the rrange the 3 pieces. Arra triangle is in rectangle. half of the

Solve It!

The figures are made up of semicircles (half circles) and straight lines.

Can you find the area of each figure? Take π = 3.14 and round off your answer to 1 decimal place.

(a)

Solve It!

2m

38

2m

Activities that require learners earners ers to apply logical reasoning problem-solving. Problems ng and nd problem-solvin problem-s hich do o not have a rou are often posed which routine strategy rners are encouraged encourag enc for solving them. Learners to think creatively and apply problem-solving y a range of probl p heuristics.

(b)

2 cm

2 cm

5 cm

2.

The figure below is made from a square of side length 6 cm.

The circular hole in the middle has a diamete r that is 2 the side length of the square. Find the 3 area of the figure. Take π = 3.14 and round off your answer to 1 decima l place.

21

as a decimal p the percentage Express Exp

4.

Looking Back

1.

Looking Back

(a) 12%

What percentage of each square is

colored?

(a)

(b)

(c)

Consolidated solidated practice where whe learners demonstrate on a emonstrate their understanding under u range ange of concepts taught taug within a unit.

(c)

(e) 70%

(d)

5.

2.

Color 14% of the square.

3.

42%

and fraction in its simplest form. (b) 28%

(d) 86%

(f)

50%

percentage.

Express the fraction as a decimal and 4 17 (b) 20 (a) 100

Color 45% of the square.

(c)

15 60

(d)

66 88

235

234

iii

Contents 2 3 12 17 27

Re ga le du ca tio n 1

iv

Integers Understanding Integers Comparing and Ordering Integers Operations on Integers Word Problems

2 Algebra Algebraic Expressions pressions sions Evaluating Algebraic Expressions Simplifying Algebraic Expressions Solving Algebraic Expressions pressions essions Word Problems

66 38 38 56 66 76 84

3 Fractions Fraction ractio Multiplying Fractions d Division Divisio Fractions and Word Problems roblems ems

98 99 116 126

4 Ratio Fraction Ratio o and Fracti Ratio and P Prop Proportion Word Proble P Problems

138 138 168 184

194 194 204 2 218

6 Mid-year Exam Section A Section B Section C

240 240 248 255

ati on

5 Percentage What Is Percentage?

Finding Percentage Percentage Increase and Decrease

v

Integers

on

1

Anchor Task

o g a c i h C Fine

–8ºC h: 4º

ºH 2 1 – : w Lo

y Monda y Tuesda sday e n d e W y Thursda Friday

2

ig

–12

4 12

–2

8

0

–2

–14

0

–7

Understanding Integers

Re ga le du ca tio n

Let’s Learn

Integers are whole numbers. The numbers on this number ber line ine are integ integers.

0

1

2

3

4

5

6

7

Are decim decimals and fractions nd fra inte integers?

8

ections ns from 0. N The number line continues in both directions Numbers to the left of 0 are less than 0. Integers that are e less than 0 are ar negative integers.

Integers that are greater than 0 are re e called positive positiv integers.

–5

–4

–3

–2

–1

0

1

2

3

4

5

We use a minus sign (–) to show an integer is negative.

We write: –5 We say: negative e five

0 is an integer nor negative. eger that hat is neither neithe positive p

negative integers nega

positive integers

0

3

Re ga le du ca tio n

When the temperature falls below 0ºC, we read the temperature as a negative number. ºC

ºC

30

30

20

20

10

10

0

0

–10

–10

–20

–20

The temperature is 5ºC.

The he tem temperature has fa fallen by 10ºC. It is now –5ºC.

The temperature erature is –5ºC. –5 10 units nits

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

Compare the thermometers. How ow has the t temperature changed? ºC

30

ºC

30

20

20

10

10

0

0

–10

–10

–20

–20

The temperature increased by 5ºC. It is now 0ºC. 4

0ºC is 5ºC warmer than –5ºC. –5ºC is 5ºC cooler than 0ºC.

Pairs of integers that are the same distance from 0 are called opposites.

5 units

Re ga le du ca tio n

5 units 2 units

–5

–4

–3

–2

–1

2 units

0

1

2

3

4

5

0 is its it own oppos opposite.

–2 and 2 are opposites

–5 and 5 are opposites

We can find the opposite of an integer by writing a min minus sign in front of it.

Integer

5 –4 12 –1 24

Opposite

The negative of a negative is a positive!

–5 –(–4) = 4 –12 –(–1) = 1 –24

Consider the opposites sites –4 and 4.

4 4 units nits

–5

–4

–3

–2 2

4 units un

–11

0

1

2

3

4

5

ntegers ers are both an equal distance of 4 units from 0.

These integers An integer's eger'ss distance from fro 0 is expressed as its absolute value.

Both an absolute value of 4. oth –4 and 4 have a

We write: |–4| We say: The absolute value of negative 4.

5

Let’s Practice Write the integer indicated on the number line. Find its opposite. pposite. te.

Re ga le du ca tio n

1.

opposite:

(a)

–4

(b)

0

2

4

0

2

4

0

10

20

0

2

4

0

20

40

opposite:

–4

(c)

–2

opposite:

–20

(d)

–10

opposite: oppos pposite: ite

–4

(e)

–2

opposite: opp

–4 –40

6

–2

–20

2.

Fill in the blanks. (a)

Re ga le du ca tio n

units

–6

–5

–4

–3

–2

–1

0

1

units to the right of 1 is

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

2

3

4

5

6

2

3

4

5

6

.

(b)

units

–6

–5

–4

–3

–2

–1

0

units to the left of 1 is

.

(c)

units nits

–6

–5

–4

–3 3

–2 2

–1

units to the right of

0

is

.

(d)

units

–6

–5

–4

–3

–2 –

–1

units to the le left of

0

1

is

.

(e)

units

–6

–5

–4

–3

–2

units to the right of

–1

0

is

1

.

7

3.

Find the opposite and absolute value of each integer.

opposite: (b) –4 opposite: (c)

5 opposite:

(d) –8 opposite:

4.

ed uc ati on

(a) 3 absolute value:

absolute value:

absolute value:

absolute e value:

stions. s.

Read and answer the questions.

(a) The temperature was 2ºC. It increa increased by 5ºC. What is the temperature now?

rature re chang cha (b) The temperature changed from –6ºC to 3ºC. How much did the

eg a

ature e rise? temperature

(c)

The e temperature mperature was 0ºC and fell 12ºC. What is the temperature

now? w?

ttemperature empe (d) The temperatu is 4ºC. How much cooler is –3ºC?

8

Solve It!

ed uc ati on

Use the diagram to answer the questions.

4m

3m

2m

4m

7m

12 m

Re ga l

(a) How deep is the ocean?

(b) How much further urther er does the diver dive need to dive to reach the treasure chest? hest?

(c)

The seagull seagull sees see the th fish and dives into the ocean to catch it.

How H w far did the fish Ho f dive through the air and water?

9

At Home Read and circle the correct integer on the number line.

Re ga le du ca tio n

1.

(a) 2 units to the right of 1.

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–2

–1

0

1

2

3

4

5

6

–2 2

–1

0

1

2

3

4

5

6

–2

–11

0

1

2

3

4

5

6

–1

0

1

2

3

4

5

6

–1

0

1

2

3

4

5

6

(b) 3 units to the left of 1.

–6

(c)

–5

–4

–3

The opposite of 5.

–6

–5

–4

–3

(d) The opposite of –4.

4 –6

–5

–4

–3

(e) 4 units to the e right of –3.

– –6

(f) f)

–4 4

–3 –

–2

5 unit unitss to th the left of 2.

–6

10

–5

–5

–4

–3

–2

2.

Find the opposite and absolute value of each integer.

Re ga le du ca tio n

(a) 2 opposite:

absolute value:

(b) –3

opposite:

(c)

absolute value:

0

opposite:

absolute value::

(d) –12

opposite:

3.

absolute te value: valu

Color to show the temperature. rature. (a) 4ºC warmer than 0ºC. ºC

(b) 10ºC cooler than 8ºC. ºC

30

30

20

20

10

10

0

0

–10

–10

–20

–20

11

Comparing and Ordering Integers rs

Re ga le du ca tio n

Let’s Learn

Chelsea measured the temperature outside 4 times during ng the he day.

She recorded her observations in the table.

Time Temperature

08 00 –2ºC

12 00 3ºC

16 00 –5ºC

20 00 –8ºC

e them em on a number num nu To compare the temperatures, we can place line.

20 00

–10

–9

–8

16 00

–7

–6

–5

08 00

–4

–3

–2

12 00

–1

0

1

2

3

4

perature ture the lowest? lowe lo (a) At what time was the temperature

er to o the left. It is the lowest number.

–8 is the furthest number est at 20 00.

00 The temperature was lowest

te ratur the highest? (b) At what time was the temperature

st number umber to the th right. It is the highest number.

3 is the furthest ature e was the highe hi The temperature highest at 12 00.

(c)

Compare at 08 00 and 12 00. e the e temperatures temperatu

eft than 3 on o the number line. It is the lower number.

–2 is more left e can n write: write –2ºC < 3ºC and 3ºC > –2ºC We

Arrange the te tempe (d) Arrange temperatures from the highest to the lowest.

Let's write each e Let's temperature as they appear from right to left on the num number line. ºC –5ºC and –8ºC. 3ºC, –2ºC,

12

5

Let’s Practice Circle the numbers on the number line. Fill in the blanks to o compare. mpare.

Re ga le du ca tio n

1.

(a) 3 and 0 –6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

0

1

2

3

4

5

6

0

1

2

3

4

5

6

0

1

2

3

4

5

6

Which number is on the left?

Which number is smaller? <

(b) –2 and 1 –6

–5

–4

–3

–2

–1

Which number is on the left? eft? ft?

Which number is smaller? ller? <

(c)

5 and –3 –6

–5

–4

–3

–2

–1

Which number right? ber is on th the rig

Which ch number umber is greater? gre greate >

(d) |–2| 2| and –6 –6

–5 5

–4

–3

–2

–1

number is on the right?

Which num

W Which number is greater? >

13

2.

Write the numbers on the number line. Then arrange them from the smallest to the greatest.

Re ga le du ca tio n

(a) -1, 5 and 1

0

<

<

(b) |–3|, 4 and -4

0

<

3.

<

Compare the numbers by writing symbols. ing ng >, < or = sym ym

(a) –4

0

(d) 1

4.

8

(c)

–6

(e)) –4

–2

(f)

5

(b) 5, |–2|, –3

>

(c)

>

>

|-1|, -1,, 0 >

>

(d) 5, –2, –3

>

>

>

Arrange rrange ge the numbers number nu from the smallest to the greatest.

(a) (a 0, –5, –10, –10 1 <

<

<

<

<

(b) 2, – –2, – –3, |–15| <

14

3

ers from m the greatest gre Arrange the numbers to the smallest.

(a) 1, 3, 0

5.

(b) b) –3

–8

5

Solve It!

n

The map below shows the temperatures across Europe on a winter's er's day.

–2ºC –1ºC

–4ºC

Re ga le du ca

0ºC

–3ºC

–2ºC

3ºC

–1ºC

7ºC

4ºC

3ºC

5ºC

(a) Which city has the lowest temperature? temperat emperat

(b) Which city has the he highest highes temperature? t (c)

Name a pairr of cities that record rrecorded the same temperature. and

(d) Name me a pair pair of cities th tha that recorded opposite temperatures. and

(e)) Which cities cities reco recorded the 4 lowest temperatures? ,

,

and

15

At Home Write the numbers on the number line.

Then arrange them from the greatest to the smallest

Re ga le du ca tio n

1.

(a) –3, –4 and 0

0

>

>

(b) 2, 5 and -5

0

>

2.

>

Circle the numbers smallerr than n –3.

– Cross the numbers that are greater than tha 3.

–4 4

0

1

–12

1 –1

4

6

–9

–3

3.

>

>

>

>

>

(b) 5, – –12, 12, 9, 9 –2 >

16

5

–5

Arrange rrange ge the numbers number nu from the greatest to the smallest.

(a) (a – –4, – –1,11, –10, 10 0

2

Operations on Integers

Re ga le du ca tio n

Let’s Learn

We can use a number line to show addition and subtraction ion of integers.

The number line below shows 3 + 2 = 5.

2 units

3 units

–1

0

1

When add adding, we move rright along the nu number line.

2

3

4

5

6

The number line below shows –5 + 3 = –2.

3 units

5 units

–6

–5

–4

–3

–2

–1

0

1

s sh The number line below shows 3 + (–6) = –3.

6 units nits nit

–6 units to the left of the number line.

3 units ni

–4

–3

–2 2

–11

0

1

2

3

4

3 + (–6) = 3 – 6 = –3

Adding a nega negative integer is the same as subtracting a positive integer.

17

The number line below shows –2 – 3 = –5.

2 units –6

–5

–4

–3

–2

–1

0

1

–2 – 3 = –2 + (–3) = –5 Subtracting an integer is the same as adding its opposite.

The number line below shows –2 – (–6) = 4.

ati on

3 units

We add the oopposite of –6, which is 6.

6 units

Re ga le du

2 units

–3

–2

–1

0

1

2

3

4

Subtracting –6 is the same as adding dding its opposite, opp 6.

–2 – (–6) = –2 + 6 = 4

The product of integers ers of the same sign is positive.

You are familiar of ar with ith the product prod o positive integers.

Let's look at some me examples examples.

3 x 4 = 12

5 x 2 = 10

16 x 3 = 48

The product of negative integers is also positive. nega Let's examples. 's look at some e exam –3 3 x (–4) = 12

18

–6 x (–5) = 30

–20 x (–8) = 160

ed uc ati on

The product of integers of different signs is negative.

Let's look at the products of integers with different signs.

–3 x 6 = –18

7 x (–3) = –21

–9 x 4 = –36

The quotient of integers of the same sign is positive. tive.

You are familiar with the quotient of positive integers. ntegers. ers Let's look at some examples.

16 ÷ 2 = 8

18 ÷ 6 = 3

20 0÷4=5

The quotient of negative integers is also positive.

Let's look at some examples.

–16 ÷ (–2) = 8

–6 ÷ (–2) = 3

–28 – ÷ (–7) = 4 –2

The quotient of integers of different differen signs is negative.

Let's look at the quotients integers with different signs. tientss of inte in –10 ÷ 5 = –2

12 ÷ (–3) = –4

–9 ÷ 3 = –3

19

Let’s Practice Write an addition equation to match the number line.

Re ga le du ca tio n

1.

(a)

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–11

0

1

2

3

4

5

6

–6

–5 5

–4

–3

–2 2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6 –6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

(b)

(c)

(d)

(e)

(f) (

20

Draw an arrow on the number line to represent the addition of integers.

Complete the addition equation.

Re ga le du ca tio n

2.

(a) –3 + 2 =

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–5

–4

–3

–2

–11

0

1

2

3

4

5

6

–4

–3

–2

–1

0

1

2

3

4

5

6

–4

–3

–2

–1

0

1

2

3

4

5

6

–4

–3

–2

–1

0

1

2

3

4

5

6

(b) –4 + 6 =

–6

(c)

–1 + 5 =

–6

(d) 2 + (–4) =

–6

–5 5

(e) –5 + 3 =

–6

(f)

–5

4 + (–7) =

–6

–5

21

3.

Write a subtraction equation to match the number line.

Re ga le du ca tio n

(a)

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2 2

–11

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4 4

–3 –

–2

–1

0

1

2

3

4

5

6

–6 –6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

(b)

(c)

(d)

(e)

(f) (

22

Draw an arrow on the number line to represent the subtraction of integers. Complete the subtraction equation.

Re ga le du ca tio n

4.

(a) 1 – 5 =

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–5

–4

–3

–2

–11

0

1

2

3

4

5

6

–5 5

–4

–3

–2

–1

0

1

2

3

4

5

6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–4

–3

–2

–1

0

1

2

3

4

5

6

(b) –2 – 3 =

–6

(c)

3–6=

–6

(d) –3 – (–4) =

–6

(e) 5 – 6 =

–6

(f)

–4 – (–7 (–7) =

–6

–5

23

5.

Add or subtract the integers. (b) 8 – 9 =

(c)

2–5=

Re ga le du ca tio n

(a) –3 + 10 =

6.

7.

24

(d) 4 + (–8) =

(e) –15 + 8 =

(f)

–9 – (–6) =

(g) 20 – 22 =

(h) 16 – 20 =

(i))

–111 + 11 =

(j)

(k)

(l)

7 + (–6) =

–4 + 19 =

6 – (– 6) =

(m) 0 – 10 =

(n) –3 – 0 =

((o) o) 16 – ((– 5) =

(p) 1 + (–11) =

(q) –50 + 49 =

(r) (r

–2 – + (– 2) =

(a) 5 x 4 =

(b) –2 x 3 =

(c)

7x5=

(d) –2 x 5 =

(e) –7 x (–5) =

(f)

3 x (–3) =

(g) 4 x 6 =

(h) h) –1 x (–2) =

(i)

–4 x (–8) =

(j)

(k))

(l)

6 x (–9) =

Find the products.

–10 x 2 =

–12 –12 x ((–5 (–5) =

(m) 3 x (–16) =

(n) –8 8 x (–8) =

(o) 100 x (–2) =

(p) 7 x (–7) =

(q) 3 x (–25) =

(r)

–9 x (–9) =

(a) 14 ÷ (–2)) =

(b) (b –18 ÷ (–6) =

(c)

–20 ÷ 5 =

(d) –4 ÷ (–2) =

(e) 42 ÷ (–7) =

(f)

–30 ÷ 15 =

(g) 16 ÷ (–4) =

(h) –81 ÷ (–9) =

(i)

–28 ÷ 4 =

(j)

(k)

(l)

–40 ÷ 20 =

Find the quotients. uotients. ts.

–6 ÷ (–2 (–2) =

24 ÷ (–6) =

(m) m) 15 ÷ (–5) =

(n) –72 ÷ 9 =

(o) –100 ÷ 10 =

(p) –45 ÷ (–9) =

(q) 48 ÷ (–8) =

(r)

–56 ÷ 7 =

At Home Write an addition equation to match the number line.

Re ga le du ca tio n

1.

(a)

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–12

–10

–8

–6

–4

–2

0

2

4

6

8

10

12

(b)

2.

Draw an arrow on the number er line to represe represent the addition of integers.

Complete the addition equation. ation. n. (a) –2 + 6 =

–6

–5

–4 4

–3

–2

–11

0

1

2

3

4

5

6

–8

–6

–4

–2

0

2

4

6

8

10

12

–30 30 –25 –20 –15 –10

–5

0

5

10

15

20

25

30

(b) –12 + 8 =

–12

(c) (

–10

25 + (–3 (–35) =

25

3.

Write a subtraction equation to match the number line.

Re ga le du ca tio n

(a)

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

(b)

4.

Draw an arrow on the number line ne to represent epresent the t subtraction of integers. Complete the subtraction action equation. equation (a) 1 – 7 =

–6

–5

–4

–3

–2

–1 –

0

1

2

3

4

5

6

–4

–3

–2

–1

0

1

2

3

4

5

6

(b) –3 – 3 =

–6

5.

26

–5 5

Complete mplete ete the equations. equation (a) 4 + (–1) (–1) =

(b) –12 + (–2) =

(c)

6 + (–12) =

(d) 12 ÷ (–2) =

(e) –18 ÷ (–9) =

(f)

–27 ÷ 3 =

(g) g) 5 – 10 =

(h) –2 – (–2) =

(i)

12 – (–5) =

(j)

(k)

(l)

5 x (–6) =

–19 x 2 =

–12 x (–3) =

Let’s Learn

ati on

Word Problems When Keira woke up in the morning, the temperature outside utside e was –8ºC. –8ºC By the time she arrived at school, the temperature wass 2ºC. Find the increase in temperature.

ºC 30 20

ºC

30 20 0

10

0

0

Re ga le d

10

–10

–10

–20

–20

4 3 2 1 0 –1 –2 –3 –4 –5 –6 –7 –8 –9

temperatu mperat To find the increase in temperature, we need to find the difference. er integer from fro the higher integer.

Let's subtract the lower

2 – (–8) = 10

erature e increased by 10ºC. The temperature

27

Re ga le du ca tio n

An office building has 8 levels above ground and 5 basement levels.

Mr. Sato took the elevator from his office on the 5th floor to the 4th basement level. How many levels down did Mr. Sato move?

8 7 6 5 4 3 2 1 0 –1 –2 –3 –4 –5

Let's find the difference.

5 – (–4) = 9

wn 9 levels.

Mr. Sato moved down

Mrs. Yi has $100 Her car repayments are deducted 0 in her savings account. acco from her savings ngs account at a rate ra of $40 per week. Find Mrs. Yi's account er 4 weeks of car c repayments. re balance after

–80 0 –60 –40 –2 –20

0

20

40

60

80

100 120

00 – (4 x 40) = 100 – 160 100 –6 = –60

Mrs. Yi's account balance is –$60. Her account is $60 in debt.

28

A car consumes 9 liters of fuel per hour. Find the change in fuel after 5 hours.

Re ga le du ca tio n

mber.

We can represent the consumption of fuel as a negative number.

–9 x 5 = –45

The car will consume 45 liters of fuel in 5 hours.

ns below. below Look at the weather map and answer the questions –19ºC

0ºC 0

–6ºC

7ºC ºC C

–3ºC

(a) How much warmer is Baltimore to Charleston? more compared com 7 – (–3) = 10

It is 10ºC warmer mer in Baltimore than th Charleston.

(b) Which city New York? y is 6ºC cooler cooler than t 0 – 6 = –6

Pittsburgh cooler than New York. tsburgh rgh is 6ºC coole

(c)

What is the temperature difference between the coolest and tem tempe warmest rmest cities? cit 7 – (–19) (–19 = 26

The temperature difference between the coolest and warmest cities is 26ºC. 29

Let’s Practice Sophie is cooking in the kitchen. She takes some chicken nuggets gets from the freezer and puts them into the oven. The freezer is set to o –16ºC and the oven is set to 180ºC. Find the difference in temperature. ature. re.

2.

Mr. Finch is exercising and dieting g to lose weight. weig He loses 2 kg every month. Express Mr. Finch's change nge in weight after aft 8 months.

3.

Ethan's basement asement ment is 4 meters me below ground level. His tree house is 3 meters rs above bove the ground. gro ground Find the difference in height of Ethan's basement ment and tree house. house ho

Re ga le du ca tio n

1.

30

The graph shows the daily profits of a noodle shop from Monday to Friday.

Noodle Shop Profits

Re ga le du ca tio n

4.

30 25

20

15

10

Profit ($)

5

0

–5

–10 –15

–20

Mon

(a) (b) (c) (d)

Tue

Wed W Day

Thur

Fri

On which h days ys was the noodle noo shop profitable?

On which ch day da ay did the noodle nood shop make the greatest profit?

On which shop make the greatest loss? hich h day day did the noodle no How w much uch profit did d the th noodle shop make from Monday to o Friday? ay?

31

At Home Mercury freezes at around –39ºC. Blake checks his temperature erature ure with a mercury thermometer. He records a temperature of 37ºC. 7ºC. How many degrees Celsius above freezing is the temperature off the e mercury?

2.

A rocket burns 50 kg of fuel every the change in mass ry second. econd. Express Expre Exp of the rocket after 1 minute.

3.

Halle owes wes her sister $25. $25 She Sh repays her $17 and then borrows another $8. How w much ch money does doe Halle owe her sister now?

Re ga le du ca tio n

1.

32

The table shows the freezing and boiling temperatures of some common solvents.

Re ga le du ca tio n

4.

Solvent Water Acetic acid Ethanol Benzene Chloroform

Freezing Point (ºC) Boiling Point (ºC) 0 100 0 17 118 –115 78 6 80 –64 61

(a) What is the difference in temperature ure between etween the freezing and boiling point of water? (a) What is the difference in temperature erature e between betwee the freezing and boiling point of chloroform? (c) Which solvent has the greatest test difference in temperature between its freezing and boiling points? nts? (d) Which solvent has the smallest difference in temperature between mallest lest differenc its freezing and boiling g points? nts

33

Looking Back Find the opposite and absolute value of each integer.

Re ga le du ca tio n

1.

(a) 7

opposite:

absolute value:

(b) –1

opposite:

(c)

absolute value:

–13

opposite:

absolute value: e:

(d) 0

opposite:

2.

absolute bsolute te value:

uestions. stions Read and answer the questions.

ture wa ºC. IIt increased by 8ºC. What is the (a) The temperature was –2ºC. ure now? temperature

emperature perature change ch (b) The temperature changed from –12ºC to –8ºC. How much did the mperature rature rise? temperature

(c) c)

was 1ºC and fell 15ºC. What is the temperature The he temperature tempe

now?

The ttemperature em (d) T is 5ºC. How much cooler is –13ºC?

34

3.

Write the numbers on the number line. Then arrange them from the smallest to the greatest.

Re ga le du ca tio n

(a) -3, 3 and 0

0

<

<

(b) |–6|, 1 and -4

0

<

4.

<

Compare the numbers by writing symbols. ing ng >, < or = sym ym

(a) –2

1

(d) 1

5.

(b) b) –4

1

(e)) |–8|

8

(c)

0

(f)

–(–2)

–8

–2

ers from m the greatest gre Arrange the numbers to the smallest.

(a) –3, 1, 0

>

(c)

(b) –3, |–2|, |–3|

>

>

4, |-1|, –2 >

6.

4

>

(d) 3, –1, –7

>

>

>

Arrange rrange ge the numbers number nu from the smallest to the greatest.

(a) (a 2 2, – –15, 15, 5 –3, 3 |-10| |-10 <

<

<

<

<

(b) 1, – –5, –3 –3, |–12| <

35

7.

Write a subtraction equation to match the number line.

Re ga le du ca tio n

(a)

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

(b)

8.

Draw an arrow on the number line ne to represent epresent the t subtraction of integers. Complete the subtraction tion equation. (a) 1 – 7 =

–6

–5

–4

–3

–2

–1 –

0

1

2

3

4

5

6

–4

–3

–2

–1

0

1

2

3

4

5

6

(b) –3 – 3 =

–6

9.

36

–5 5

Complete mplete ete the equations. equation (a) 8 + (–2) =

(b) –15 + (–7) =

(c)

3 + (–12) =

(d) 18 ÷ (–2) =

(e) –14 ÷ (–7) =

(f)

–36 ÷ 9 =

(g) g) 5 – 8 =

(h) –4 – (–5) =

(i)

12 – (–1) =

(j)

(k)

(l)

7 x (–6) =

–16 x 2 =

–10 x (–3) =

Re ga le du ca tio n

10. The temperature inside an ice box is –3ºC. As the ice melts, the temperature increases by 2ºC per hour. Find the temperature ure of the t ice box after 10 hours.

11.

A motorcycle consumes 4 liters of fuel every hour.

Find the change in fuel after 4 hours. urs.

12.

At the South uth Pole, the minimum minimu min average temperature in October is –54ºC and temperature is –48ºC. nd the maximum maximu average a Find the temperature. he difference ference in te tem

37

Algebra

Algebraic Expressions Anchor Task

on

2

= 10 = 12 =7

38

Let’s Learn

Re ga le du c

on

Riley and Sophie are collecting seashells.

Riley has 5 seashells. Sophie has some seashells in a bucket.. How w many ma seashells do they have altogether?

x

5

?

The number of seashells in the bucket is an a unknown quantity. We can use the letter represent the er x to repre number of seashells in Sophie's bucket.

So Sop bucket

Riley and Sophie have ave (5 + x)) seashells. seash sea 5 + x is an algebraic of x. aic expression in terms te

In an algebraic expression, letters are used to represent unknown quantities. qu

Riley finds 3 more re seashells. seashel She now has 8 seashells in n all.

3+5+x=8+x

x

8

(8 + x)

Riley and Sophie have (8 + x) seashells in all.

39

I am 3 years older than Riley.

ed uc ati on

Let's use algebraic expressions to express the age of Riley and her siblings.

I am 2 years yea ounger than younger y.

Riley.

Jessica

Riley

Joanne Jo

ress Riley's age a as y.

Riley's age is unknown. We can express We can say, Riley is y years old.

arss older than th Riley.

We know that Jessica is 3 years We can express Jessica's age in terms ca's a's ag erms of Riley's age. ars old.

Jessica is (y + 3) years

We know that Joanne than Riley. nne is 2 years yea younger yo We can express ess Joanne's age ag in terms of Riley's age. ( – 2) years old.

Joanne iss (y

(y + 3) ye years old

40

y years old

(y – 2) years old

Re ga le du ca tio n

Each brick has the same mass. The mass of each brick is unknown. Let's use z to represent the mass of 1 brick.

z

z

z

z

z

z

?

We can say that each brick has a mass ss of z kg.

There are 6 bricks in all. We can express of the bricks using press the total mass m an algebraic expression. z+z+z+z+z+z=6xz

We write 6 x z as 6z.

The total mass of the bricks is 6z kg. bri brick kg

A ribbon has a length ngth h of t m. Wyatt Wy cut c the ribbon into 3 pieces of equal length. Express the length of the ribbon in terms of t. ngth of 1 piece p tm

?

The he length of 1 p piece o of ribbon is t ÷ 3.

We write t ÷ 3 as

t .

3

41

Re ga le du ca tio n

Blake cooked a pizza. He cut the pizza into b slices. He ate 2 slices then shared the remaining slices equally among his 4 friends. How many sslices of pizza did each friend receive?

b

?

(b – 2) ÷ 4 =

2 slices

(b – 2) 4

Each friend received

(b – 2) slices lices of pizza pizza.

4

Mrs. Johnston has 5 green to the grocery store and buys een apples. les. es. She goes g 3 bags of red apples. Each ach bag ba contains nta c apples. How many apples does Mrs. Johnston have in all?

c

5

?

5 + c + c + c = 5 + 3c 3

Mrs. Johnston has 5 + 3c apples in all.

42

Let’s Practice Write an algebraic expression.

Re ga le du ca tio n

1.

(a) Chelsea has x strawberries. She gives 3 strawberries ies to Sophie.

Sophi How many strawberries does Chelsea have now? w?

(b) Ethan scored m points on his science quiz quiz. Blake scored 5 points more than Ethan. What was score? as Blake's score scor

(c)

Riley kilometers on Saturday. She cycled 9 kilometers y cycled cled for n kilome kil on n Sunday. day. How far fa did d Riley cycle on the weekend?

43

Re ga le du ca tio n

(d) A packet contains b candies. The candies are shared equally among 5 friends. How many candies does each friend get?

(e) Riley used c beads to make a bracelet. racelet. et. She made ma 12 such bracelets. m How many beads did she use e in all? ll?

(f)

44

The total tal length gth of 12 ident iidentical tiles is d centimeters.

What at is the length of o 1 tile? ti

2.

Write an algebraic expression.

Re ga le du ca tio n

(a) 10 is added to x.

(b) Multiply m by 7.

(c)

20 is divided by y.

(d) There are 12 groups roups of t.

(e) e) 88 is subtracted subtra from z.

45

3.

Write the algebraic expression. (a)

p

Re ga le du ca tio n

5

t

(b)

(c)

m

m

f

(d)

9

(e)

12

h

46

m

Choose your own letter to write an algebraic expression. s. (a) A large bottle of orange juice is poured equally into 4 cups.

How much orange juice is in each cup?

Re ga le du ca tio n

4.

(b) Blake has a jar of marbles. Ethan gives ves him m 100 mo more marbles.

How many marbles does Blake have ave now?

(c)

Mrs. Jenkins withdraws draws ws some m money mo from the bank and spends $425 on groceries. How does Mrs. Jenkins have now? w much money m

(d) 12 childr children hildren en raised the tth same amount of money in a school much did they raise in total? fundraiser. ndraiser. How m

47

5.

Write the algebraic expression.

Re ga le du ca tio n

(a) The distance around a running track is d meters. Ethan n walks 500 5 meters then runs 8 laps around the track. What is the he total otal distance distanc Ethan covered?

(b) A tub of popcorn costs $p. An costs $2 more than a tub of n ice-cream cream cos popcorn. What is the cost of 2 tubs of pop popcorn and an ice-cream? popc

(c)

48

Riley received eived $ $x x for her birthday. She kept $5 and gave the rest to herr 3 sisters amounts. How much did each sister get? sters in equal eq a

6.

Write an algebraic expression.

Re ga le du ca tio n

(a) 10 is subtracted from the product of x and 5.

(b) The sum of y and 5 is divided by 12.

(c)

19 is subtracted from om the produc produ product of m and 10.

(d) 25 is adde added d to 12 g groups of t.

49

At Home Write an algebraic expression.

Re ga le du ca tio n

1.

(a) A monkey has g bananas. It finds 6 more bananas. s. How w many man bananas does the monkey have now?

(b) In a soccer match, Jordan scored m goals. goals Wyatt scored 2 fewer goals than Jordan. How many score? any goals did Wyatt W

(c)

50

Every q kilometers. How far does Mr. Whyte ry day, ay, Mr. Whyte Whyt drives dr drive ive in 2 weeks?

Re ga le du ca tio n

(d) Keira has a piece of ribbon that is h meters in length. She cuts the ribbon into 12 equal lengths. What is the length of 1 piece ce of ribbon? r

(e) Halle is r years old. Chelsea is 5 years younger than t Halle.

How old is Chelsea?

(f)

There are w players in a fo football team. There are 25 football teams.

How many in all? any players are a there t

51

2.

Write an algebraic expression.

Re ga le du ca tio n

(a) 15 is multiplied by x.

(b) Multiply m by 11.

(c)

29 is added to a.

(d) There are g groups roups of 8.

8

(e) e) 14 is subtr subtracted a from u.

52

3.

Write the algebraic expression. x

Re ga le du ca tio n

(a)

(b)

x

e

(c)

13

(d)

65

a

53

4.

Write the algebraic expression.

Re ga le du ca tio n

(a) An egg costs p¢. A 50% discount is applied when buying ng eggs in a 12-pack. What is the cost of a 12-pack of eggs?

ber n and doubles it. (b) A computer program takes a number e result.

It then subtracts 6 from the

(c)

54

A bottle tle contains d liters lit of o water. The water is poured in equal amounts 3 liters of water remain in the bottle. ounts ts into 4 glasses. glasses gla ow much uch water is i in each glass?

How

5.

Write an algebraic expression.

Re ga le du ca tio n

(a) 10 is added to the product of w and 50.

(b) The sum of d and 12 is divided by 12.

2.

(c)

The product of a and 10 is added adde to 14.

(d) 5 iss subtr subtracted acted from fro the quotient of 12 and t.

55

Let’s Learn A car tire has a mass of x kg.

Find the mass of 4 tires in terms of x. x kg + x kg + x kg + x kg = 4x kg

Re ga le du ca

The mass of 4 tires is 4x kg.

Find the total mass of 4 car tires when x = 6.

When x = 6 4x = 4 x x =4x6 = 24

The total mass of 4 car tires is 24 kg..

Let's evaluate te these algebraic aic expressions! xpre

d 6a 6 – 5 when h a = 7.

Find

When a = 7, 6a – 5 = 6 x 7 – 5 = 42 – 5 = 37

Find d 24 + 5y y when y = 4.

When y = 4, 4 24 + 5y 5 = 25 + 5 x 4 = 24 + 20 = 44

56

Find F

(15 – 2r) when r = 3.

3

When r = 3,

(15 – 2r) (15 – 2 x 3) = 3 3 15 – 6 = 3 9 = =3 3

n

Evaluating Algebraic Expressionss

Let’s Practice Find the value of each expression when w = 3.

Re ga le du ca tio n

1.

(a) 25 – 2w

(b) 10w + 22

(c)

15 – w 4

(d)

6w + 18 6

(e) (e 6 62 2 + 12w

57

2.

Find the value of each expression when x = 12.

Re ga le du ca tio n

(a) 21 + 2x

(b) 7x + 16

(c)

88 – 7x 2

(d)

6x + 33 5

(e)) 52 2 + 12 12x x

58

3.

Complete the tables. (a)

Value of Expression when s = 2

pressio Value of Expression ns=7 when

E Value of Expression when t = 10

Value of Expression when t = 15

Re ga le du ca tio n

Expression 20s

12s – 24 7s + 7 7

122 – 12s 10s + 14 2

(b)

Expression 0 7t + 100

2tt – 24 12t 6 + 6t 6 3

12tt – 19 12

22tt + 15 5

59

4.

Evaluate the algebraic expressions when x = 12 and y = 8.

Re ga le du ca tio n

(a) 5x – 4y + 5

(b) 62 – 3x – 2y

(c)

500 – 11x + y

(d) 9x – 12y 12 2y

60

5.

Evaluate the algebraic expressions when r = 3 and s = 15.

Re ga le du ca tio n

(a) r + 5s

(b) 10s + 5r – 100

(c)

18 – r + 5s

(d) 2s – 8 8rr + 13

61

At Home Find the value of each expression when z = 9.

Re ga le du ca tio n

1.

(a) 25 – z

(b) 9z – 11

(c)

144 – z 9

(d)

16z – 12 12

(e) (e 21 2 21zz + 2z

62

2.

Find the value of each expression when d = 8.

Re ga le du ca tio n

(a) 21 + 8d

(b) 2d + 96

(c)

98 – 7d 7

(d)

11d + 2 9

(e)) 24 24d 4d + 327

63

3.

Complete the tables. (a)

Value of Expression when k = 12

pressio Value of Expression when k = 9

E Value of Expression when c = 6

Value of Expression when c = 10

Re ga le du ca tio n

Expression 8k +8

12k – 44 56 + 2k 2

204 – 12k 12k + 8 4

(b)

Expression 6c + 4 8

100c 0c – 550 18 8 + 3c 3 6

122 + 12 12cc 4c + 14 2

64

4.

Evaluate the algebraic expressions when v = 9 and w = 12.

Re ga le du ca tio n

(a) v + 3w

(b) 10w + 10v – 100

(c)

12w – 7v 9

(d)

3w 4v + 3w v

65

Simplifying Algebraic Expressions ns

Re ga le du ca tio n

Let’s Learn

Riley, Sophie and Halle each have $x.

How much money do they have in all? $x

$x

$x

They have ($x + $x + $x) in all.

We can simplify ($x + $x + $x) as $3x.

Riley, Sophie and Halle have $3x in all..

Jordan has 3y computer games. es. Wyatt yatt has 2y 2y computer games.

How many computer games do they hey have in all?

3y

2y

They have (3y + 2y) in all.

2y) computer omputer games game ga

We can simplify plify (3y (3 3y + 2y) 2y) as 5y.

5 .

5y

Jordan and games in all. nd Wyatt have 5y 5y computer co

Simplify plify 4z 4z + 6z 6z – 2z.

2z.

2z

4zz + 6z 6z – 2z 2 = 10z 10 – 2z 2 = 8z

66

Simplify 4a + 8 + 2a + 4.

Re ga le du ca tio n

First, group the numbers together and group the unknowns together.

Then simplify.

4a + 2a = 6a 8 + 4 = 12

4a + 8 + 2a + 4 = 4a + 2a + 8 + 4 = 6a + 12

Simplify 8b + 6 – 3b + 5b – 2.

8b – 3b + 5b = 5b + 5b = 10b 6–2=4

First, group the numbers together and group the unknowns together.

Then simplify.

8b + 6 – 3b + 5b – 2 = 8b – 3b + 5b + 6 – 2 = 10b + 4

Simplify c + 2d + 4c + 12 – d – 9.

First, group the numbers and ers together to tog d the unknowns together.

Then simplify.

c + 2d + 4c + 12 – d – 9 = c + 4c 4c + 2d 2d – d + 12 – 9 = 5c 5c + d + 3 There aree 2 different unknowns, nknowns, d c and d.

c + 4c = 5c 2d – d = d 12 – 9 = 3

67

Let’s Practice Simplify. Show your working.

Re ga le du ca tio n

1.

(a) a + a + a

(b) 10x – x

(c)

y+y+y+y+y+y

(d) 3e + 13e

(e) 30 30g g – 21g 21g

68

2.

Simplify. Show your working.

Re ga le du ca tio n

(a) 2s + 3s – 7

(b) 2r + 3r + 38

(c)

10v – v – 6

(d) 19 + 2x + 4 – x

(e) 1111 + 5 5p p + 13 – 2p 2p

69

3.

Simplify. Show your working.

Re ga le du ca tio n

(a) 5d + 3d – c – d – 3d

(b) 5m + 5n + 23m + n + 5m

(c)

50b – 14b + 60c – 10c – 30c 0c + 6

(d) 2g – g + 6 + 5h –2h –2 2h + 6

(e) 70 + 2i 2i + 5 5ii – 12j 2 – 6j + 15

70

Solve It! A number is represented by v. The number is multiplied by y 3.

5 is subtracted from the result. The new result is divided d by 5.

Re ga le du ca tio n

1.

(a) Write an algebraic expression in terms of v. (b) What is the value of the final expression when n v = 5?? (c) What is the value of the final expression when hen en v = 50?

2.

A number is represented is divided by 9. ed by w. w. The number n 12 is added to the result. is multiplied by 6. es The he e new result re (a) Write an algebraic expression in terms of w. gebraic aic ex expr (b) What is the he value alue of the final fina expression when w = 18? (c) What is the v value alue of the th final fin expression when w = 81?

71

At Home Simplify. Show your working.

Re ga le du ca tio n

1.

(a) 12 x a

(b) 10p – p – p

(c)

d + d + d + d + d + d+ d + d

(d) 23h + 3h

(e) 30 30q q – 20q 20q – 5q

72

2.

Simplify. Show your working.

Re ga le du ca tio n

(a) 110 – 7e – 5e – 43

(b) 12y + 4 – 3y – 3

(c)

7b + 23b + 6 + 40

(d) 19 + 15x + 14 4 – 5x 5x

(e) 12 112m m – 3 – 2 - m + 6m

73

3.

Simplify.

Re ga le du ca tio n

(a) 15s + 3s – t – s – 13s

(b) 15m + 3n + 23m – 2n + 15m

(c)

50c – 14c + 23d – 10d – 10d 0d d + 6+ 1

(d) 8x – x + 6 x + 15h 15 5h –12 –12h h + 9 + 4 +4

(e) 26 + 12i 12 + 15i 15 – 112j – 6j + 15i – 14

74

Solve It! A number is represented by x. The number is multiplied by 5.

12 is subtracted from the number. The number is divided d by 6.

Re ga le du ca tio n

1.

(a) Write an algebraic expression in terms of x. (b) What is the value of the final expression when n x = 12?

2? (c) What is the value of the final expression when hen en x = 6?

2.

A number is represented is divided by 8. ed by y. y. The number nu 12 is added to the number. is multiplied by 15. u . The number num (a) Write an algebraic expression in terms of y. gebraic aic ex expr (b) What is the he value alue of the final fina expression when y = 64? (c) What is the value value of the th final fin expression when y = 8?

75

Solving Algebraic Equations

Re ga le du ca tio n

Let’s Learn

Halle had $x. Her grandmother gave her $10 and she had d $23 3 in all.

How much did Halle have at first? $x

$10

?

Do you know what w value of x will make equat true? the equation

x + 10 = 23 x + 10 – 10 = 23 – 10 = 13

Halle had $13 at first.

owns in an equation have been found, When the values of the unknowns quation. we say we have solved the equation.

A baker baked 48 bread read rolls. olls. He H sold y bread rolls and had 15 bread rolls left. How many bread sell? ead rolls did the baker ba 48

15 5

y + 15 5 = 48 y + 15 – 15 5 = 48 – 15 y = 33

76

y

If you change 1 side of an equation, you must always change the other side in the same way!

We can also use the balance method to visualize the solving of algebraic equations. Let's represent the equation 2x + 5 = 11 on a balance.

1 1 1 1 1 1

Re ga le du ca tio n 1 1 1 1 1

x x

1 1 1 1 1

2x + 5 = 11

Remove 5 units from both sides.

1 1 1 1 1

1 1 1 1 1

x x

1

Divide each side into equal parts. rts.

x

x

1 1 1

1 1 1

h sides es are equal. equa The parts on both x

1 1 1

Subtract both sides. ubtract ract 5 from b

1 1 1 1 1

2x + 5 – 5 = 11 – 5 2 2x = 6

Divide both sides by 2.

D 2x 6 = 2 2

We have solved the equation. x=3

Solve e the equation 2w 2 + 8 = 14.

2w w + 8 = 14 2w w + 8 – 8 = 14 – 8 2w = 6 2w ÷ 2 = 6 ÷ 2 w=3

77

Let’s Practice

Re ga le du ca tio n

Solve the equations. Show your working. (a) t + 16 = 30

(b) 14c + 10 = 52

(c)

12z – 6 = 42

(d) 4y – 20 = 100

(e) 5a 5a – 7 + 2 = 5

78

2m – 100 = 0

Re ga le du ca tio n

(f)

(g) 4e – 8 = 20

(h) 36 – 2h = 10

(i)

54 – 9u = 9

(j)

5w + 100 00 = 125 12

79

At Home

Re ga le du ca tio n

Solve the equations. Show your working. (a) 2b + 16 = 30

(b) 14t – 52 = 4

(c)

12p + 51 = 99

(d) 9w – 7 = 38

(e) 5x 5x – 50 = 450

80

123 + 2m = 149

Re ga le du ca tio n

(f)

(g) 12e – 8 = 136

(h) 126 – 14h = 0

(i)

54 + 6 + 3u = 69

(j)

25v 5v + 125 25 – 25 = 225

81

Solve It! Sophie has a piece of red ribbon which is 245 cm in length. h. She he has a piece of green ribbon which is x cm shorter than the red d ribbon. bon She has a piece of yellow ribbon which is 3 times longer ger than the green ribbon.

Re ga le du ca tio n

1.

(a) Express the length of the green ribbon in terms rms ms of x. x. (b) Express the length of the yellow ribbon in terms rms of o x. (c) Find the total length of the green and yellow when x = 12. ellow w ribbons wh w

82

Mrs. Brown earns $m per month. Mrs. Williams earns 4 times as much as Mrs. Brown.

Re ga le du ca tio n

2.

e over er a period (a) Express Mrs. Brown's and Mrs. William's total income of 6 months. (b) How much does each person earn over 1 year when n m = $1,250? $1,250

83

Word Problems

Re ga le du ca tio n

Let’s Learn

The figure below shows the length of the fences of Mr. Rolland's olland's d's paddock. paddock paddoc

2x m

x + 18 m

xm

3x m

Express the perimeter of Mr. Rolland's of x. nd's d's paddock in terms t Perimeter = 2x + x + 18 + x + 3x = 7x + 18

Find the perimeter of Mr. Rolland's when x = 13. olland's nd's paddock paddo padd

When x = 13, Perimeter = = = =

7x + 18 7 x 13 + 18 8 91 + 18 109 09 m

When x = 13, 3, the perimeter of Mr M Rolland's paddock is 109 meters.

Find the paddock when x = 25. e perimeter rimeter of Mr. Rolland's Ro When en x = 25, Perimeter rimeter = 7x + 18 = 7 x 25 + 18 = 175 + 18 = 193 m

When x = 25, the perimeter of Mr Rolland's paddock is 193 meters.

84

The table shows the number of pupils in Grades 1 to 6 attending Bright Sparks STEM School.

Re ga le du ca tio n

Student Enrollments – Bright Sparks STEM School Grade

Students Enrolled

1

w

2

w – 25

3

3w + 2

4

w + 22

5

2w

6

8 2w + 18

Express the number of students in Grades des 1 to 3 in terms term of w.

Number of students in Grades 1 to 3 = w + w – 25 + 3w + 2 = 5w 5w – 25 +2 = 5w 5w – 23 2

Express the number of students entss in Grades 4 to 6 in terms of w.

Number of students in Gradess 4 to 6 = w + 22 + 2w + 2w + 18 = 5w + 22 + 18 = 5w + 40

Find the total number students en enrolled in Grades 1 to 6 at Bright Sparks mberr of student STEM School when hen w = 85.

When w = 85, Total students ents enrolled nrolled = = = = = = =

in Grades 1 to 3 + students in Grades 4 to 6 sstudents tu tud 5w 5w – 23 + 5w + 40 10w – 23 + 40 10w + 17 10 x 85 + 17 850 + 17 867

When w = 85 85, tthe total number of students enrolled in Grades 1 to 6 at Bright Sparks STEM School is 867.

85

Let’s Practice Keira has $x. Riley has 3 times as much money as Keira. Riley's 's mother mothe gives Riley an additional $20.

Re ga le du ca tio n

1.

(a) Express the amount of money Riley has in terms ms of x. x. (b) How much money does Riley have when x = 50?

2.

Blake, Wyatt and Sophie ie each ach have a pet cat. Blake's pet cat weighs 2z kg. Wyatt's cat is twice e as heavy as a Blake's cat. Sophie's cat is half the weight of Blake's cat. at at. (a) Express the e total otal weight of o the cats in terms of z.

1 (b) Find the weight ght of each cat when z = 2 .

2

86

The width of a rectangle is r cm.

The length of the rectangle is 3 times its width.

Re ga le du ca tio n

3.

(a) Express the perimeter of the rectangle in terms of r. (b) Find the perimeter of the rectangle when r = 8. (c) Draw the rectangle and label the length and width dth when r = 3.

87

4.

The side lengths of a triangle are t cm, (t + 5 cm) and 2t cm.

Re ga le du ca tio n

(a) Express the perimeter of the triangle in terms of t. (b) Find the perimeter of the triangle when t = 15. de when t = 5. (c) Draw the triangle and label the length of each side

88

Halle, Sophie and Chelsea put their savings together. Halle contributed $r. Sophie contributed 5 times more than Halle. Chelsea contributed ntribut $30 less than Sophie.

Re ga le du ca tio n

5.

(a) Express the amount each child contributed in terms ms off r. r. (b) Find the amount each child contributed when r = 15.. (c) Find the total amount of money when r = 126.

89

Blake bought d cans of cat food for $6 each. He paid for the cat food with a $100 note.

Re ga le du ca tio n

6.

(a) Express the change Blake will receive in terms of d. (b) What change will Blake receive when d = 9? (c) Blake decides to also purchase a cat collar for $20.

Express the change Blake will receive in termss of o d. d

90

Solve It!

Re ga le du ca tio n

The perimeter of a rectangle is 56 cm.

The length of the rectangle is p cm. (a) (b) (c) (d)

Express the width of the rectangle in terms of p.

Express the area of the rectangle in terms of p.

Find the area of the rectangle when p = 18.

What shape is formed when p = 14?

91

At Home Riley has 5 kg of apples. She goes to the supermarket and d buys uys 5 bags bag of apples, each with a mass of u kg.

Re ga le du ca tio n

1.

(a) Express the mass of the apples Riley has in terms ms of u. u. (b) What is the total mass of the apples Riley hass if u = 2?

2.

Chelsea wanted to buy y f movie tickets for her friends. The tickets cost $13 each. The cashier told old Chelsea she sh s needed another $6 to buy the movie tickets. (a) Express the Chelsea had in terms of f. e amount ount of o money m (b) How much have if f = 5? ch money did Chelsea Chel C

92

Wyatt, Jordan and Ethan each have a length of rope. Wyatt's rope is 8p cm length. Jordan's rope is half as long as Wyatt's. Ethan's n's rope is 40 cm longer than Jordan's rope.

Re ga le du ca tio n

3.

(a) Express the length of each child's rope in terms off p. (b) Find the length of each child's rope if p = 22. (c) Find the total length of all the ropes if p = 17 .

93

Looking Back Mr. Whyte buys a used car for $x. He sells the used car to o his brother for $2,350 less than he paid for it. Express the price Mr. Whyte's te's brother paid for the car in terms of x.

2.

Ethan scored m points on his mathematics matics cs quiz. Blake scored 23 points more than Ethan. What was Blake's score in terms of o m?

3.

Evaluate the algebraic expressions pressions ressions when a = 3 and b = 15. (a) 75 – b + 5a

Re ga le du ca tio n

1.

b – 8a 8a + 33 – 11 (b)) 8b

94

4.

Complete the table.

Value of Expression when d = 12

ssion Value of Expression when d = 5

Re ga le du ca tio n

Expression 2d +8

12d – 24 – 3 10 + 4d 2

331 – 12d 12d + 8 4

5.

rking. g.

Simplify. Show your working. (a) 33 + 2x + 4x – x

2 + 9p 9 + 13 – 2p 2p – p (b) 22

95

Riley and Chelsea are making pancakes for dessert.

Chelsea made w pancakes. Riley made 16 more pancakes than Chelsea.

C They made 56 pancakes in all.

Re ga le du ca tio n

6.

(a) Express the number of pancakes Riley made in terms ermss of w. w. (b) How many pancakes did Riley make?

7.

A bricklayer has 25 kg of cement. ement. He buys bu 12 bags of cement.

Each bag has a mass of h kg. kg (a) Express the total mass of the cement in terms of h. tal m ta (b) What is the ttotal mass otal al mas ma of the cement if h = 32? (c) What is the e total tal mass of o the cement if h = 11?

96

Ethan has $44. He buys q donuts for his friends for $3 each and a cupcake for his sister for $2.

Re ga le du ca tio n

8.

ms of q. q. (a) Express the amount of money Ethan has left in terms (b) How much money does Ethan have left if q = 5? (c) How much money does Ethan have left if q = 13??

97

on

3

Fractions

Anchor Task

Cheesecake Recipe cipe Serves 10 people e Prep Time e Cook Time Cooling Time ime

30 minut minu es minutes 1 hour minutes 15 minute minut e

Ingredients Cheesecake 1 2 lbs cream m cheese 2 3 our cream eam cup sour 4 1 p salt tsp 2 3 ps sugar 1 cups 4 1 3 tbsp flour 2 5 eggs eggs 2 egg e g yolks yo

98

Crust 14 1 Graham Crackers 1 1 cup pecans 2 4 tbsp butter 1 cup sugar 4 1 tsp cinnamon 2

Let’s Learn

ati on

Multiplying Fractions Riley is making fruit punch for the school fair.

The recipe requires a

4 cup of lemon juice per liter.

5

Re ga le d

She plans on making 4 liters of fruit punch.

How much lemon juice will she need in total?

4 5

4x 4 5

Multiply ply the numerator erator by the whole hole number.

Then simplify. hen simpl

31 5 4x 4 = 4x4 5 5 16 = =31 5 5

When a fraction by a whole number, we multiply the numerator hen multiplying m by the whole Then simplify if possible. who number. num Riley needs 3

1 cups of lemon juice in total.

5

99

Multiply

7 by 6.

8 Remember to w write the fraction in its simplest form.

Re ga le du ca tio n

7 6x7 = 6x 8 8 42 = 8 1 2 =5 =5 4 8

When multiplying a proper fraction by a whole number, is less mber, the product pro than the whole number.

A bowling ball has a mass of 4

5 kg. Find the he mass of 3 such su bowling balls.

7

?

4

5 7

12

3x4

33 5 =3x 7 7 3 x 33 = 7 9 99 = 7 1 = 14 7

3 bowling balls have hav a total mass of 14

15 = 2 1 7 7

5 4 7 is a mixed number. So we expect the product to be greater than 3!

1 kg.

7

When multiplying a mixed number by a whole number, we convert the ult ultiplyi mixed number into an improper fraction. Then we multiply and simplify.

100

7x5

7x5

3 43 =7x 8 8 7 x 43 = 8 301 = 8 5 = 37 8

3 8 3 0 2 4 6 5

3 5 = 37 8 8

du ca tio n

3 Find 7 x 5 .

8

7 1

3 40 + 3 43 3 58 = 8 = 8

1 6 5

3 cm. A total of 18 such textbooks are 4 stacked in a pile in a bookstore. Find the height of o the stack of books.

Re ga

A textbook has a width of 3

3 by 18 to find the total height of the books.

4 270 ÷ 4 is 67 R 2.

3 12 + 3 1 x 18 3 x 18 = The product is 67 2 .

4 4 15 5 1 5 6 7 = x 18 x 1 8 4 2 7 0 4 270 1 2 0 2 4 = 1 5 0 3 0 4 2 7 0 2 8 1 2 = 67 = 6 67 2 2 4 1 The stack of books b is 67 cm high.

2 Let's multiply y3

1 01

Blake bought

3 of an apple pie to school to share with his friends. s. They T ate 5

Re ga le du

ati on

2 of the pie Blake brought. What fraction of the whole apple pie did Blake 3 and his friends eat?

3 of whole 5

When both factors are proper fractions, the product is less than both factors.

3 2 of 5 3

3 2 3x2 x = 5 3 5x3 6 2 = = 15 5

When by a fraction, multiply the numerators and the en multiplying a fraction frac denominators. simplify if possible. enominators. Then T s Blake ake and an his friends frien ate

1 02

2 of the whole apple pie.

5

Find

1 3 of .

4 4

Re ga le du ca tio n

3 1x3 1 x = 4 4 4x4 3 = 16

3 4

3 3 1 of is .

4 16 4

3 1 of 4 4

Find the product of 4 3 4x3 x = 5 8 5x8 12 = 40 3 = 10

3 4 and .

8 5

3 8

3 4 of 8 5

The product of

4 3 3 and nd is .

5 8 10 1

Find the product oductt of o

8 3 and .

5 4

3 8 3x8 x = 4 5 4x5 24 2 = 2 20 1 =1 5

The product ct of o

8 5 is an improper fraction.

3 8 1 and is 1 .

4 5 5

1 03

Let’s Practice Complete the following. Show your working and write your ur answer nswer in its simplest form.

Re ga le du ca tio n

1.

(a)

4 x3 5

(b) 4 x

(c)

2 x6 3

(d) (d d)

3 x8 7

3 4

(f)

7 x 13 12

(e) 12 x

104

5 8

Multiply the fractions. Show your working and write your answer in its simplest form. (a) 8 x

2 3

(b)

(c)

3x

8 9

(d) 6 x

3 4

(e)

7 x5 12

((f)

8x

3 5

7 20

(h)

4 x 12 15

6 x4 7

Re ga le du ca tio n

2.

(g) 4 x

1 05

3.

Multiply the mixed numbers. Show your working and write your answer in its simplest form.

1 x4 2

Re ga le du ca tio n

(a) 3

(b) 5 x 3

(c)

106

2

3 4

7 x3 8

Multiply the mixed numbers. Show your working and write your answer in its simplest form. (a) 5

2 x2 7

(b) 2

2 x6 3

Re ga le du ca tio n

4.

(c)

4x4

(e) 5

5 6

(d) 5 x 12

3 x 15 5

(g) 20 x 4

2 3

(f) (

6

1 2

7 x4 10

(h) 12 x 3

4 5

1 07

Color parts of the rectangle to show the product of the fractions.

Write the product in its simplest form. (a)

2 1 x 2 3

(b)

3 1 x 3 4

(c)

1 2 x 3 3

(d) (d

4 2 x 5 5

(e)

7 2 x 8 3

(f)

4 5 x 5 6

Re ga le du ca tio n

5.

108

Multiply the fractions. Show your working and write your answer in its simplest form. (a)

2 1 x 2 3

(b)

1 4 x 5 2

(c)

5 3 x 9 4

(d)

5 4 x 3 5

(e)

9 2 x 2 3

(f) (

7 2 x 9 3

(g)

5 4 x 3 111

(h)

5 8 x 6 3

Re ga le du ca tio n

6.

1 09

Hands On

Re ga le du ca tio n

Work in pairs.

(a) Use the grid below to draw a rectangle. Lightly shade rectangle blue. Color

1 off the 2

1 of the shaded part green. Write the fraction fractio of fract 3

the rectangle that is colored green.

w to draw d recta (b) Use the grid below a rectangle.

Lightly shade w. Color olor rectangle yellow.

3 of tthe shaded part red. Write the fraction of o 4

le that at is colored colore red the rectangle red.

110

1 of the 4

(c)

Use the grid below to draw a rectangle. Lightly shade

2 of the shaded part blue. Write the fraction raction of o 3

Re ga le du ca tio n

rectangle yellow. Color

5 of the he 6

the rectangle that is colored blue.

(d) Use the grid below to draw shade w a rectangle. ectangle. Lightly L rectangle red. Color

5 of the 8

3 of the he shaded shade part green. Write the fraction of 4

co d green. green the rectangle that is colored

111

At Home Complete the following. Show your working and write your ur answer nswer in its simplest form.

Re ga le du ca tio n

1.

(a)

4 x5 7

(b) 6 x

(c)

5 x 12 8

(d) (d d) 10 x

(e) e)) 6 x

112

3 7

(f)

2 3

5 6

7 x5 9

Multiply the mixed numbers. Show your working and write your answer in its simplest form. (a) 3

2 x5 5

Re ga le du ca tio n

2.

(b) 2 x 3

5 8

(c)

2 3

4x2

(e) (e 12 x 4

1 8

(d) 7 x 3

3 5

(f)

3 12

8x5

1 13

Color parts of the rectangle to show the product of the fractions.

Write the product in its simplest form. (a)

3 2 x 4 3

(b)

1 2 x 7 4

(c)

4 3 x 7 4

(d)

5 3 x 8 4

Re ga le du ca tio n

3.

4.

Multiply the e fractions. actions. Show Sho your yo working and write your answer in its simplest est form.

(a)

114

9 5 x 4 6

(b)

17 3 x 4 7

Solve It! 1 as many y figurines gurines as 4 1 many w has as m Sophie. Sophie gives Chelsea 20 of her figurines. Chelsea now 2

tio n

Sophie and Chelsea collect figurines. Chelsea has

Re ga le du c

ave ve originally? figurines as Sophie. How many figurines did Sophie have

1 15

Fractions and Division

Re ga le du ca tio n

Let’s Learn

Blake pours 2 liters of mineral water into 5 cups. Each cup hass the same volume of water. Find the volume of mineral water in each ach cup. cup

2 ÷ 5 = 1 of 2 5 = 2 5

Each cup contains contain c

116

2 5 of 1,000 ml 2,000 2 = = 400 ml 5

2 liters of mineral water.

5

1 of 16 12 16 = 12 1 4 = =1 3 3

16 ÷ 12 =

Each cookie contains 1

ati on

A recipe requires 16 teaspoons of cocoa powder to make 12 cookies.

How many teaspoons of cocoa powder are there in each cookie? e?

1 teaspoons of cocoa powder. wder. er.

3

Sophie has a strip of paper

3 m in length.

4

She cuts the strip into 6 pieces of equal length to make bookmarks. k.

Find the length of each bookmark.

Re ga le d

3 m 4

3 1 of m 4 6

3 1 3 ÷ 6 = of 4 6 4 1 3 = x 6 4 1 3 = = 24 8

Each ach bookmark has a length of

Dividing by 6 is the same as 1 multiplying by 6 !

1 m.

8

1 17

3 ÷ 5.

4

Re ga le du ca tio n

Find

1 3 3 ÷ 5 = of 5 4 4 3 1 = x 5 4 3 = 20

3 4

3 1 of 4 5

3 3 ÷5= 20 4

Keira is baking chocolate chip cookies.

Each cookie requires

2 cup of chocolate late chips. chip hip ps.

7

ps. s Keira has 3 cups of chocolate chips.

How many whole cookies can she he make?

2 7

2 7

2 7

1 cup

3÷

2 7

2 7

2 7

1 cup

2 7 =3x 7 2 7x3 = 2 21 = 2 1 = 10 2

Keira can an m make 10 whole cookies.

118

2 7

2 7

2 7

2 7

1 cup

2 Dividing by 7 is the same 7 as multiplying by !

2

A phone charger requires

5 m of cable. How many chargers can n be b made 8

Re ga le du ca tio n

with 10 m of cable?

10 ÷

5 8 = 10 x 8 5 80 = 5 = 16

16 phone chargers can be made with th 10 m of cable. cab

Find 8 divided by

8÷

3 .

5

3 5 =8x 3 5 40 = 3 1 = 13 3

8 divided d by y

3 1 = 13 5 3

1 19

Let’s Practice Complete the following.

Show your working and write your answer in its simplest st form. m.

Re ga le du ca tio n

1.

(a) 9 bags of flour are used to make 36 cakes. What at fraction action of a bag b of flour is used in 1 cake?

(b) 16 mini pizzas are ordered to feed ed 6 guests at a party. Each guest received an equal amount of pizza. much pizza does each a. How muc guest receive?

(c)

12 ÷ 8

(e) 8 ÷ 20

120

(d) 10 ÷ 4

(f)

9 ÷ 27

Use the model to help divide the fractions.

Write the answer in its simplest form. (a)

(b)

7 ÷4 8

Complete the following. Show your ur working rking and write w your answer in its simplest form. (a)

1 ÷ 12 2

(c)

(b) (b)

3 ÷6 8

8 ÷4 3

(d)

11 ÷9 5

4 ÷8 7

(f)

2 ÷ 12 3

Re ga

3.

2 ÷5 3

ed uc ati on

2.

(e) (e)

121

Use the model to help divide whole numbers by fractions.

Write the answer in its simplest form. (a) 4 ÷

2 5

(b) 4 ÷

2 9

Re ga le du ca tio n

4.

5.

122

Complete the following. win Show your working your ng g and write w o answer in its simplest form.

(a) 10 ÷

1 2

(b) 8 ÷

3 4

(c)) (c

2 5

(d) 9 ÷

2 7

12 ÷

Solve It!

Re ga le du

tio n

3 of a packet of nuts. He divides the nuts equally into 12 bags.

8 He gives 6 bags to his family and 2 bags to his friends. He keeps eps the rest for fo himself. What fraction of the original packet does Ethan keep himself? p for himse Ethan has

123

At Home

(a) 3 ÷ 12

(c)

4 ÷ 20

Re ga l

(e) 12 ÷ 8

ed uc ati on

Complete the following. Show your working and write your answer nswer er in its simplest form. (b) 10 ÷ 6

(d) 6 ÷ 16

(f)) (f

4 ÷ 18

(g)

2 ÷6 3

(h) (

6 ÷8 7

(i)

7 ÷ 10 2

(j)

8 ÷4 3

1 24

5 ÷4 4

(l)

3 ÷ 12 10

Re ga le du ca tio n

(k)

(m)

14 ÷7 3

(n)

4 ÷ 12 9

(o) 2 ÷

4 7

(p) 8 ÷

2 3

(q) 12 ÷

5 3

((r)) (r

7÷

3 4

(s)

7 3

(t)

10 ÷

9÷

18 7

125

Word Problems Riley has

n

Let’s Learn 2 liters of milk. She uses 7

3 of the milk to make a milkshake.

4

2 3 of 7 4

ed uc at

How much milk did she use?

3 2 x 4 7 6 = 28 3 = 14 =

Riley used

2 l 7

3 l of milk.

14

2 3 of l 7 4

Re g

24 mini pizzas are divided equally ally among amon 18 people. How many mini pizzas does each person receive receive? eive? ve?

24 1 18 4 = 3 1 =1 3

24 ÷ 18 8=

Each person receives 1

1 26

1 mini pizzas.

3

A plank of wood has a length of

3 m. It is cut into 6 pieces 4

Re ga le du ca tio n

of equal length. Find the length of each piece of wood.

We need to divide

3 m by 6.

4

3 3 1 ÷ 6 = of 4 4 6 1 3 = x 6 4 1 3 = = 24 8

as a length le Each piece of wood has of

A can of beans has a mass of 2

1 m.

8

4 kg.

5

Find the mass ss off 12 such ca cans.

14 4 12 x 2 = 12 x 5 5 12 x 14 = 5 168 = 5 3 = 33 5

x

1 1 4 1 2 1 6

2 4 8 0 8

3 5 1 6 1 5 1 1

12 cans of beans has a mass of 33

3 8

8 5 3

3 kg.

5

127

2 of his money on 5 1 some new ear pods. He then spent of his 8

Re ga le du ca tio n

Ethan had $360. He spent

remaining money on a phone case. Find the cost of the ear pods and the phone case. $360

spent on ear pods

1 3 He spent 8 of 5 of his money on a phone case.

?

spent on phone case

?

Let's find the cost of the ear pods.

2 x 360 2 of 360 = 5 5 720 = 5

1 5 7 5 2 2

= 144

44.

The ear pods cost $144.

4 4 2 0

2 0 2 0 2 0 0

pho case. c Now let's find the costt of the phone Method 1 Find

1 of the remaining maining money. mon mo 8

4 = $216 $360 – $144 1 x 216 1 of 216 2 = 8 8 = 27

The phone ne case costs $27.

1 28

2 8 2 1 1 6 5 5

7 6

6 6 0

Method 2 1 3 of the money Ethan had originally.

8 5

3 1 3 1 of = x 5 8 5 8 3 = 40 3 x 360 3 of 360 = 40 40 1080 108 = = 4 40 = 27

ed uc ati on

Find

The ear pods cost $144 and the phone case e costs $27.

Sophie is pouring water from a cooler The cooler contains 18 liters ooler ler into cups. T of water and each cup can hold old

3 liters iters of wa water. How many cups can she 7

Re ga

waterr? fill with the 18 liters of water?

We need to divide de 18 by

18 ÷

3 .

7

7 3 = 18 x 3 7 126 = 3 = 42

Sophie can fill 42 cups of water.

129

Let’s Practice 3 of them are girls.

4 (a) How many boys are at the museum? (b) How many more girls than boys are there?

56 children are at the museum.

Re ga le du ca tio n

1.

2.

130

1 Keira made 168 cookies to o sell at the fair. She S sold of them on 3 5 Saturday and of the remaining maining cookies cook coo on Sunday.

8 How many cookies did she e sell on Sunday?

S

3 min to complete a full rotation.

4 How long does it take to complete 12 rotations?

It takes a Ferris wheel 8

Re ga le du ca tio n

3.

4.

Mr. Timmins is tiling his bathroom which has a length of 4 hroom m floor wh w

3 m and 8

a width of 4 m.

mmins' b (a) Find the area of Mr. Timmins' bathroom. pe square uare meter. How much will it cost to tile (b) The tiles cost $28 per om? the bathroom?

1 31

5.

Michelle spent

1 3 of her savings on a telescope. She spent of the 4 5

Re ga le du ca tio n

remaining money on a chess set. (a) What fraction of her total savings did she spend on n the chess set? (b) She had $102 left over. How much did the telescope ope cost?

6.

132

1 Halle picked some flowers wers in her gard ga garden. of the flowers were roses, 3 1 of them were tulips ulips and ulip a the he rest res were daisies. If she picked 20 4 daisies, how many any flowers did she pick in all?

At Home 3 Ethan has 140 toy cars. He gives his brother of the cars. s. How ow many 7 cars does he have left?

Re ga le du ca tio n

1.

2.

4 Sophie is making a poster er for or a school p presentation. She colors of the 9 3 poster blue and of the he remaining emaining part p green. What fraction of the 4 poster is green?

1 33

3.

Mrs. Laycock's lawn is rectangular in shape with a length of 12 m 5 m.

6

Re ga le du ca tio n

and a breadth 7

(a) Find the area of Mrs. Laycock's lawn. (b) She plans on replacing the lawn with synthetic grass. s.

The synthetic grass costs $17 per square meter. er.

How much will it cost to replace the lawn?

4.

1 34

2 min tto complete a full rotation.

3 How long does it take complete 36 rotations? ake to co It takes a merry-go-round d1

Looking Back Multiply the fractions and mixed numbers.

Show your working and write your answer in its simplest st form. m.

Re ga le du ca tio n

1.

(a)

4 x6 9

(c)

8x3

(e)

7 5 x 2 6

2 7

(b) 2

7 x5 8

(d) (d) 12 x 4

(f)

3 4

7 2 x 12 3

1 35

Complete the following.

Show your working and write your answer in its simplest form. rm. (a) 8 ÷ 12

(b) 24 ÷ 18

3 4

(d) 10 ÷ 36 6

Re ga le du ca tio n

2.

(c)

1 36

3÷

2 3

(e) 14 ÷ 16

(f)) (f

12 ÷

(g)

3 ÷4 5

(h)

6 ÷3 7

(i)i))

8 ÷ 10 3

(j)

9 ÷7 4

3.

39 liters of olive oil is to be poured into smaller bottles. Each bottle can 3 liters of oil. How many bottles can be filled with 39 liters? s?

4

Re ga le du ca tio n

hold

4.

2 of the apples were red.

7 (a) How many green apples pples les were there? t (b) How many more or green ore n apples apple than red apples were there?

Sophie bought 182 red and green apples. apple

1 37

Ratio

Ratio and Fraction Anchor Task

1 38

on

4

Let’s Learn

Re ga le du ca tio n

In Blake's fruit bowl there are 2 oranges and 5 pears.

Let's use a model too represent tthe mber of each e number type of fru fruit.

oranges pears

The ratio of the number of oranges to the he number mber of pears pe p is 2 : 5.

The ratio of the number of pears to the is 5 : 2. e number mber of oranges ora We can also express ratio as a fraction. ction. tion Number of oranges 2 = 5 Number of pears

The number of oranges is

2 the e number numbe o of pears.

5

5 Number of pears = Number of oranges 2

The number of pearss is

5 the n number of oranges. numb 2

e 7 fruits uits in the bowl bow in total. The ratio of oranges to the total There are

er of fruits is 2 : 7. So, So S number

2 of the fruits in the bowl are oranges.

7

he ratio of pears tto the total number of fruits is 5 : 7. The number of pears The is

5 of the he total t number of fruits in the bowl.

7

1 39

Riley and Keira put their money together to buy a $50-giftcard for Sophie.

Riley contributed $30 and Keira contributed ributed $20.

0.

1 unit

Re ga le du ca tio n

Each ch unit has a alue of $10 value $10.

Riley

$50

Keira

The ratio of Riley's contribution to Keira's ra's contribution contributio butio on on is 3 : 2.

2 The ratio of Keira's contribution to Riley's contribution tion iss 2 : 3.

Each unit has a value of $10. The total cost of the giftcard $50 = 5 units. iftcard is $ The ratio of Riley's contribution to the totall costt of the giftc giftcard is 3 : 5. gift Riley paid for

3 of the giftcard.

5

The ratio of Keira's contribution to the he total cost of the giftcard is 2 : 5.

Keira paid for

2 of the giftcard.

5

ts the he lengths o The model below represents of 2 pencils.

1 cm

Yellow pencil

Green pencil

The yellow pencil cil is 6 cm in length. leng The green pencil is 8 cm in length.

The ratio of the length of the th yellow pencil to the length of the green pencil is 6 : 8.

We express the ratio rati 6 : 8 in its simplest form by dividing each term in the ratio common factor. tio by the greate greatest c

÷2

6 : 8

÷2

= 3 : 4

The ratio 6 : 8 in its simplest form is 3 : 4.

140

Re ga le du ca tio n

In its simplest form, the ratio of the length of the green pencil to the length of the yellow pencil is 4 : 3.

The green pencil is 4 the length of the 3 yellow pencil. pencil

The yellow pencil is 3 the length of the 4 green pencil.

The total length of both pencils is 14 cm cm. m m. Let'ss find tthe ratio of the length of each pencil to the total length of the e pencils.

The length of the yellow pencil to the e total length lengt leng of the pencils is 6 : 14.

The length of the green pencill to the of the pencils is 8 : 14. e total length leng len ÷2

6 : 14

÷2

= 3 : 7

÷2

8 : 114

÷2

= 4 : 7

The length of the yellow ellow w pencil to the th total length of the pencils is 3 : 7.

The yellow pencil cil is

3 off the total o tot length le of the pencils.

7

e green pencil pe tto the total length of the pencils is 4 : 7.

The length of the

en pencil encil is The green

4 off the total length of the pencils. o 7

1 41

The mass of Ethan and his siblings is shown in the model.

Re ga le du ca tio n

1 unit Mike

Ethan

Jessica

than iss 3 : 7.

The ratio of the mass of Jessica to the mass of Ethan Jessica's mass is

3 of Ethan's mass.

7

The ratio of the mass of Mike to the mass ss of Ethan is 11 : 7.

Mike's mass is

11 of Ethan's mass.

7

a to the he mass of o Mike 3 : 11.

The ratio of the mass of Jessica Jessica's mass is

3 of Mike'ss mass. ass.

11

ma of 21 units.

Ethan and his siblings have a total mass

Jessica's mass is ÷3

3 : 21

3 of the total mass mas m of the siblings.

21

÷3

= 1 : 7 1 of the total tota mass of the siblings. to Jessica's mass iss of 7

Ethan's n's mass is ÷7

7 : 21

7 off the total mass of the siblings. o 21

÷7

= 1 : 3

ma is Ethan's mass

142

1 of the total mass of the siblings.

7

The ages of Riley, Tejal and Zhang Wei is in the ratio 2 : 3 : 6.

Re ga le du ca tio n

Riley Tejal

Zhang Wei

3 The ratio of the age of Riley to the age of Zhang Wei is 2 : 6 = 1 : 3.

The age of Riley is

1 the age of Zhang Wei. 3

The age of Zhang Wei is

3 the age of Riley. y.

1

We can say Zhang Wei is 3 times olderr than n Riley.

The ratio of the age of Tejal to the e age ge of Zhang Wei is 3 : 6 = 1 : 2.

The age of Tejal is

1 the age of Zhang ng Wei. 2

The age of Zhang Wei is

2 the he age of Tejal.

Te 1

Zhang Wei is twice ass old as Tejal.

Te

The ratio of the age age of Tejal and Zhang Wei ge of Riley to the th combined co is 2 : 9.

The age of Riley is

2 the combined combin age of Tejal and Zhang Wei. co 9

The ratio tio off the age of Zhang Zha Wei to the combined age of all three children is 6 : 11.

11

he age of Zhang Zha Wei W is The

6 the combined age of all three children.

11

1 43

Let’s Practice Complete the tables.

Re ga le du ca tio n

1.

(a) Express each ratio as a fraction.

Ratio

Fraction n

1:4

7:9

9:5 2:5

12 : 5

(b) Express each fraction n as a ratio. ratio Fraction 7 1 1 3 5 7 13 3 9 4 5

1 44

Ratio

Express each of the following quantity comparisons as a ratio in its simplest form. Show your working.

Re ga le du ca tio n

2.

(a) $70 to $90

(b) 12 cm to 50 cm

(c)

27 kg to 9 kg

(d) 35 ml to 140 ml

(e) 15 min to 1 h

(f)

14 in to to 63 in i

1 45

3.

An eraser has a length of 3 cm. A stapler has a length of 7 cm.

stapler

ed uc ati on

eraser

(a) The ratio of the length of the eraser to the length gth of the stapler staple stap is

:

.

(b)

Length of eraser = Length of stapler

(c)

The length of the eraser is

the he length of the stapler.

(d) The ratio of the length of the e stapler apler to the length of the eraser is

:

.

(e)

Length of stapler = Length of eraser

(f)

The length off the stapler is sta

the length of the eraser.

eg a

he length of o th (g) The ratio of the the eraser to the total length of both ts is objects

:

.

he length ngth of the eraser eras is e (h) The (i)

length of the stapler to the total length of both The he ratio ratio of the le

objects is

(j)

146

the total length of both objects.

:

The length of the stapler is

. the total length of both objects.

Farmer Joe keeps cows and horses in the ratio 9 : 11.

horses

le du ca tio n

cows

(a) The ratio of the number of horses to cows is (b)

Number of horses = Number of cows

(c)

The number of horses is

(e)

Number of cows = Number of horses

(f)

The number of cowss is

:

.

the number of cows. c

(d) The ratio of the number of cows ws to horses orses is

:

.

the th number of horses.

(g) The ratio of the number umber er of cows cow to the total number of animals is

:

.

(h) The number ber of cows is (i)

is (j)

the total number of animals.

atio of the num number of horses to the total number of animals The ratio

g

4.

:

.

horses is The he number number of ho

the total number of animals.

1 47

5.

The mass of a goose is 3 kg. The mass of a lamb is 9 kg.

Re ga le du ca tio n

goose lamb

Express all ratios in simplest form.

(a) Find the ratio of the mass of the lamb to the e mass of the th goose.

The mass of the lamb is

the e mass of th the t goose.

(b) Find the ratio of the mass of the goose oose to the th mass of the lamb.

The mass of the goose oose e is

(c)

the mass of the lamb.

the goose to the total mass of Find the ratio of the mass of th both animals.

ss of of the goo goose is The mass

the total mass of both animals.

e ratio ratio of th the mass of the lamb to the total mass of (d) Find the th animals. both

mas of the lamb is The mass

148

the total mass of both animals.

6.

The model shows the number of flowers in Chelsea's garden.

Re ga le du ca tio n

1 unit roses

tulips

daisies

Express all ratios in simplest form.

(a) Find the ratio of the number of rosess to daises. aises.

The number of roses is

the numb number num of daisies.

(b) Find the ratio of the number mber er of tulips to roses.

The number of tulips iss

(c)

the number of roses.

o off the number nu n Find the ratio of roses and daisies to the number of tulips.

numb mber er of tulips tuli is The number

the number of roses and daisies.

(d) d) Find nd the ratio ratio of the number of daisies to the total number of Chelsea's garden. flowers in C Chel

number of daisies is Th nu The Chelsea's garden.

the total number of flowers in

1 49

7.

The capacity of 3 beakers is in the ratio 8 : 2 : 4

Re ga le du ca tio n

Beaker A Beaker B

Beaker C

Express all ratios in simplest form.

er B to the capacity capa cap (a) Find the ratio of the capacity of Beaker of Beaker C.

The capacity of Beaker B iss

the he c capacity of Beaker C.

(b) The capacity of Beakerr C iss

tthe capacity of Beaker B.

(c)

times the capacity of

The capacity of Beaker akerr C is Beaker B.

acity of Beaker B to the capacity of (d) Find the ratio of tthe capacity Beaker A.

150

The Beaker B is e capacity pacity of Be Beake

the capacity of Beaker A.

e capacity capacity of Be (e) The Beaker A is

the capacity of Beaker B.

(f)

times the capacity of

of Beaker A is The capacity capac o Beaker B.

8.

Sophie has

1 the savings of Halle.

4

Re ga le du ca tio n

Sophie

Halle

(a) Express Halle's savings to Sophie's savings ass a ratio.

(b) Express Halle's savings as a fraction savings. on of Sophie's sa

(c)

9.

Halle has

Jordan ran

timess more ore savings that t Sophie.

7 the distance nce Ethan ran.

1

Jordan

Ethan

J (a) Express the distance Jorda Jordan ran to the distance Ethan ran as a ratio. o.

(b) Express distance Ethan ran as a fraction of the distance b) Exp press the dist Jordan ra ran.

(c)

Jordan d ran

times further than Ethan.

1 51

Re ga le du ca tio n

10. Riley read 15 books. Blake read 5 books.

(a) Find the ratio of the number of books Riley ley read to the number n of books Blake read. Write the ratio in its simplest plest form.

(b) Express the number of books oks Blake read as a a fraction of the number of books Riley read. in its simplest form. ead. d. Write the fraction fr fra

(c)

Express the number of books Riley read as a fraction of the um umbe number of books oks Blake Bla Blak read. Write the fraction in its simplest form.

(d) Riley read ad

times t the number of books Blake read.

e) Exp press the number num (e) Express of books Blake read as a fraction of the total number of boo books read by Riley and Blake. Write the fraction in its simplest form simples form.

152

Ethan has 5 times more money than his brother, Steve.

Re ga le du ca tio n

11.

(a) What is the ratio of Ethan's money to Steve's eve'ss money?

(b) What fraction of Ethan's money ney does oes Steve have? h

(c)

Ethan and Steve put their heir money mon together.

What fraction of the from Ethan? e money came ca

(d) What money came from Steve? at fraction action of the mo

153

Chelsea's height is

4 Sophie's height.

5

Re ga le du ca tio n

12.

(a) What is the ratio of Sophie's height ht to Chelsea's height? h

(b) Express Sophie's height of Chelsea's height. ht as a fraction o

(c)

Express Chelsea's elsea'ss heig height as a ratio of Chelsea and Sophie's combined d height. eight.

(d) Exp Express height as a fraction of Sophie's and Chelsea's press Sophie's h height. combined heigh

154

Hands On Place 20 blue cubes into a cup. Place 20 red cubes into another nother er cup Take turns to close your eyes and remove a small handful ful off cubes from each cup..

Re ga le du

tio n

1.

(a) Express the number as en mber of blue bl e cubes c bes to the number n mber of red cubes c a ratio in its simplest form. m. (b) Express the number of red cubes as a a fraction of the number of blue cubes.

Repeat 5 times. s.

2.

Use blue and each ratio in the table. d red d cubes to show s Remove the the ratio in its simplest form. he cubes ubes to show th Blue Cubess

Red Cubes C

3

9

6

18

2

12

5

20

4

20

Ratio

Ratio in Simplest Form

155

At Home Complete the tables.

Re ga le du ca tio n

1.

(a) Express each ratio as a fraction.

Ratio

Fraction n

2:7

13 : 9 7:9 5:1

12 : 7

(b) Express each fraction n as a ratio. ratio Fraction 9 1 1 5 5 11 13 3 7 5 3

156

Ratio

Express each of the following quantity comparisons as a ratio in its simplest form. Show your working.

Re ga le du ca tio n

2.

(a) 6 cm to 48 cm

(b) 20 min to 2 h

(c)

36 lb to 6 lb

(d) 90 l to 10 l

(e) 500 g to 2 kg

(f)

to 2 ft 10 in to

157

3.

Keira is 11 years old. Her younger sister, Shanice, is 7 years old.

Re ga le du ca tio n

Shanice Keira

(a) The ratio of Shanice's age to Keira's age is

:

.

:

.

(b) Shanice's age = Keira's age (c)

the age of Keira. ra.

Shanice is

nice'ss age is (d) The ratio of Keira's age to Shanice's

(e)

Keira's age = Shanice's age

(f)

Keira is

the e age e of Shanice.

Shanice

ce's age to th the combined age of her and her (g) The ratio of Shanice's sister is

:

(h) Shanice iss her sister. er. (i)

(j)

1 58

.

the age of the combined ages of her and

The e ratio io of Keira's age to the combined age of her and her

sister er is

:

Keira eira is her sister.

the age of the combined ages of her and

.

The mass of a bag of apples is 2 kg. The mass of a bag of potatoes is 10 kg.

Re ga le du ca tio n

4.

apples

potatoes

Express all ratios in simplest form.

(a) Find the ratio of the mass of the apples to the he mass of the potatoes.

The mass of the apples is

the mass o of the potatoes.

(b) Find the ratio of the mass of the potatoes to the mass of the apples.

toes is The mass of the potatoes

(c)

the mass of the apples.

Find the ratio of tth the mass ss of tthe apples to the total mass of apples and potatoes. tatoes

The off the apple apples is e mass as o a and potatoes. tatoes.

the total mass of apples

(d) d) Find nd the ratio rat of the mass of the potatoes to the total mass of potatoes. apples and po

The of the potatoes is e mass m and potatoes.

the total mass of apples

159

5.

The model shows the fish Wyatt caught on a boat trip.

eg al ed uc ati on

1 unit cod

mullet

bream

Express all ratios in simplest form.

(a) Find the ratio of the number of cod to o the e number of o bream.

The number of cod Wyatt caught gh is bream he caught.

the number of

(b) Find the ratio of the number umber er of mullet mulle to the number of bream.

The number of mullet mulle Wyatt mu att caught is bream he caught. ght.

(c)

Find the number of cod and bream to the number e ratio o of the nu numb of mullet. ullet.

The number numbe of cod and bream Wyatt caught is mullet Wyatt caught. number o of m

160

the number of

the

To make a fruit punch, Halle mixed orange juice, apple juice and pineapple juice in the ratio 12 : 4 : 3

ga le du ca tio n

6.

Orange

Apple Pineapple

Express all ratios in simplest form.

(a) Find the ratio of the volume of orange of ge juice ce to the volume vo v apple juice.

The volume of orange juice ce e is

the volume of apple juice. th

(b) The volume of apple juice is

the volume of orange juice.

ch there here is In Halle's fruit punch as apple juice.

times as much orange juice

(c)

(d) Find the ratio volume o of the v e of pineapple juice to the volume of orange juice. ce.

The pineapple juice is he volume ume of pinea pin orange ange juice. (e) The volume volume of orange juice is juice. pineapple juic (f)

In Halle's ffruit punch there is pineapple juice. as pinea

the volume of

the volume of

times as much orange juice

1 61

7.

The mass of a hamster is

1 the mass of a kitten.

3

Re ga le du ca tio n

hamster kitten

he mass of o (a) What is the ratio of the mass of the kitten to the the hamster?

(b) Express the mass of the kitten as fraction mass of ction of the m the hamster.

The kitten is

8.

Riley's house is

timess heavier than th t the hamster.

6 the distance istance nce from sc school compared to Michelle's house.

1

Riley

Michelle

(a) Expresss the distance of Michelle's house from school to the distance o M of Riley's school as a ratio. ley'ss house from sch

distance of Michelle's house from school as a fraction of (b) Express Express the dis the dista distance of Riley's house from school.

Michelle's house is

1 62

times closer to school than Riley's house.

Jordan picked 18 apples. Blake picked 6 apples.

Re ga le du ca to n

9.

(a) Find the ratio of the number of apples es Jordan rdan picked picke to the number of apples Blake picked. Write rite the ratio in itits i simplest form.

(b) Express the number of apples pples ples Blake picked pick picke as a fraction of the number of apples Jordan an picked.

Write the fraction in itss simplest plest form. form

(c)

Express the number of apples umbe o l Jordan picked as a fraction of the number picked. er off apples Blake Bla B Write the form. e fraction action in its simplest simp s

Jordan rdan picked

times as many apples than Blake.

(d) Express Express the num number of apples Blake picked as a fraction of the total num number of apples picked by Jordan and Blake.

W rite the th fr Write fraction in its simplest form.

1 63

Re ga le du ca tio n

10. Chelsea has 10 times more marbles than Halle.

(a) What is the ratio of the number of Chelsea's elsea's a's marbles to the number of Halle's marbles?

(b) Express the number of marbles has as a fraction of the rbles Halle ha number of marbles Chelsea elsea a has.

(c)

Chelsea and d Halle alle pu put all of their marbles in a jar.

What fraction ction n of the marbles ma marb in the jar are Halle's?

(d) What hat ffraction raction of tthe marbles in the jar are Chelsea's?

1 64

Ethan has

7 the money Wyatt has.

13

Re ga le du ca tio n

11.

(a) What is the ratio of Wyatt's money to Ethan's an's money? mone

(b) Express Wyatt's money as a fraction of Ethan's money.

E

(c)

Ethan and Wyatt together to buy a sandwich. yatt put p their ir money ir mo What fraction is Ethan's? on of the money m

What fraction raction of th the money is Wyatt's?

1 65

Solve It! Riley has

5 the amount of savings Chelsea has.

6

Re ga le du ca tio n

1.

(a) Draw a model to compare the amount each child has. ountt of money e

(b) Express Chelsea's savings ngs as a fra fraction of Riley's savings.

(c)

Chelsea ea gives

1 of o her savings to Riley.

3

What hat is the rratio atio ti of Chelsea's savings to RiIey's now?

1 66

Ethan has

3 the number of computer games Wyatt has.

11

Re ga le du ca tio n

2.

(a) Draw a model to compare the numberr of video games game each child has.

(b) Express the number of video eo games Wyatt has as a fraction of the number of video games ames es Ethan has. has

(c)

3 of his v video games to Ethan. Write the ratio of the 11 number mber of video games g Ethan has to the number of video games Wyatt yatt has now. E Express the ratio in its simplest form.

Wyatt yatt gives giv

1 67

Ratio and Proportion

on

Let’s Learn

Sophie is making orange soda. To do so, she mixes orange e juice ce with soda sod water at the ratio 1 : 2. To make a small jjug cup g of orange g soda, da, she mixes 1 c of orange juice with 2 cups of soda water. ter. orange juice

Re ga le du ca

soda water

Sophie would like to make orange soda for all of her fr friends. To do so, she'll frien need to mix orange juice and soda water ter in n the same ratio, but in different amounts. She uses the table below to o help her. er.

Cups of Orange Juice

1

2

4

8

Cups of Soda

2

4

8

16

If Sophie uses 2 cups of orange nge e juice, she'll need 4 cups of soda water.

If she uses 16 cups of soda water, 8 cups of orange juice. To er, she'll need ne n make different amountss of orange ange nge soda, sod the amount of orange juice and soda water she uses change changes, hang but the ratio and proportion are the same.

To paint a tennis court, needs to mix yellow paint and blue paint in rt, a painter painte nee the ratio 1 : 4. yellow paint

blue paint

Tins of Yellow P Paint

1

5

10

20

ns of Blue Paint Tins

4

20

40

80

If the painter uses 5 tins of yellow paint, he'll need 20 tins of blue paint. nter us nte If he uses 80 tins of blue paint, he'll need 20 tins of yellow paint.

168

Re ga le du ca tio n

To make cupcakes, a baker needs to mix flour and sugar. The table shows the quantity of each ingredient needed to make differentt quantities of cupcakes.

Number of Cupcakes

5

10

15

20

25

Flour (kg)

3

6

9

12

15

Sugar (kg)

1

2

3

4

5

3:1

3:1

3:1

3:1

3:1

3 1

3 1

3 1

3 1

3 1

Flour : Sugar

Fraction of Flour to Sugar

orr each amount amoun of flour the baker uses, The ratio of flour to sugar is 3 : 1. For 1 he'll need as much sugar.

3 If the baker wants to make 15 5 cupcakes, upcakes, he'll he'l need 9 kilograms of flour and 3 kilograms of sugar. Iff he wants to make 25 cupcakes, he'll need 15 m kilograms of flour and 5 kilograms rams ams of sugar. su

To make rice soup, Mr. Lim uses 2 cups of rice and 3 cups of water. c Using the same proportion, ortion, how many ma cups of water will Mr. Lim need if he uses 8 cups of rice? x4

Cups of rice = Cups off water ater

2 3

=

8 12

x4

If Mr. Lim uses 8 cups of rice, he will need 12 cups of water. c

1 69

Re ga le du ca tio n

To make purple paint, Halle mixes 14 milliliters of red paint with 8 milliliters of blue paint. The ratio of red paint to blue paint is 14 : 8 = 7 : 4.

Using the same proportion, how much blue paint is needed if 35 milliliters of red paint are used?

35 ml

red paint

blue paint

?

7 units 1 unit 4 units

35 35 ÷ 7 = 5 4 x 5 = 20

d, 20 ml ml of blue b When 35 ml of red paint are used, paint are needed.

Using the same proportion of paint, how much red paint is needed when 36 ml of blue paint nt are a used? sed? ed? ?

red paint

blue paint

36 ml ml

4 units 1 unit 7 units nits

36 36 ÷ 4 = 9 7 x 9 = 63 6

ml of blue paint are used, 63 ml of red paint are needed.

When 36 ml

170

Re ga le du ca tio n

Mr. Fong mixed tea, peach juice and syrup in the ratio 9 : 8 : 3 to make 1 liter of iced peach tea. How much of each ingredient did he use?

tea

peach juice

1l

syrup

20 units 1 unit

1 l = 1,000 ml 1,000 ml ÷ 20 = 50 ml ml

Mr. Fong used 9 units of tea.

9 x 50 ml = 450 m ml 9 units Mr. Fong used 450 ml of tea.

Mr. Fong used 8 units its of peach juice. juice ju 8 units 8 x 50 0 ml ml = 400 m mll Mr. Fong used 400 mll of peach juice. juic

Mr. Fong used 3 units of syrup. syrup syr 3 units 3 x 50 ml ml = 150 ml Mr. Fong ong used sed 150 m ml of ssyrup.

o make 1 liter o of ice To iced peach tea, Mr. Fong used 450 ml of tea, 400 ml of peach 150 ml of syrup. ch juic juice and 15

1 71

Let’s Practice To make jelly, Chelsea needs 1 cup of jelly crystals for every ry 3 cups of water.

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1.

(a) Complete the table.

2.

Cups of Jelly Crystals

1

Cups of Water

3

2

4

9

(b) The ratio of jelly crystals to water is

:

(c)

the amount of water

The amount of jelly crystals used d is used.

.

The table shows the amount off flour and suga sugar used to make donuts. (a) Complete the table.

Cups of Flour

5

Cups of Sugar

2

10

15

25

6

8

(b) To make donuts, nuts, uts, the rratio off flour to sugar is (c)

3.

The amount ount of sugar is

:

the amount of flour.

ter mixes 7 buckets buck A concreter of cement for every 5 buckets of sand. (a) Complete ete the table. table

14

Buckets uckets of Cem Cement Buckets of Sand

5

(b) The ratio of sand to cement is (c)

172

amount of cement is Th a The

35

10

15

:

20

.

the amount of sand.

.

On Mr. McKenzie's farm, the ratio of chickens to sheep is 3 : 5. There are 150 sheep. How many chickens are on Mr. McKenzie's farm??

Re ga le du ca tio n

4.

5.

At a school camp, the number Chelsea took to the ber of photographs photograp photogra number of photographs Riley took ook was in the ratio 5 : 7. In all, they took 180 photographs. How many y photographs photograp did each child take?

1 73

To make 1 serving of pasta sauce, Riley uses 5 cups of tomato paste for every 2 cups of water.

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6.

er does oes she use? use (a) Riley uses 15 cups of tomato paste. How much water

(b) Riley uses 12 cups of water. How much uch tomato paste past pas does she use?

(c)

Riley uses 50 cups of tomato omato paste. paste How much water does she use?

(d) Riley ley needs eeds to make mak 22 servings of pasta. How much tomato paste need? and water will she n

174

To make a blueberry pie, Mrs. Williams uses 150 grams of pastry and 200 grams of blueberries.

Re ga le du ca tio n

7.

mount nt of (a) What is the ratio of the amount of pastry to the amount ss the he ratio in its blueberries used to make a blueberry pie? Express simplest form.

(b) Mrs. Williams uses 450 grams off pastry. try. How much m mu blueberries does she need? How many blueberry ueberry erry pies can ca she make?

(c)

Mrs. Williams uses 1 kilogram ogram o of blueberries. How much pastry does she need?

How many pies can she make? ed? d? Ho H any blueberry b

(d) Mr Mrs. rs. Williams needs need to make 15 pies for a school fundraiser. How and blueberries will she need? much uch pastry past an

1 75

Wyatt is cooking a vegetable stir fry. He mixes 250 grams of peas, 200 grams of carrots and 100 grams of corn.

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n

8.

(a) Write the ratio of peas to carrots to Express the o corn orn Wyatt used. us ratio in its simplest form.

(b) Using the same ratio tio of vegetables, vegetab vegeta Wyatt used 1 kilogram of peas.

How much carrots ro and d corn did d he use?

(c)

176

Using sing the ssame ame ratio ra of vegetables, Wyatt used 1 kilogram of corn.

How Ho ow much peas and carrots did he use?

The ratio of the Ethan's money to Jordan's money to Wyatt's money is 3 : 8 : 5. The total amount of money is $560.

Re ga le du ca tio n

9.

(a) How much money did each child have?

1 3 of his money ey to Wyatt Wyat Wya and of his money to 4 4 Ethan. How much doess each child have now?

(b) Jordan gives

1 77

At Home The table shows the amount of syrup and tea used to make ake iced tea. tea

Re ga le du ca tio n

1.

(a) Complete the table.

Syrup (ml)

10

20

Tea (ml)

150

40

50 5

450

ea is (b) In its simplest form, the ratio of syrup to tea

(c)

2.

60

The amount of syrup used is

:

the amoun amount of tea used.

The table shows the ratio of the length of a rectangle. ngth and width o (a) Complete the table.

Length (cm)

9

Width (cm)

7

18

36

72

28

42

(b) The ratio of the length width of the rectangle gth h to the w wi is

(c)

3.

:

.

The length is gth of the rectangle rectan

the width.

atballs, Mrs. Jen To make meatballs, Jenkins needs to mix minced beef and flour. o of minced b beef to flour is shown in the table.

The ratio (a) Complete mplete the table. table Minced Be Beef (g)

250

Flour ((g)

500 240

(b) b) The Th ratio rati of minced beef to flour is (c)

178

The amount of flour is

1,000

360

600

:

.

the amount of minced beef.

.

Gordon and Annie bought a waterfront block of land with an area of 520 m2. They divided the block of land into 2 parts. The ratio atio of the area of Block A to the area of Block B was 7 : 6.

Find the area of each block.

Re ga le du ca tio n

4.

5.

Broadbeach College hass a total The ratio of boys to otal of 960 students. s girls is 6 : 9. (a) How many boys attend Broadbeach College? ys a y d Broad

(b)) Ho How attend Broadbeach College? ow many girls a

1 79

To make a protein shake, Mrs. Sender uses 30 grams of bananas and 20 grams of protein powder.

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6.

e amount (a) What is the ratio of the amount of protein powder to the ess the he ratio in its of bananas used to make a protein shake? Express simplest form.

(b) Mrs. Sender uses 120 grams of protein rotein n powder.

How many grams of protein powder owder does she need?

How many protein shakes is she making?

(c)

Mrs. Sender uses 270 0 grams of bananas.

How many grams rams am of o protein otein tein powder does she need?

How many protein is she making? tein shakes sh sha

(d)) Mr Mrs. rs. Sender need needs to make 14 protein shakes for the college football much protein powder and bananas will she need? team. How mu

180

A baker decorates a cake with 12 strawberries and 9 chocolates. ess the ratio in (a) What is the ratio of chocolates to strawberries? Express its simplest form.

Re ga le du ca tio n

7.

(b) The baker uses 60 strawberries to decorate ate cakes in i the same ratio. How many chocolates will the many cakes he baker aker need? How H is the baker decorating?

(c)

The baker uses 72 chocolates ocolates to t decorate cakes in the same ratio.

How many strawberries aw es will the th baker need? How many cakes is ecorating corati the baker decorating?

(d)) The he baker baker needs to decorate 12 cakes. How many strawberries and choco chocolates will the baker need?

1 81

To make a bottle of aroma oil, Halle mixes 30 ml of lavender oil, 60 ml of rose oil and 40 ml of sandalwood oil.

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tio n

8.

(a) Write the ratio of lavender to rose e to sandalwood oil.

Express the ratio in its simplest form.

(b) To make the same oil,, she uses 180 ml of lavender oil. How much rose oil and sandalwood does she use? How many bottles of an anda d oil do an she make? m aroma oil can

(c))

182

Ha Halle alle wants to m make a bottle of the aroma oil for each of her 14 How much of each type of oil does she need? classmates. Ho classmates

The ratio of lemon to orange to strawberry flavor candies in a pack is 2 : 5 : 3. There are 160 candies in the pack.

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tio n

9.

(a) How many of each flavor candy are in the pack?

1 1 (b) Michelle eats of the lemon emon candies, of the orange candies and 8 2 1 of the strawberry rawberry erry candies. can c 4 ny of each flavor flavo candy c How many are left?

1 83

Word Problems

Re ga le du ca tio n

Let’s Learn

On Magnolia Ranch, the ratio of the number of horses to the number of cows is 7 : 4. There are 186 more horses can cows.

How many horses are on Magnolia Ranch?

How many cows are on Magnolia Ranch?

How many horses and cows are there altogether Ranch? er on n Magnolia R ?

horses cows

186

The number of horses more than n the e number of cows is 3 units.

3 units 1 unit

186 186 ÷ 3 = 62

s The number of horses iss 7 units.

7 units

62 x 7 = 434 34

orses on Magno R There are 434 horses Magnolia Ranch. uni The number of cows is 4 units.

4 units

62 62 x 4 = 248

e are 248 cows on M There Magnolia Ranch.

34 + 248 = 682 434

e are 6 682 ho There horses and cows on Magnolia Ranch.

184

Re ga le du c

tio n

Sophie's family went on a road trip. The amount of money they spent on accommodation, food and gas was in the ratio 9 : 7 : 4. They spent ent $1,800 $1,8 on accommodation. How much did they spend on accommodation, ion, food and an gas combined?

800 $1,800

accommodation food

$?

gas

9 units 1 unit

$1,800 0 $1,800 00 ÷ 9 = $200

Food accounted d for 7 units of the th cost. c 7 units 7 x $200 = $1,400 $1,4 Sophie's family mily spent $1,400 $1,40 on food.

Gas accounted of the cost. ounted nted for 4 units o 4 units 4 x $200 = $800 Sophie's hie's family spen spent $800 $8 on gas. $1,800 1,800 + $1,400 + $800 $80 = $4,000

Sophie's 's fam family sspent $4,000 on accommodation, food and gas.

1 85

Let’s Practice At a technology conference, the number of attendees on n Saturday urday 4 was the number of attendees on Sunday. There were re 105 more 9 attendees on Sunday.

Re ga le du ca tio n

1.

(a) How many people attended the conference e on Saturday? (b) How many people attended the conference ce on Sunday? (c) How many people attended the conference weekend? ence e on the wee

186

A carpenter cuts a plank of wood into 2 pieces – Plank A and Plank B. The ratio of the length of Plank A to Plank B is 2 : 9. Plank A is 273 cm shorter than Plank B.

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2.

(a) Find the length of Plank A. (b) Find the length of Plank B. as cut into 2 pieces? piec pie (c) What was the length of the plank before it was

1 87

At Home 7 the amount of prawns on Saturday day than on 3 Sunday. On Saturday he caught 252 kilograms more prawns wns than on Sunday.

A fisherman caught

Re ga le du ca tio n

1.

n Saturday? (a) What was the mass of the prawns caught on ht on n Sunday? (b) What was the mass of the prawns caught (c) What was the total mass of prawns caught ughtt on the weekend? wee

188

The ratio of the mass of a deer to the mass of a hippopotamus is 2 : 11.

The mass of the hippopotamus is 693 kilograms more than n the deer. d

Re ga le du ca tio n

2.

(a) What is the mass of the deer? (b) What is the mass of the hippopotamus? (c) What is the combined mass of both animals?

1 89

Solve It!

Re ga le du ca tio n

The rectangular prisms below are made up of unit cubes.

Prism A

Prism B

Prism C

(a) Find the ratio of the volume of Prism A to the of Prism B to the he volume o volume of Prism C. Express the ratio in n its simplest form. for fo

(b) Find the ratio of the volume lume e of Prism A to the total volume of the 3 prisms. Express the ratio in its simplest form. simple simpl

(c)

190

Another her row ow of unit cube cubes cub is added to Prism C. Find the ratio of the volume ume e of Prism A to tthe volume of Prism B to the volume of Prism C. Express xpress the ratio rati in its it simplest form.

Looking Back The ratio of the capacity of 3 beakers is 4 : 6 : 10

Re ga le du ca tio n

1.

Beaker A Beaker B

Beaker C

Express all ratios in simplest form.

er B to the capacity c (a) Find the ratio of the capacity of Beaker of Beaker C.

er B iss (b) The capacity of Beaker

the t capacity of Beaker C.

eaker er C is The capacity of Beaker

the capacity of Beaker B.

(c)

(d) The capacity of B Beaker err C is Beaker B.

times the capacity of

o of the cap capacit (e) Find the ratio capacity of Beaker B to the capacity of Beaker A.

(f)

capacity Beaker B is The he c apacity of B

the capacity of Beaker A.

(g) The capacity capa of Beaker A is

the capacity of Beaker B.

capacity of Beaker A is (h) The capac B. Beaker B Be

times the capacity of

1 91

In an orchard, the ratio of apple trees to orange trees is 3 : 7. There are 180 apple trees. How many orange trees are in the orchard??

Re ga le du ca tio n

2.

3.

192

Blake and Wyatt collect baseball eballll cards. The number of baseball cards Blake has to the number of baseball seball cards card Wyatt has is in the ratio 5 : 6. In all, they have 506 cards. How many baseball cards 6 baseball seball cards does Wyatt have?

To paint her bicycle pink, Michelle mixed red paint, white paint and blue paint in the ratio 9 : 7 : 2 to make 450 milliliters of pink paint.

Re ga le du ca tio n

4.

Find the amount of each paint Michelle used.

5.

The ratio of Sophie's savings to o Halle's savings to Michelle's savings is 3 : 11 : 7. Sophie has $27. alle have? have (a) How much savings does Halle

e gives ves $10 to M Miche (b) Halle Michelle. How much savings does Michelle ave now? ow have

1 93

What Is Percentage?

Anchor Task

1 94

on

5

Percentage

Let’s Learn

Re ga le du ca tio n

The grids below are each made up of 100 squares.

What percentage of the squares are coloured?

4 out of 100 squares are green. en.

4 reen. een.

100 of the squares are green.

ares are green.

4 percent of the squares 4% of the squaress are green.

50 out of 100 squares are pink.

50 he squares square are pink.

100 0 of the

50 percent of the t squares are pink.

50% % of the squares are pink.

97 o out of 100 squares are blue.

9 97 100 of the squares are blue.

97 percent of the squares are blue.

97% of the squares are blue.

1 95

Re ga le du ca tio n

There are 100 magnets on the whiteboard.

What percentage of the magnets ets are re blue?

29 out of 100 of the magnets are blue.

29 percent of the magnets are blue.

29 29% = 0.29 = 100

What percentage are green? e of the magnets magnet ar 56 out of 100 off the magnets are a green. g 56 percent of the magnets are ar green. g 56 23 56% = 0.56 6 = 100 00 = 50

What at percentage of o the magnets are red?

10 out of 100 of the are red. t magnets m 10 are red.

0 percent of the th magnets m 10 1 10% = 0.11 = 100 = 1 10

196

Re ga le du ca tio n

2 There are 5 apples. 5 of the apples are green.

What percentage of the apples are green?

Method 1

x 20

2 5

=

40 = 40% 100

x 20

Method 2

2 2 = x 100% 5 5 2 x 100% = 5 200% = 5

Multiply th the minator by a factor of 100 denominator to make the de denominator 100.

Multiply th the numerator by the same sa factor.

Multiply the fraction by 100%.

Then simplify.

= 40%

Method d3

2 = 0.4 5 = 0.4 x 100% = 40%

Convert the fraction to a decimal. Then convert the decimal to a percentage by multiplying it by 100%.

1 97

Let’s Practice What percentage of each square is colored?

Re ga le du ca tio n

1.

2.

1 98

(a)

(b)

(c)

(d)

Color 33% % off the square. square

3.

Color 12% of the square.

Write the percentage, fraction and decimal represented by the colored part of the square. (a)

(b)

(c)

(d)

(e)

(f)

Re ga le du ca tio n

4.

1 99

5.

Express the percentage as a decimal and fraction in its simplest form. (b) 3%

Re ga le du ca tio n

(a) 21%

(c)

75%

(e) 12%

6.

20 0

(d) 48% 8%

(f)

88%

Express the fraction ion on as a decimal d ma and percentage.

(a)

8 100

(b)

3 4

(c)

2 2 25

(d)

4 12

At Home What percentage of each square is colored?

Re ga le du ca tio n

1.

2.

(a)

(b)

(c)

(d)

Color 91% % of the square.

3.

Color 19% of the square.

2 01

4.

Express the percentage as a decimal and fraction in its simplest form. (b) 9%

Re ga le du ca tio n

(a) 35%

(c)

52%

(e) 80%

5.

(f)

44%

Express the fraction ion on as a decimal d ma and percentage.

3 (a) 100

(b)

5 20

2 12 60 6

(d)

18 60

(c)

20 2

(d) 18% %

Solve It!

du ca tio n

What do you call a sleeping bull?

To find the answer, convert the fractions and decimals to percentages and centages a an write the matching letters in the boxes below.

e

2 8

o

d b

3 20

12 50

r

l

0.43

14 28

z

4 5

0.6

0.06

a

l

3 4

Re g

u

0.35

75%

24% 43 % 6% 50% 15% 35% 80% 25% 60%

2 03

Finding Percentage

Re ga le du ca tio n

Let’s Learn Riley has $350. She spends 65% of her money on a new bicycle. How much does Riley spend on the bicycle? $350

bicycle ?

Method 1

Method 2

100% = 350

100%

65% =

=

65 x 350 100

65 x 350 100

65 x 35 = 10

350 0

1%

$350 ÷100 ÷10 = $3.50 $3.5

65%

$3.50 x 65 $3 = $227.50

2275 10 = 227.5 =

Try using u Method 1 Metho 2 first. Show or Method wor your working, then check your answer with a calculator.

Riley spent $227.50 on the bicycle.

20 4

al ed uc ati on

At an interschool soccer match, 75 of the spectators were wearing caps and 50 of the spectators were not wearing caps. Find the percentage entage of spectators wearing caps.

100%

wearing caps ?

not wearing caps

Method 1 Total spectators = 75 + 50 = 125

Method hod 2 100% 00%

75 Spectators wearing caps = x 100% 125 = 60%

1%

125 12

100% ÷ 125 4 = 5

75 7 spectators

ng caps.

60% of the spectators are wearing

4 x 75 5 = 60%

25% of a chocolate bar is cut ut off. ff. The length lengt len of the piece cut off is 3 cm.

What is the original length of the chocolate bar? he choco chocola 100% 00%

3 cm 25%

We know that at 25% 5% of the chocolate choco bar is 3 cm.

3 cm.

So, 1% of the chocolate bar is 25 3 100% of the chocolate chocola bar ba = cm x 100 25 = 12 cm

The original length of the chocolate bar is 12 cm. origina lengt

2 05

Re ga le du ca tio n

18% of the applications on Jack’s tablet computer are games.

He has 36 games. How many applications are on Jack’s tablet computer altogether?

100%

games 18%

?

We know that 36 is 18% of the total number of applications. ications.

36 18 2 = =2 1

1% of the applications =

100% of the applications = 2 x 100

There are 200 applications on Jack’s ack’ss tablet computer. com

Dan spent 10% of his money and had $360 left. y on a pair of shoes sh How much money did Dan have ve at first?

100%

$360 90%

10%

100% – 10% = 90% %

90% of Dan’s an’s money is $360 $360.

1% of Dan’s money =

360 4 = = 4 1 90

100% 0% of Dan’s m money = 4 x 100 = $400

206

Let’s Practice A department store is offering a 35% discount on all sofas. s. A 3-seater sofa normally costs $400. What is the price of the sofa during ng the sale?

2.

Mrs. Cole's monthly salary month she spends $1,920 on ary is $6,400. $6,400 Each E rent. What percentage Cole's salary is spent on rent? ag of Mrs. Cole

Re ga le du ca tio n

1.

2 07

A total of 250 students took part in a fun run. 48 students did not finish the fun run. What percentage of students finished the fun run? un?

Re ga le du ca tio n

3.

4.

208

A bicycle shop sells mountain ountain ain bikes for $1,400. During a sale, the shop discounted the price of mountain bikes by $98. What percentage was ountain bik b the discount?

Mr. Peters spend $1,250 on a holiday. He spent $300 on food and the rest on accommodation. What percentage of the money spent pent was w on food?

Re ga le du ca tio n

5.

6.

Halle's science quiz had She answered 45 questions d 72 questions. S correctly. Express Halle'ss score core on the th science quiz as a percentage.

2 09

A sports store is offering a store-wide sale. During the sale, all merchandise is discounted by 35%. The normal price of an exercise exerc bike is $400. What is the price of the exercise bike during the sale?

Re ga le du ca tio n

7.

8.

210

A cinema has 120 seats. premiere, 65% of the seats s. During uring a movie mov m were occupied. How many the premiere? any y people attended a

During a sale, a computer is discounted by 32%. The sale price of the computer is $714. What is the normal price of the computer? r?

Re ga le du ca tio n

9.

10. In a sports stadium, 38% spectators are male. There are 456 % off the specta spec male spectators. How many ny female spectators are there?

211

784 children visited the aquarium which accounted for 56% of the total number of visitors. How many people other than children visited sited the t aquarium?

Re ga le du ca tio n

11.

12.

212

On a farm 7% of the animals sheep. There are 210 sheep. nimals als are shee sh How many other animals ls are on the farm?

Solve It! What animal always carries an umbrella?

To find the answer, find the values in each box and write the matching letters in the boxess below.

Show your working.

8% of 8,000

i

25% of 852

r

52% of 125

e

80% of 110

n

10% of 3,240

r

40% of 720

d

32% of 1,750

e

45% of 80

Re ga le du c

e

288

36

213

324 560 640

88

65

213

At Home The cost of a first-class ticket from New York to Los Angeles eles iss $1,300.

The cost of an economy ticket is 35% cheaper than the first-class ticket.

-class ticke What is the cost of an economy ticket?

2.

In a fruit bowl, 25% of the apples. There are 18 apples in the he fruits ruits are a fruit bowl. How many ny pieces ess of fruit frui are in the fruit bowl?

Re ga le du ca tio n

1.

21 4

There are 140 cars in a parking lot. 15% of the cars are white and 20% of the cars are black. Find the number of white and black cars. s.

Re ga le du ca tio n

3.

4.

Chelsea scored 88% on herr mathemat mathematics test. There were a total of 50 questions in the test.t. How did Chelsea ow many questions q answer correctly?

215

A farmer picked 250 pieces of fruit from his orchard. 24% of the fruits were apples. 40% of the fruits were pears. The rest of the fruits uits were w mangoes. How many mangoes did the farmer pick?

Re ga le du ca tio n

5.

6.

21 6

A total of 1,260 people visited isited ed the zoo zo in 1 week. Adults accounted for 45% of the visitors. The How many children visited T rest were children. c the zoo in 1 week??

There are 172 donuts in a bakery. 25% of the donuts are chocolate.

How many donuts are chocolate?

Re ga le du ca tio n

7.

8.

During a sale, a television by 36%. The sale price of the n iss discounted discount discoun television is $1,200. What is the normal price of the television?

W Wh nor

217

Percentage Increase and Decrease

Re ga le du ca tio n

Let’s Learn The price of a bicycle last year was $120.

The price of the bicycle this year is $150.

What is the percentage increase in the price of the bicycle? $120

original price new price

$3 $3 $30

increase i

$150

Method 1

Increase in price = $150 – $120 = $30 0

Percentage increase =

increase ease x 100% 10 original nal price

30 x 100% 00% 120 1 1 = x 100 100% 4 =

= 25% 5%

Method 2

Increase in price = $150 – $120 $ = $30

Percentage increase = 30 ÷ 120 x 100% centage increas = 0.25 x 100% = 25%

The percentage increase in the price of the bicycle is 25%. rcen rcentage

21 8

On Monday, 440 people visited Gardens by the Bay. On Tuesday, 616 people visited Gardens by the Bay. What is the percentage increase in the number of visitors from Monday to Tuesday?

Re ga le du ca tio

440 Monday

176

Tuesday

increase

616

Method 1

Increase in the number of visitors = 616 – 440 40 = 176

Percentage increase =

increase x 100% original price ce

176 x 100% 440 2 = x 100% 0% 5 =

4 = 40%

Method 2

Increase in the number umber of visitors vis = 616 – 440 = 176 Percentage increase rea = 176 ÷ 440 x 100% = 0.4 x 100% = 40%

e percentage in increa The increase in the number of visitors is 40%.

219

Re ga le du ca tio n

During a sale, the price of a sofa was reduced from $820 to $492. What is the percentage decrease in the price of the sofa? $820

normal price

decrease

sale price

$328 32

$492

Method 1

Decrease in price = $820 – $492 = $328

Percentage decrease =

decrease x 100% 0% original price

328 x 100% 820 2 = x 100% 5 =

= 40%

Method 2

Decrease in price = $820 820 – $492 = $328

Percentage decrease ease = 328 ÷ 820 x 100% = 0.4 x 100% = 40% 4

The percentage rcentage ntage decrease in the price of the sofa is 40%.

22 0

$30

Re ga le du ca tio n

On Saturday, 1,800 tourists visited the Great Barrier Reef. On Sunday, 1,350 tourists visited the Great Barrier Reef. What is the percentage decrease ecreas in the number of tourists from Saturday to Sunday?

1,800

Saturday

decrease

Sunday

450

1,350

Method 1

Decrease in the number of tourists = 1,800 – 1,350 = 45 450

Percentage decrease =

decrease a x 100% tourists Saturday aturday

450 x 100% 0% 1,800 1 = x 100% 00% 4 =

= 25%

Method 2

Decrease in the number mber of tou tourists = 1,800 – 1,350 = 450 Percentage decrease 450 ÷ 1,800 x 100% crease = 4 = 0.25 0.2 x 100% = 25% 2

The percentage decrea decrease in the number of tourists is 25%. dec

221

Let’s Practice Blake took 580 photographs on a school camp. Riley took k 35% % more photographs than Blake. How many photographs did Riley iley take? ake?

2.

At an end of year clearance rance ce sale, the th price of a car was decreased by 12%. The original p price off the car was $16,000. What is the sale price of the car?

Re ga le du ca tio n

1.

222

During peak periods an airline increases the price of all tickets by 45%.

The off-peak price for a flight from Dubai to Cairo is $450. What is i the price of a ticket from Dubai to Cairo during peak periods?

Re ga le du ca tio n

3.

4.

s In a reptile park there are 65% more snakes than lizards. sna There are 40 lizards. How many snakes are in the reptile park?

223

Wyatt put all of his savings into a high-interest bank account with an interest rate of 5%. After one year, the balance of the account was unt wa $840.

Re ga le du ca tio n

5.

(a) How much was Wyatt's savings before he put it in n the e bank? (b) How much interest did Wyatt earn? (c) If Wyatt does not withdraw any money, what will the balance balanc of the account be after another year?

224

A set of golf clubs was on sale for 40% off the normal price. The normal price for the set of golf clubs is $250. Blake purchased the set of g golf clubs and was charged an extra 8% tax.

Re ga le du ca tio n

6.

(a) What was the sale price of the golf clubs? (b) How much tax did Blake pay? (c) How much did Blake pay for the golf set?

225

The price for a tablet computer is $600. Halle bought the tablet computer on sale for $420. What percentage discount did Halle receive?

Re ga le du ca tio n

7.

8.

22 6

Jim's Steakhouse charges ges a 10% service fee on all meals. Once the service fee has been added, customers are charged 5% tax. ed, custom custome Halle ordered a steak for $50.

50. What Wha was the total cost of the steak?

A fruit-picker picked 360 apples on Monday, 480 apples on Tuesday and 240 apples on Wednesday.

Re ga le du ca tio n

9.

d between twe (a) What was the percentage increase in apples picked Monday and Tuesday? (b) What was the percentage decrease in apples picked ed between Tuesday and Wednesday?

227

Solve It!

Re ga le du ca tio n

What kind of tree fits in your hand?

To find the answer, find the values in each box and write the he matching letters in the boxes below. Show your working.

e

90% more than 90

a

8% more than 50

l

5% less th than 560

m

20% more than 75

a

55% more than 320

p

12% 2% less les than 300

e

65% less than 840

t

20% less than 75

r

30% more than 150 30

54

22 8

264 496 532

90

60

195

294

171

At Home Sophie had $50. She received some money from her grandmother andmother mother and had $70 in all. What was the percentage increase in n the e amount of money Sophie had?

2.

Last year, the cost of Atlantic ntic salmon salmo was $15. This year, the price was raised to $18. What was increase in the price of as the percentage pe Atlantic salmon?

Re ga le du ca tio n

1.

229

Sophie ran 2,500 m on Tuesday. On Wednesday, she ran 3,200 m. What was the percentage increase in the distance e Sophie ran on Wednesday compared to Tuesday?

Re ga le du ca tio n

3.

4.

2 30

Riley had $1,200 in savings. ngs. She bought a set of headphones for $60.

What was the percentage age decreas decrease iin Riley's savings after buying the headphones?

Mr. Jenkins sold 180 kg of fish on Saturday. He sold 162 kg of fish on Sunday. What was the percentage decrease in the fish sold between Saturday and Sunday?

Re ga le du ca tio n

5.

6.

Wyatt bought a pair of shoes es on sale for fo $95. The original price of the shoes was $125. What percentage did Wyatt receive? centage discount d dis

2 31

90% of the runners in a marathon completed the race. 300 runners did not complete the race.

Re ga le du ca tio n

7.

(a) How many runners completed the marathon? (b) How many runners participated in the marathon??

23 2

Solve It!

Re ga le du ca tio n

Joe is kayaking to Stradbroke Island. He left the Broadwater Pier ier and paddled for 1,200 m. He still had 38% more to paddle before reaching ching the island.

(a) How much further does Joe need to paddle? (b) What is the distance from the Broadwater Pier to o Stradbroke Island?

Isla

2 33

Looking Back What percentage of each square is colored?

Re ga le du ca tio n

1.

2.

2 34

(a)

(b)

(c)

(d)

Color 14% % of the square.

3.

Color 45% of the square.

4.

Express the percentage as a decimal and fraction in its simplest form. (b) 28%

Re ga le du ca tio n

(a) 12%

(c)

42%

(e) 70%

5.

(d) 86% 6%

(f)

50%

Express the fraction ion on as a decimal d ma and percentage.

17 (a) 100

(b)

4 20

5 15 60 6

(d)

66 88

(c)

2 35

A sports store is offering a 28% discount on all tennis equipment. A tennis racket costs $150. What is the price of the racket during ing the sale?

Re ga le du ca tio n

6.

7.

23 6

A total of 280 students took ook an English exam. e 15% of the students did not pass the exam. How w many students student studen passed the exam?

Mr. Sax spent $1,600 on a new computer and monitor. He spent $240 on the monitor and the rest on the computer. What percentage tage of o the money spent was on the computer?

Re ga le du ca tio n

8.

9.

Riley scored 92% on herr English glish test. There Th were a total of 75 questions in the test.. How did Riley w many questions q answer correctly?

2 37

Re ga le du ca tio n

10. Michelle put all of her savings into a high-interest bank account with an interest rate of 8%. After one year, the balance of the account was $648. unt wa he bank? (a) How much was Michelle's savings before she put it in the (b) How much interest did Michelle earn? (c) If Michelle does not withdraw any money, whatt will the balance of the account be after another year?

238

A fruit-picker picked 240 pears on Monday, 276 pears on Tuesday and 207 pears on Wednesday.

Re ga le du ca tio n

11.

d between wee (a) What was the percentage increase in pears picked Monday and Tuesday? (b) What was the percentage decrease in pears picked cked d between Tuesday and Wednesday?

2 39

6

Re ga le du ca tio n

Mid-year Exam

Section A – Multiple Choice Questions Questions 1 – 20 carry 1 mark each. There are 4 options given en in each a question – (a), (b), (c) and (d). Shade the letter that best matches the answer.

1.

What is the opposite of the integer indicated ed on the number numb num line?

–4

(a) (b) (c) (d)

2.

–2

0

2

0 3 –3 |3|

4

A

B

C

D

A

B

C

D

Evaluate the algebraic raic expression ression ession w when y = 7.

24 – 2y

(a) (b) (c) (d)

10 14 –10 22 2

Sub-total 240

3.

Find 8 x

2 . Give your answer in its simplest form.

3

Re ga le du ca tio n

(a) 16

1 5 1 (c) 5 3 2 (d) 8 3 (b) 3

4.

B

C

D

A

B

C

D

A

B

C

D

Express 15 cm to 60 cm as a ratio in itss simplest mplest form. (a) (b) (c) (d)

5.

A

15 : 60 15 cm : 60 cm 1:4 3 : 12

What percentage of the square quare grid gri is colored?

% (a) 46% 4 b) (b) 6 (c 54% (c) 5 46 (d) 100

Sub-total 241

6.

Find –14 ÷ (–7).

–7 7 –2 2

Re ga le du ca tio n

(a) (b) (c) (d)

7.

B

C

D

A

B

C

D

A

B

C

D

Find 2 – (–8).

(a) (b) (c) (d)

8.

A

6 10 –6 –10

Evaluate the algebraic expression ression when a = 8 and b = 3.

3a + 2b b

(a) (b) (c) (d)

30 6 5 10

Sub-total 242

9.

Find

2 3 x . Give your answer in its simplest form.

5 4

Re ga le du ca tio n

3 10 2 (b) 3 6 (c) 20 6 (d) 9 (a)

A

B

C

D

10. The capacities of beakers A, B and C is in the rra ratio 7 : 5 : 4.

What is the ratio of the capacity city y of beaker C to the capacity of all the beakers combined? Give your our answer in its simplest form. (a) 4 : 12

(b) 1 : 4 (c)

16 : 4

(d) 1 : 3

A

B

C

D

Sub-total 24 3

11.

12 Express 60 as a percentage.

Re ga le du ca tio n

(a) 20% 1 (b) 5 (c) 12% (d) 0.2

12.

A

B

C

D

A

B

C

D

A

B

C

D

Arrange the numbers from the smallest st to o the greatest. greates

2, –8, –3, |-10|, 0

(a) (b) (c) (d)

13.

–8, –3, 0, 2, |-10| –8, –3, –10, 0, 2 |-10|, 2, 0, –3, –8 0, –8, –3, 2, |-10|

Simplify the algebraic braic raic e exp expression. on

14 + 5w – 11 – 2w

(a) (b) (c) (d)

w 14 + 5w 7w 3 + 3w 3w 25 + 3 w

Sub-total 244

Find 5 ÷

3 . Give your answer in its simplest form.

4

1 3 2 (b) 6 3 15 (c) 4 2 (d) 3 3 (a) 5

15.

A

B

C

D

A

B

C

D

A

B

C

D

What is 20% of 240? (a) 25 (b) 50 (c)

45

(d) 48

16.

le du ca tio n

14.

The temperature re is 12ºC. How Ho much cooler is –11ºC? (a) 1ºC (b) –1ºC (c)

23ºC 3ºC

R

(d) –23ºC 3ºC

Sub-total 24 5

17.

Simplify the algebraic expression.

Re ga le du ca tio n

12x + 2y – y – 2x + 5y (a) (b) (c) (d)

18.

10x + 6y 10y + 6x 10x + 8y 14x + 6y

A

B

C

D

A

B

C

D

C

D

Express 12 ÷ 8 as a mixed number in its simplest mplest form.

2 3

(a)

1 2 1 (c) 1 4 1 (d) 1 2 (b) 2

19.

Match the addition ion on the number n line with an addition equation.

–6

(a) (b) (c) c) (d)

–5 5

–4

5 + (–1) 1) = 4 –5 5+9=4 –9 9 + 4 = –5 –4 – 1 = –5

–3

–2

–1

0

1

2

3

4

5

6

A

B

Sub-total 246

20. Solve the equation.

Re ga le du ca tio n

3w – 20 = 4 (a) (b) (c) (d)

w=5 w=6 w=7 w=8

A

B

C

D

End of Section nA

Sub-total 247

Re ga le du ca tio n

Section B – Short Answer Questions Questions 21 – 40 carry 2 marks each. Show your working and write ite your yo answer in the space provided.

21.

The temperature changed from –12ºC to –8ºC. How much ch did the temperature rise?

Answer:

22. Evaluate the algebraic expression n when a = 7 and an b = 3.

75 – 4a + 3b

Answer:

23. Multiply.

3

4 x7 7

Answer:

Sub-total 248

le du ca tio n

24. Write the ratio 10 : 15 : 40 in its simplest form.

nswer: er: Answer:

st form. rm.

25. Express 48% as a fraction in its simplest

Answer:

26. Arrange the numbers mberss from tthe greatest to the smallest.

R

5, –12, 9, –2

Answer: Sub-total 249

27. Simplify the algebraic expression.

Re ga le du ca tio n

9x + 18 – 2x – 7

Answer: wer

28. Divide.

8÷

4 5

Answer:

29. The ratio of the mass of a mango mang m to the mass of a lemon is 8 : 3.

The mass of the he mango is 200 g. Find the mass of the lemon.

Answer:

Sub-total 250

Re ga le du ca tio n

30. What is 42% of 600?

Answer: swer: r:

31.

Write a subtraction equation to match the number line. line

–6

–5

–4

–3

–2

–1

0

1

2

3

4

5

6

Answer:

32. Solve the equation. ation. n.

78 – 13y = 0

Answer:

Sub-total 251

Re ga le du ca tio n

33. What is two-fifths of three-sevenths?

Answer: wer:

34. The ratio of Sophie's savings to Halle's savings vings to Riley's ssavings is 2 : 5 : 3. Sophie has $58. How much money have? ney y does Halle h

Answer:

35. Blake's science quiz had 84 8 questions. He answered 63 questions correctly. Express ess Blake's score sco on the science quiz as a percentage.

Answer:

Sub-total 252

36. Simplify the algebraic expression.

Re ga le du ca tio n

14m + 3n – 4m – 2n + 5m

nswer: wer: Answer:

1 min to complete mplete e a full rotation. rota ro 4 How long does it take to complete e 8 rotations? otations?

37. It takes a Ferris wheel 3

Answer:

38. Solve the equation. uation. on.

20 – 3x = 2 2x x

Answer:

Sub-total 253

21 as a percentage.

60

Re ga le du ca tio n

39. Express

Answer: wer

40. What is 12% of 75?

Answer:

E of End o Section B

Sub-total 254

Re ga le du ca tio n

Section C – Word Problems Questions 41 – 50 carry 4 marks each. Show your working and write ite your y answer in the space provided.

41.

The width of a rectangle is t cm. The length of the rectangle ectangle ngle is 4 times tim its width. (a) Express the perimeter of the rectangle in terms rms of o t. t. (b) Find the perimeter of the rectangle when en t = 8.

8

Answer (a): (b):

Sub-total 255

42. Look at the weather map and answer the questions below.

Re ga le du ca tio n

–12ºC

0ºC 0 ºC

–4ºC

4ºC C

0ºC

(a) Which city is 4ºC cooler than n New ew York? (b) What is the temperature difference between the coolest and betw warmest cities?

Answer (a): (b):

Sub-total 256

1 Sophie is making a poster for a school presentation. She colors olo of 3 3 the poster yellow and of the remaining part green. Whatt fraction 8 of the poster is green? Give your answer in its simplest est form. orm.

Re ga le du ca tio n

43.

Answer:

Sub-total 257

Re ga le du ca tio n

44. To paint his bicycle green, Jordan mixed yellow paint, blue paint and white paint in the ratio 9 : 6 : 2 to make 680 milliliters of green en paint. pain Find the amount of each paint Jordan used.

Answer :

Sub-total 258

Re ga le du ca tio n

45. A farmer picked 450 pieces of fruit from his orchard. 24% of the fruits were apples. 32% of the fruits were oranges. The rest of the e fruits were mangoes. How many mangoes did the farmer pick?

Answer:

Sub-total 259

ed uc ati on

46. Wyatt bought n cans of beans for $4 each. He paid for the beans with a $50 note.

R

(a) Express the change Wyatt received in terms of n. (b) What change will Wyatt receive when n = 12?

Answer (a): (b): Sub-total

260

Re ga le du ca tio n

47. The temperature inside an ice box is –8ºC. As the ice melts, the temperature increases by 1.5ºC per hour. Find the temperature ure of the ice box after 6 hours.

Answer:

Sub-total 261

48. Mrs. Hanlan's lawn is rectangular in shape with a length of 8 m and 3 m.

4

Re ga le du ca tio n

a breadth of 5

(a) Find the area of Mrs. Hanlan's lawn. (b) She plans on replacing the lawn with synthetic grass. s.

The synthetic grass costs $14 per square meter. er.

How much will it cost to replace the lawn?

Answer (a): (b):

Sub-total 262

49.

The model shows the flowers Sophie picked in her garden.

Re ga le du ca tio n

1 flower tulips

daisies

roses

Express all ratios in simplest form.

he number numbe of daisies. (a) Find the ratio of the number of tulipss to the aisiess to the num (b) Find the ratio of the number of daisies number of roses. num nu (c) Find the ratio of the number of roses to the number of tulips.

Answer (a): (b): (c):

Sub-total 263

During a sale, a smart phone is discounted by 22%.

The sale price of the smart phone is $936.

What is the regular price of the smart phone?

Re ga le du ca tio n

50.

Answer:

End of Exam

Sub-total 264

Mid-year Exam Results Correct

Score

Comments

Re ga le du ca tio n

Section

A

/ 20

/ 20

B

/ 20

/ 40

C

/ 10

/ 40

Total

/ 50 0

/ 100

265

Re ga le du ca tio n

Re ga le du ca tio n

Re ga le du ca tio n © Bluee Ring Media dia Pty Ltd ACN 161 16 590 496 2013 - 2021.

This his publication lication would not have been possible without the tireless effort of our production team.

Special thanks to: Daniel Cole, Matthew Matthe Cole, Col Wang Hui Guan, Kevin Mahoney, Winston Goh, Jesse Singer, Joseph eph Anderson, Anderson Halle Taylor-Pritchard, Sophie Taylor-Pritchard, Tejal Thakur, Nakapat,Varasinun Mathanattapat, Kanungnit Pookwanmuang, Saijit Lueangsrisuk Natchanuch Nak Nakapat,V

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