1
Rational Numbers
Let's Recall In our previous grades, we learned about natural numbers and whole numbers. We know that integers are positive and negative whole numbers. We also know how to read, write and represent fractions and decimals. We know that these are all rational numbers. Let us recap how to add and subtract integers, fractions and decimals. Adding Integers
Subtracting Integers
Like Sign
Unlike Sign
Add and keep the sign.
Subtract and keep the larger number sign.
4+2=6
Change the sign of subtrahend and solve. –3 – (–2) = –3 + 2 = –1
(–4) + 3 = –1
(–4) + (–2) = –6 Adding and Subtracting Decimals • Convert to like decimals • Align the decimal points • Add or subtract 1 2 .6 3 + 1 .3 0 1 3 .9 3
1 2 .6 3 − 1 .3 0 1 1 .3 3
Adding and Subtracting Fractions Like Fraction
Unlike fraction
• Add or subtract the numerators.
• Convert to like fraction by taking the LCM.
• Keep the common denominator.
• Add or subtract as done in like fractions.
2 4 2+4 6 + = = 5 5 5 5
2 2 10 − 6 4 – = = 3 5 15 15
Letʼs Warm-up Fill in the blanks. 5 + 2 4 3
__________________
2 52.31 + 61.2
__________________
3 − 23 + 69
__________________
4
7 − 5 8 4
__________________
5 89 − 47.25
__________________
1
I scored _________ out of 5.
Understanding Rational Numbers Shubhi and Rishabh collected data on the temperature of different cities across the world in December. Rishabh: Shubhi, can you convert the temperature of Yakutsk and Dubai in fractions? Shubhi: Let me try Rishabh! We can write −25.5°C as −255 °C and 22.5°C as 255°C. 10 10 Rishabh: But −255°C is not a fraction! 10 Both the kids got confused about what should they call this number. Let’s help them out!
Temperature in ℃
Real Life Connect
30
22.5
20 10 0
-10 -20 -30
–14.2
–18.3 –25.5 Yakutsk
Dubai
Fraser
City
Golmud
Rational Numbers On converting − 25.5°C and 22.5°C to fractions, Shubhi got −255°C 10 and 255°C. These numbers are called rational numbers. 10 A rational number is a number that is expressed as p, where p and q q are integers and the denominator should never be equal to zero.
Rational
Any number that can be written as function –5.25 56 12
For example, 12.5, − 15, 25, 6 and −8 are all rational numbers. 7 9
Whole
–12 10 7.5
All positive integers and zero 0, 1, 2, 3, 4,....
Natural
Identifying and Representing Rational Numbers
All positive integers, excluding zero 1, 2, 3, 4,....
We know how to represent integers, fractions and decimals on a number line. Let us see how to represent rational numbers on a number line. We have marked the number line in equal intervals of 0.5 keeping 0 in the center.
Integers
All positive and negative whole numbers, and zero ....-3, –2, –1, 0, 1, 2, 3,....
Rational Number −1 = − 0.5 2 −1 2
−2 −1.5 −1 −0.5
Integer
Example 1
2
0
0.5
1
1.5
2
2.5
Decimal Number
Represent 5 and − 4 on the number line. 3 3 We can write 5 as 1 1. So, it will lie between 1 and 2 on the number line. Divide the space between 1 and 2 3 3 into 3 equal parts and mark 5. 3
We can write −4 as − 11. So, it will lie between -1 and –2 on the number line. Divide the space between –1 3 3 and –2 into 3 equal parts and mark –4. 3 −5 3
Example 2
−4 3
−3 −2 3
−1
3
3
0
1
3
2
3
3
3
4
3
5
3
Mark − 2, 1, 1.75 and –2.25 on the number line.
−2
0
−1
1
2
The number line has been marked at equal intervals of 1 in both directions. 4 4 7 − 8 − 9 −2= , 1 = , 1.75 = and − 2.25 = 4 4 4 4
−10 −9 4 4 Do It Together
Represent
−8 4
−7 4
−6 4
−5 4
−1 −3 −2 4 4
−1 4
0
1 4
2 4
0
1 8
2 8
3 4
4 4
5 4
6 4
7 4
2
9 4
10 4
−2 5 , , −1 and 0.5 on the number line. 8 8
–7 8
–6 8
–4 8
–1 8
4 8
Comparing and Ordering Rational Numbers Rational numbers can be compared by changing them to the same form. Let us recall how to write rational numbers in standard form and equivalent rational numbers. Standard Form of Rational Numbers A rational number is said to be in standard form if: • both the numerator and denominator contain no common factors other than 1. • the denominator has a positive integer. To convert rational numbers to standard form: •
divide both numerator and denominator with their HCF.
For example, the standard form of Equivalent Rational Numbers
6 = 6 ÷ 3 = 2. 9 9÷3 3
Think and Tell
Is pi a rational number?
Two rational numbers are equivalent if their standard forms are equal. To find the equivalent rational number of a rational number, we multiply or divide the numerator and denominator by the same natural number. 1 1×2 2 1×3 3 2 3 1 = ; = . So, and are equivalent to . For example, = 4 4 × 2 8 4 × 3 12 8 12 4 Two or more rational numbers can be compared and ordered using the steps given.
Step 1: Write all the rational numbers in the same form. Chapter 1 • Rational Numbers
3
p q
Step 2: If the rational numbers are in form and their denominators are different, find the LCM of the denominators. Step 3: Find equivalent rational numbers with denominator as the LCM. Step 4: C ompare the numerators of the equivalent rational numbers. The rational number with the greater numerator is the greater rational number.
2 5 and using the above steps. 3 7 Both the rational numbers are already in the same form, so we will find the LCM of 3 and 7 = 21. 2 2 × 7 14 5 5 × 3 15 = = and = = 3 3 × 7 21 7 7 × 3 21 14 15 As, 14 < 15. So, < . 21 21 2 5 Hence, < . 3 7 Now what if the rational numbers are in different forms? How do we compare a fraction with a decimal or a decimal with an integer? Let us compare
To compare different kinds of rational numbers, we convert them into the same form. −7 . 3 p −7 The comparison can be done by either converting − 2.5 to form or by converting to integers. q 3 Let us compare − 2.5 and
Method 1
Method 2
−25 −5 = 10 2 LCM of 3 and 2 = 6
−7 = −2.33 3 As, − 2.5 < − 2.33
− 2.5 =
−5 −5 × 3 −15 −7 −7 × 2 −14 . = = ; = = 2 2×3 6 3 3×2 6 −15 −14 < As, −15 < −14. So, 6 6 −7 . Hence, −2.5 < 3
So, −2.5 <
Let us now arrange the rational numbers LCM of 7, 3, 9 and 5 = 315 The rational numbers can be written as
−7 . 3
−5 7 3 , , and in ascending order. 7 3 5
−5 × 45 −225 7 × 105 735 2 × 35 70 3 × 63 189 = , = , = and = . 3 × 45 315 3 × 105 315 9 × 35 315 5 × 63 315
Compare the numerators of the rational numbers. −225 < 70 < 189 < 735 We can write the rational numbers in ascending order as
−225 70 189 735 < < < . 315 315 315 315
−5 2 3 7 < < < . 7 9 5 3 −4 Give any three equivalent rational numbers to . 5
So, the rational numbers in ascending order are Example 3
−4 −4 × 2 −8 = = 5 5 × 2 10 So, the three equivalent rational numbers of 4
−4 − 4 × 3 −12 = = 5 5×3 15 −4 −8 −12 −16 are , , and . 5 10 15 20
−4 −4 × 4 −16 = = 5 5×4 20
Example 4
4 Suman and Kamal bought equal amounts of fruit in their lunch box. Suman ate of the fruit while 9 8 of the fruit. Who ate more fruit? Kamal ate 11 To compare the given numbers, we will find the LCM of 9 and 11. LCM (9,11) = 99 Write the equivalent rational numbers. Compare the numerators 44 < 72
4 × 11 44 8 × 9 72 = and = . 9 × 11 99 11 × 9 99
Think and Tell How is
4 8 < . 9 11 Therefore, Kamal ate more fruit than Suman.
−4 different from 4 ? 5 −5
Hence,
Do It Together
Arrange the rational numbers − 3,
13 8 , , 1.7 in descending order. 6 3
p form. q Therefore, the rational numbers can be written as: _____________________ To compare and order the rational numbers, first convert them in
Find the LCM of the denominators and write the equivalent rational numbers. The LCM of 1, 6, 3 and 10 is _____________________. The equivalent rational numbers are: ____, ____, ____, ____ Compare the numerators. _____________________ Descending order = ____>____>____>____
Rational Numbers Between Two Rational Numbers Rational numbers between two rational numbers can be found using their mean or equivalent rational numbers. By Finding the Mean
By Finding Equivalent Rational Numbers
p m p m If q and n are two rational numbers and q < n ,
• Find the LCM of the denominators and write the equivalent rational numbers.
1 p m p m + is a rational number between q and n . 2 q n
• Write the rational numbers in increasing order. For example, to find the rational numbers between
For example, a rational number between −21 +10 −3 2 −11 and = 1 −3 + 2 = 1 = 5 7 2 5 7 2 35 70
(
Example 5
)
1 and 8, we find the LCM of 3 and 9 and write the 3 9 1 3 8 8 equivalent rational numbers as = and = . 3 9 9 9 3 8 H ence, the rational numbers between and are 9 9 4, 5, 6 and 7 . 9 9 9 9
Find two rational numbers between 2 and 3. The rational number between 2 and 3: =
Chapter 1 • Rational Numbers
1 2 + 3= 2
(
)
5 5 ;2 < < 3 2 2
5
The rational number between 2 and 5: = ;2 < < 2 + = = 2 2 2 2 4 4 2 2 1
5
19
9
9
5
Hence, two rational numbers between 2 and 3 are 5 and 9. 2 4 Give four rational numbers between 1 and 1 . 3 2 LCM of 3 and 2 = 6.
Example 6
Hence,
1 2 1 3 = and = 3 6 2 6
Here, the new equivalent rational numbers do not have any numbers between them. So, we multiply and divide each of them by 5. 2 2 × 5 10 3 3 × 5 15 = = and = = 3 6 × 5 30 6 6 × 5 30
Four rational numbers between Do It Together
10 15 11 12 13 14 and = , , and . 30 30 30 30 30 30
− 4 and 7 . 12 9 LCM of 9 and 12 = ____, hence ____ − 4 = and 7 = ____ 9 12 Five rational numbers between − 4 and 7 = ________________________________ 9 12
Give five rational numbers between
Do It Yourself 1A 1 Represent the rational numbers on the number line. a
4 5
b -
2 4
c
5 3
d -
6 5
2 Which rational number does point Q represent on the number line? Q
-2
3 Convert the rational numbers into their standard form. a
3 9
b
-2 16
0
-1 c
8 -28
d
-16 -42
d
15 -21
4 Give three equivalent rational numbers for the rational numbers. a
5 8
b
5 Compare and fill in the blanks.
6
-9 12
c
6 8
a
1 3 ______ 4 5
b
-5 7 ______ 8 12
c - 0.4 ______
-3 14
d
2 16 ______ 9 72
e
-6 -5 ______ 11 12
f
14 ______ 2.8 5
g -3.2 ______
18 5
h
53 19 ______ 7 2
6 Arrange the rational numbers in ascending order. 9 1 8 5 a -3.5, , 2 , , 5 4 7 2
d
-3 2 7 , 1.5, 4.2, 2 , 2 5 5
b
5 1 , -0.75, 1.25, 1 , 7 9 6 10
c
8 2 , -2, 1.25, 1 , 13 9 4 4
e
16 1 , 3 , -2.5, 3, -18 3 4 5
f
3 -25 36 17 , 2.1, , , 5 2 5 2
7 Circle the rational numbers that lie between -3 and -2. -13 5
a
b
-16 8
c
-21 6
d
-18 8
-14 -26 and . 9 12 1 Five times the reciprocal of a number plus is 8. Find the number. 2
8 Give four rational numbers between 9
Word Problem 1
7 5 2 m of red cloth and m of green cloth. She used of the total cloths. Which 8 6 5 property can be used to find the amount of cloth used? Use the property and verify your answer.
Simran bought
Operations on Rational Numbers Real Life Connect
Shubhi and Rishabh now started looking for the temperature of their city over the past 5 days. They recorded the data in the form of rational numbers. Both of them performed various operations on the rational numbers and found different data. Let us see how!
Day
Temperature
Monday
-3 °C 2 16 °C 5 9 °C 5 23 °C 5 -3 °C 4
Tuesday Wednesday
Addition and Subtraction of Rational Numbers
Thursday
Addition of Rational Numbers
Friday
The addition of rational numbers is similar to the addition of fractions. However, we first have to convert all the rational numbers to be added into rational numbers with positive denominators. Let us use the data collected by Rishabh and Shubhi and look at the cases and steps of addition of rational numbers. Same Denominators
1
16 9 + 5 5 (16 + 9) 25 = = =5 5 5
Different Denominators
2 −3 + 9
2 5 LCM of 2 and 5 is 10.
Write the equivalent rational number with the
denominator as LCM. −3 × 5 −15 9 × 2 18 and = = 10 2×5 5 × 2 10 −15 18 −15 + 18 3 + = = 10 10 10 10 Chapter 1 • Rational Numbers
7
Adding Rational Numbers with Negative Denominators Let us add 16 and 3 5 −4
Converting 3 into a rational number with a positive denominator, we get 3 × −1 = −3 −4 −4 −1 4 LCM of 5 and 4 = 20 Write the rational numbers with denominators as 20. 16 × 4 = 64 and −3 × 5 = −15 5 × 4 20 4×5 20 So, 16 + (−3) = 64 + −15 = 64 − 15 = 49. 5 4 20 20 20 20 Example 7
Add
18 −19 , and 2.5 4 8
2.5 can also be written as 25. 10 LCM of 4, 8 and 10 is 40.
Write the rational numbers with the denominator as 40. 18 × 10 = 180, −19 × 5 = −95 and 25 × 4 = 100 4 × 10 40 8 × 5 40 10 × 4 40 Hence, 18 + −19 + 25 4 8 10 180 + −95 + 100 = 180 − 95 + 100 = 185 = 37 40 40 40 40 40 8 Example 8
A shopkeeper sold 2
1 3 2 kg, 5 kg 1.5 kg and 4 kg of rice to four customers. Find the total amount of 4 8 3
rice sold by the shopkeeper.
The weights can be expressed in rational numbers as: 2
1 9 3 43 15 3 2 14 = , 5 = , 1.5 = = and 4 = 4 4 8 8 10 2 3 3
Total weight of rice sold =
9 43 3 14 + + + 4 8 2 3
LCM of 4, 8, 2 and 3 = 24 54 129 36 112 54 + 129 + 36 + 112 331 Hence total rice sold = + + + = = kg 24 24 24 24 24 24 Do It Together
1 Add 15, 23 and 3.6 4 5
2 Add 5, 6 and ‒ 2.8 3 ‒13
Subtraction of Rational Numbers We know that subtraction is the opposite or inverse of addition. To subtract two rational numbers, we simply add the additive inverse of the rational number to be subtracted to the first rational number. Also, the rational numbers should have a positive denominator. 17 7 17 -7 34 + (-21) 34 - 21) 13 For example - = + = = = 6 6 3 2 3 2 6
8
What if the rational numbers are in different forms? We will first convert the numbers into the same form. 23 Let us subtract from 7.85 5 Method 1
Convert
23 = 4.6 5
Method 2
Convert 7.85 to a fraction.
23 to decimal. 5
7.85 =
785 100
785 23 785 - 23 = + 100 5 100 5
7.85 - 4.6 = 3.25
785 + (- 460) 785 - 460) 325 13 = = = 100 100 100 4
What if we have more than two rational numbers with different signs between them? Let us find out! Simplify
2 5 1 4 + - + 3 9 6 3
To simplify the above statement, we will find the LCM of all the denominators, and convert them in like rational numbers. LCM of (3, 9, 6) = 18 Hence,
2 5 - 1 4 12 + 10 - 3 + 24 43 + + + = = 3 9 6 3 18 18
Simplify
Example 9
7 8 -3 -9 + + 6 9 12 15
Example 10
7 8 -3 -9 7 8 -3 + + = + + additive inverse of 6 9 12 15 6 9 12 7 8 3 -9 ⇒ + + + (LCM of 6, 9, 12 and 6 9 12 15 15 is 180.)
Do It Together
521 kg. Suppose 12 the empty drum weighs 12.5 kg. What is the weight of the sugar in the drum? 521 Weight of drum with sugar = kg 12 125 25 Weight of empty drum = 12.5 kg = = kg 10 2 A drum full of sugar weighs
⇒
210 160 45 - 108 + + + 180 180 180 180
Weight of sugar =
⇒
210 + 160 + 45 - 108 307 = 180 180
=
521 - 150 + (LCM of 12 and 2 is 12.) 12 2
=
521 - 150 371 = kg 12 12
a Subtract 12 from 6.25 5
Chapter 1 • Rational Numbers
521 25 521 - 25 - = + 12 2 12 2
b Subtract from 5 from 31 ‒2 2
9
Properties of Addition and Subtraction of Rational Numbers Closure Property If
For example,
a c a c a c and are rational numbers, then + and - are also rational numbers. b d b d b d
-5 3 -7 -1 + = or (Rational number) 7 14 14 2
Commutative Property
For Addition If
For example, - 4.5 -
a a c c a c and are rational numbers, then + = + . b b d d b d For example,
If
1 = -5 (Rational number) 2
For Subtraction
a c a c c a and are rational numbers, then - ≠ - . b d b d d b
5 3 13 3 5 13 = and + = + 7 14 14 14 7 14
For example,
8 7 1 7 8 -1 - = but - = 9 9 9 9 9 9
Associative Property
For Addition
For Subtraction
a c e a c e a + + = + If , and are rational numbers, then b d f b d f b c e + . d f For example,
If
a c e a c e , and are rational numbers, then - b d f b d f a c e ≠ - - . b d f
2 2 32 2 3 7 + + = + = 5 5 10 10 5 10
For example,
2 3 5 2 28 32 + = + + = 5 10 2 5 10 10 Additive Identity If
For example, Example 11
If
a a is a rational number, then + b b -a = 0. b For example,
Verify if m - n is a rational number if m and m-n=
Example 12
-2 -2 +0= 3 3
4 3 5 4 -1 9 - - = = 5 5 10 5 10 10
Additive Inverse
a a is a rational number, then + 0 b b a a =0+ = . b b
5 4 3 1 5 –3 – = = 10 5 5 5 10 10
5 -5 + =0 8 8
Subtraction Property of Zero If
a is a rational number, then b a a -0= . b b For example,
–9 6 and n = . 11 22
-9 6 -18 - 6 -24 -12 = = = = rational number 11 22 22 22 11
5 3 7 and are three rational numbers, then prove that they are not associative under subtraction. 4 8 12
If ,
a c e a c e - ≠ b d f b d f 5 3 7 7 7 7 5 3 7 5 - 5 35 So, - = = and - = = 4 8 12 8 12 24 4 8 12 4 24 24
Associative property for subtraction states that
7 35 ≠ ; hence rational numbers are not associative under subtraction. 24 24
Do It Together
Fill in the blanks using the properties of rational numbers. 1 5 - _______ = 5 2 2
10
3 3 -0= 5 5
2 5 + -3 = -3 + _______ 2 9 9
3 12 + _______ = 0 25
Do It Yourself 1B 1 Solve. a
4 6 + 7 7
b
d -2 + 4 + 2.5
e
c
e 8 + (-24) + 14.5
2 Solve. a
5 -7 + 9 11
8 -2 11 11
b
9 2 - 1.3 7 -9
f
9 38 13 39
3 What decimal should be added to
f 17 + 18.44 + (-12) -5 7 + 12 15
c
8 -14 7 15 11 5
-12 -13 + 2 15
d
13 9 - 2.1 -4 16
g
h
-13 -6 8 15
17 - 16 2 5 3 7
21 -27 to get ? 5 6
4 What fraction should be subtracted from 8.5 to get (-12.25)? 3 7
2 3
5 A rope is 5 m long. Another rope is 1 m shorter than it. What is the total length of the ropes? 6 Ravi earned ₹480 in 1 day. He spent ₹
153 253 on tea and snacks, ₹ on food and the rest he saved. How much did he save? 2 2
7 Name the property illustrated through each of the operations. a
a c - = a rational number b d
b
-12 7
b
a c e a c e + + = + + b d f b d f
c
a a -0= b b
13 2
c
5 21 d -9 14
8 Find the additive inverse of the rational numbers. a
9 Simplify
5 -3 6 -3 2 8 + - + - 6 8 7 14 5 -15
8 9
5 3
7 6
10 Find the rational number that must be added and subtracted so that they will make the sum + - to the nearest whole number.
11 Era has the given weight of grains in her pantry.
Wheat 25.5 kg
Rice
Corn
2 kg 3
37 kg 2
12
a How much wheat and rice does Era have?
b What is the difference between the weight of wheat and corn?
Word Problems 1 2
A carpenter bought a piece of wood 12
How long is the piece of wood now?
2 5 feet long. He then cut off feet from both ends. 3 8
1 2 7 kg in the first week, 4 kg in the second week, 4 kg 5 3 8 in the third week and the rest in the last week. How much flour did he use in the last week?
Jimmie bought 18 kg of flour. He used 3
Chapter 1 • Rational Numbers
11
3 4
15 kg of apples 2.6 kg of bananas and some oranges from the market. 4 Suppose she purchased 9 kg of fruit. What is the weight of the oranges purchased? 2 A school prepared 25.5 litres of juice for its players. The players drank 8 litres in the first 5 3 half. In the second half, the players drank 2 litres more than in the first half. What is the 8 volume of the juice left at the end? Suhani purchased
Multiplication and Division of Rational Numbers Multiplication of Rational Numbers The multiplication of rational numbers is similar to that of fractions. Product of numerators Product of two rational numbers = Product of denominators For example,
12 5 60 7 -3 -21 -7 × = and × = = 13 7 91 9 8 72 24
What if the rational numbers are in different forms? -4 and 0.56 Let us multiply 13 56 14 We will first convert them into a similar form as 0.56 = = . 100 25 (–4) 14 (–56) × = 13 25 325 Example 13
Find the product of
13 and -8.5. 25
13 -85 -1105 × = 25 10 250
Example 14
-1105 -221 = 250 50
145 231 The length and width of a rectangular park are m and . 3 9 What is the perimeter of the park? Perimeter of rectangle = 2 × (L + B) = 2 2×
145 231 + 3 9
435 + 231 2 666 1332 = × = = 148 m 9 1 9 9
Hence, the perimeter of the rectangular park is 148 m. Do It Together
1 Multiply 5 and 16 9 -7
12
The product of two rational numbers with similar signs is positive and with opposite signs is negative.
Error Alert!
p -85 Write the numbers in the form ; hence, –8.5 = . q 10 HCF of 1105 and 250 = 5; hence
Remember!
2 Multiply 2.5 and -19 -3
p Convert the numbers into q form and then multiply. 2 8.1 × 2.7 = 3 3
2 27 54 18 × = = 3 10 30 10
Division of Rational Numbers We know that division is the opposite of multiplication. To divide a rational number by another rational number, we multiply the dividend by the multiplicative inverse of the divisor. That is
a c a d ÷ = × b d b c Divisor
Dividend
Did You Know?
15 3 15 6 90 15 ÷ = × = = 4 6 4 3 12 2 To divide rational numbers into different forms we first convert them to the same form and then divide them.
For example,
For example, Example 15
8 951 m of cloth for ₹ What is 3 4 the cost of the cloth per meter?
Divide:
Example 16
Simplify
12 13 -5 3 9 × ÷ × ÷ 5 18 8 5 8
Solve the numbers inside the brackets first,
8 951 m of cloth = ₹ 3 4
Cost of 1 m of cloth =
Do It Together
the size of Jupiter.
1 times 11
13 13 2 13 1 13 ÷ 2 = ÷ = × = 5 5 1 5 2 10
Meeta bought
Cost of
The size of Earth is
12 13 -5 3 9 × ÷ × ÷ 5 18 8 5 8
951 8 951 3 2853 ÷ = × = m. 4 3 4 8 32
=
12 13 -8 3 8 × × × × 5 18 5 5 9
=
12 -104 24 -29952 -1664 × × = = 5 90 45 20250 1125
–15 21 -19 ÷ Divide: 3.5 ÷ 8 5 25
Properties of Multiplication and Division of Rational Numbers Closure Property For Multiplication
For Division
a a c c and are rational numbers, then × will also be b d b d a rational number. If
For example,
−6 −3 2 × = (rational number) 5 2 10
a a c c and are rational numbers, then ÷ is not b d b d necessarily a rational number. If
For example,
6 ÷ 0 = not defined (not a rational number) 11
Commutative Property For Multiplication If
a c c a a c and are rational numbers, then × = × . b d d b d b
For example, 1 × 2 = 2 and 2 × 1 = 2 6 7 6 42 7 42 Chapter 1 • Rational Numbers
For Division If
a c c a a c and are rational numbers, then ÷ ≠ ÷ . d b d d b b
For example,
5 2 25 20 2 5 = = ÷ ÷ but 10 5 5 10 20 25
13
Associative Property For Multiplication If
a, c
b
d
and
e
For Division
are rational numbers, then
f
a
×
7
×
7
×
7
×
b For example,
2 2
∴
2
c
×
2
×
2
×
2
×
d 5 5
5
c
e
e
=
1
=
14 1 14 × = 10 2 20
1
=
2 14 7 × = 2 10 20
1
=
2 1 7 × × 2 2 5
f
2
2
2
a
b
×
d
×
f
a, c
If
b
d
and
e
are rational numbers, then
f
a
.
c
÷
b
d
e
÷
≠
f
a
b
c
÷
d
e
÷
.
f
For example, 3 ÷ 2 ÷ 1 = 21 ÷ 1 = 21 8 4 7 8 8 3
4
3
∴
2
÷
4
7
2
÷
1
÷
8 ÷
7
1
8
=
16 21 3 ÷ = 4 7 96
≠
2 1 3 ÷ ÷ 4 7 8
Distributive Property Multiplication over addition and subtraction a, c
If
b
d
and
e
f
are rational numbers, then a
a, c
If
b
a e a c × + × b f b d and a c a e e a c × − × − × = b d f b f b d b
×
c
Division over addition and subtraction
d
+
e
d
and
e
f
=
f
a
× 1 = 1 × 3
7
×1=
a
b
=
If
a.
14
a
b
÷1=
a
b
c
d
c e a e ÷ + ÷ d f b f and c e e a e ÷ = ÷ − ÷ d f f b f
÷
e
f
÷
2 −1 2 −4 = ÷ = 4 4 4 8 and 1 2 3 2 4 12 −4 ÷ ÷ = = 2 4 4 8 4 4 8 1 3 2 4
÷
Multiplicative Inverse
a is a rational number, then b b
3
=
2 5 2 20 = ÷ = 4 4 4 8 and 1 2 3 2 4 12 20 ÷ + ÷ = + = 2 4 4 8 4 4 8
a
b
a
× 0 = 0 ×
b
If
a is a rational number, then b
= 0.
a
×
3
×
b
6 ×0=0 11
7
5
2
a
b
If ÷ (−1) =
b
a
5
3
= 1. =1
Division by itself and inverse
a is a rational number, then b
1
−
For example, 1 + 3 2 4
Division by 1 and −1 If
a
c
d
Multiplicative Property of 0
a is a rational number, then b b
+
b
1 1 3 2 1 1 × = × = 2 2 4 4 8 4 and 2 1 3 1 3 2 1 × × = - = 2 4 8 8 8 2 4 Multiplicative Identity
a
b
For example, 1 × 3 + 2 = 1 × 5 = 5 2 2 4 4 8 4 and 2 1 3 1 3 2 5 × + × = + = 2 4 8 8 8 2 4
If
are rational numbers, then
−a b
a is a non-zero rational number, then b
1
a
b
÷
a
b
=1
2
a
b
÷
−a b
= −1
Example 17
6÷ 8 ÷ 7 – 8 using the distributive property of division over subtraction. 5 11 12 11 a c a c e e e − − ÷ ÷ = The distributive property of division over subtraction states that ÷ b d b d f f f 6 8 8 8 6 7 7 So, ÷ − = − ÷ ÷ 5 11 11 12 12 5 11 Solve
8 = 11
⇒ 6 × 12 − 7 × 5 ÷ 60
⇒ Example 18
÷
8 11
8 37 11 37 407 × ÷ = = 60 60 11 8 480
Give the multiplicative inverse of
−2 . 7
7 −2 = . 7 −2
Multiplicative inverse of Do It Together
72 − 35 60
Fill in the blanks using the properties of rational numbers. 1
5 3 − 8 2
÷
1 3 = ____ ÷ ____ − ÷ ____ 4 2
6 ____ × = 0 8
2
13 18
3 ____ × 1 =
Do It Yourself 1C 1 Multiply: 2
18 7
b
−5 and 5 8
c
38 16 and 7 −19
f
23 and −3.2 5
g
a
5 14 by 3 5
b
−13 by 3.6 6
c
e
−28
f
34 51 by 7 19
g
a e
and
5
2 Divide:
by
15
30 −7
d
13 and −2.5 15
h
−51 and −28 5
12 25 by 5 −4
d
31 by −7 14
−36
h
−12 and 4.2 7 11
19
and
10
−27 44
by 4.8
48
17
by
65 −14
3 If the product of two rational numbers is 28 and one of them is −17 , find the other. 3
4 Fill in the blanks. a d
3
2
×
−9 −7
5 If a =
5 4
1
8 ×
× −6 5
,b=
−7 9
÷ 3 8
=
2
b −4.5 ÷
____
−13
=
e
____
and c =
6
7
12
25 4
×
5
7
16 5
×
4
9
=
____
÷ 1.5 =
____
c
−8
11
×
f 3.4 ÷
−7 3
8
3
×
÷
14 5
=
____
−4 = ____ 9
then verify
a a × (b × c) = (a × b) × c
b (a − b) ÷ c = a ÷ c − b ÷ c
c a×c=c×a
d (a − b) − c ≠ a − (b − c)
6 Sunita has a square plot of length 185 m. What is the area and perimeter of the plot? 3
7 A ribbon of length 357 m has been cut into 24 equal pieces. What is the length of each piece? 8
Chapter 1 • Rational Numbers
15
8 A zoo ticket usually costs ₹ 250 for elders. For kids, they are priced at 4
for 5 kids’ tickets?
5 8
of the usual cost. How much will it cost
9 The area of a room is 389 m2. If its length is 247 m, find its breadth. 4
10 Simplify:
3
5 2 9 31 17 34 11 ÷ ÷ × × + 8 8 7 13 15 21 7
11 During the summer holidays, Mansi read a book. After reading pages are there pages in the book?
5 8
of the pages, 120 pages were left. How many
Word Problems 1 2
Kavya bought buys 8 rolls of paper towels. The total length of all the rolls was is 88 is the length of 5 such rolls?
1 8
m. What
2 3 Kapil earns ₹18,000 per month. He spends of the total income on rent, of the remaining on 9 7 1 food and of the rest on shopping. How much will he be able to save in a year? 4
Points to Remember • A rational number is a number that is expressed as p , where p and q are integers and the q denominator should never be equal to zero. A negative rational number is always less than a positive rational number. • To compare two or more rational numbers, make their denominators the same and compare the numerators. •
There is an infinite number of rational numbers between two rational numbers.
• The operations on rational numbers are similar to fractions. The signs need to be kept in mind like we do in integers.
Math Lab Setting: In groups of 4 Materials Required: Cards with properties of operations written on them, paper and pen. Method: • Prepare cards with different properties of operations written on them for example, 1 3 5 1 5 3 5 + = ÷ + ÷ ÷ 2 5 7 2 7 5 7 •
16
Distribute an equal number of cards among the groups.
• Each group identifies the property and writes its name along with the solution in on the paper. •
The group that finishes all the sums first correctly first, wins!
Chapter Checkup 1 Which rational number does point P represent on the given number line? P
−2
0
−1
1
2
2 Give a rational number equivalent to −9 having 87 as the denominator. 29
3 Compare and fill in the boxes. a
4
4
−5 7
b
8
1
3
3
9
12 32 e −16 7.2 f 5 8 −6 Arrange the given rational numbers in descending order. a 1.2,
5 2
, −1
1 −7 15 , , 2 6 6
8
b
3
c
−12
g
9
, 1.75, −0.85, −1
8
1
,
−12 7
b
−8 3
6 Give four rational numbers between 7 If m =
c −12 5
and
1 −3 8 ,n= and o = then verify 6 5 10
a m ÷ (n ÷ o) ≠ (m ÷ n) ÷ o
18 4
21 19
1.75
13
h
29 3
51 6
7
17
5 Circle the rational numbers that do not lie between −2 and −1. a
d
10
5
12
−15
c
3
18 5
, −5, 3.25, 2
−10
d
8
3 21 , 6 5 −15 8
.
b m × (n + o) = m × n + m × o
c m−n≠n−m
d (m − n) + o = m + (n + o)
8 Fill in the blanks. a The sum of a rational number and its additive inverse is __________. b Reciprocal of a negative rational number is __________. c The rational number __________ does not have any reciprocal. d Adding __________ to a rational number gives the rational number itself.
9 The sum of two rational numbers is −
8 5
10 Which rational number should be subtracted from
12 7
to get
−15 4
11 Find the area of a triangle with a base of 156 m and a height of 12 Solve and find the multiplicative inverse. 7 −2 + 6 5
b
e −13 ÷ 25 26 5
f
a
Chapter 1 • Rational Numbers
3 7
×
19 17
5 14 −51
g
9 2
m.
2
−
17 6
.
191
5
c
8 +
9 , find the other. 17
, if one of the numbers is
−8 3
÷
24 59
d
−12 7
×
−21 4
h −24 × −29 13 39
17
13 One roll of ribbon is 15
2 3
m long. What is the length of 5 full and one-fourth of such ribbons?
14 Find (m + n) ÷ (m − n), where m =
5 6
and n =
−8 9
.
15 The given graph shows the time taken by different runners in a 500 m race. What is the average time taken by all the runners to complete the race in minutes?
Time taken in sec
105 100
100.26
95
91.63
90.75
90 85
98.55
Satish
Itisha
Rohan
Rhea
Runner
16 Simplify
12 7
×
−5 9
+
2 8
+
7 5
÷
15
19
−
8
15
×
Between which two whole numbers does the result lie?
−5
16
Word Problems 1 5 hours last week. He practised 2 hours 8 6 this week. For how many hours did Rohit practice practise cricket this week? 1 2 Nikita distributed 52 kg of rice equally among 15 families. How much rice did 4 each family get? 175 256 m and width m while Shweta has a 3 Vivek has a rectangular plot of length 3 5 189 square plot of length m. Whose plot has a larger area? Also, find the difference 2 in between the areas of both the plots. 1 Rohit practiced practised cricket for 12
5 3 packets of chips, Alex bought more packets than Richa and 6 4 1 Prince bought 1 times as many packets as Richa. What is the total number of 2 packets of chips bought by all of them?
4 Richa bought 1
3 3 , Ronita divided it by . If the difference 5 5 between the wrong and the correct answer is 80, then find the number. 2 4 6 Shilpa had ₹750 with her. She spent of the money on purchasing a book and of 5 9 the remaining amount on stationary. How much money is left with her? 5 Instead of multiplying a number by
18
Solving Equations in One Variables
2 Let's Recall
Neeta is learning about the ages of all the family members in her family. She asks her mother about her brother’s age. She says that she is 5 years more than 5 times her brother’s age. She also tells her that she is 40 years old. Neeta wants to calculate her brother’s age. We know how to solve the equations of the type ax + b = c by the balancing method and transposing method. Let us recall the two methods. Balancing Method
Transposing Method
5x + 5 = 40
5x + 5 = 40
Subtracting 5 from both sides of the equation, we get,
Transposing 5 from the LHS to the RHS, we get, 5= x 40 − 5
5 x + 5 − 5 = 40 − 5
Simplifying the RHS of the equation, we get,
Simplifying both sides of the equation, we get,
5 x = 35
5 x = 35
Transposing 5 from the LHS to the RHS, we get,
Dividing 5 on both sides of the equation, we get,
x =7
5 x 35 = 5 5 Simplifying both sides of the equation, we get, x =7
So, Neeta’s brother is 7 years old.
Letʼs Warm-up
Match the linear equations with their solutions. Linear Equation
Solution of the equation
a 2x + 4 = 8 b
3x = 9 2
x=7 x = −6
3x − 6 = 15
x=6
d −5x = 40
x=2
c
e
−
x = 3 2
x = −8
I scored _________ out of 5.
Mean, Median Mode Solving Linearand Equations Real Life Connect
Vimala and Jaya are raising money by selling tickets for a hockey tournament. Both Vimala and Jaya have sold the same number of tickets. Jaya has sold thirty-six tickets more than Lata. Vimala has sold three times as many as Lata has. So, how many tickets has Lata sold?
Transposing Method Let us denote the number of tickets sold by Lata be x. So, the number of tickets sold by Jaya are 36 + x. And the number of tickets sold by Vimala are 3x. Now, according to the situation, Vimala and Jaya sold the same number of tickets. 3x = 36 + x
Let us now solve this equation using different methods.
Balancing Method Imaging the expressions on two sides of a simple balance.
For the balancing method, we perform the same mathematical operations on both sides of the equation, so that the balance is not disturbed. The equation is 3x = x + 36 Subtract x from both sides, 3x − x = x + 36 − x ⇒ 2x = 36 Divide both sides by 2,
2x 36 = 2 2
⇒ x = 18
Thus, Lata sold 18 tickets. Solve: 5x + 21 = 2x + 39 5x + 21 – 21 = 2x + 39 – 21 5x = 2x + 18
⇒ 5x – 2x = 2x + 18 – 2x
3x = 18
3 x 18 = 3 3
x=6
20
⇒
Let us understand some important terms,
Equation: 2x + 2 = 6 + x An expression is a combination of terms (variable or numbers or both) that are connected using mathematical operations.
An equation is a mathematical statement with an equal symbol between two expressions.
2x + 2 Expression
=
6+x Expression
Linear equation in one variable is an equation that has linear expressions with only one variable. Variable
2x
+
LHS: 2x + 2
2
=
6x
Equal to symbol
+
2
Equation
RHS: 6x + 2
Solution of a Linear Equation The method of finding the value of the variable so that it satisfies the equation is called solving a linear equation. The value of the variable obtained from the method is called the solution of the linear equation. Let us understand the transposition method. Transposition is the process of shifting a term of an equation to the other side simply by changing its ‘sign’ or by applying an opposite operation. When any term of an equation can be taken to the other side of the equation with the change of its sign then this method is called transposition. Solve for x: 5x + 12 = 3x + 4 Steps for the transposition method:
Step 1
Step 2
Step 3
Identify the variable and constants
Remove the brackets to simplify the LHS and RHS.
Move all variable terms on one side and all the constant terms on the other side.
5x + 12 = 3x + 4
No brackets here
Variable: x Constants: 12, 4
5x – 3x = 4 – 12
Step 4 Simplify the LHS and RHS to solve for the variable. 2x = – 8 x = –4
Remember! While transposing, keep the variable terms on one side of the equation and the constants on the other side.
Chapter 2 • Solving Equations in One Variables
21
Example 1
Solve using the balancing method. 1
2
3x + 47 = 95
5x + 15 = 3x + 33
3x + 47 – 47 = 95 – 47
5x + 15 −15 = 3x + 33 − 15
3x = 48
5x = 3x + 18
⇒ 5x − 3x = 3x + 18 − 3x
2x = 18
2 x 18 = 2 2
x = 16
⇒
x=9
Remember! The mathematical operation is reversed when we move terms from one side to the other side of the equal symbol.
Example 2
Solve using the transposition method. a
Do It Together
8x – 14 = 3x + 11
b
Error Alert!
6x – 25 = 4x – 13
8x – 3x = 14 + 11
6x – 4x = 25 – 13
5x = 25
2x = 12
x = 5
x=6
Add: x + 3 = 7 On solving the equation:
x=7+3 x = 10
Find the value of x. 2
1
x=7–3 x=4
3x + 4(x – 1 ) = 10 3x + __________________ = 10 __________________ __________________ __________________
Simplifying Equations to Linear Form Remember, Lata, Jaya and Vimala sold hockey tickets. They sold the tickets online as well as offline. The
Do It Yourself 2A 1 Find the value of the unknown weights. a
22
b
c
2 Find the value of x using the balancing method. a 6x + 4 = 22
b 2x + 3 = x + 18
c 7x + 45 = 4x + 78
d 2x + 17 = 9x – 32
e 11x + 1 = 3
f 5x – 11 = 2x + 7
3 Find the value of x using the transposition method. a 3x – 5 = 2x + 97
b 5(x – 7) – 2x = 13 – x
c 15x – 2(18 + x) = 29
d x + 2(3 + x) = 5(x – 6)
e 9x + 3 = 13(2x + 8)
f 5(x –17) = 7(x –19)
4 Solve for the value of x in the following equations. a 3x + 4(x – 2) = 41
b x – 1.5(x – 3) = 2(x − 3)
c 3(1.5x + 1) = 16.5
d 2.4x + 4.8 = 1.2x + 6
e 4x + 7 = 1.5(x + 12)
f 1.3(x – 3) + 2.6 = x
5 Solve for the value of x. 18 cm
a
x
b
x Perimeter = 54 cm 12 cm 15 cm
x+3
2x + 3
c 4
3x
Perimeter = 56 x – 1
Perimeter = 35 x+4 x+3
2(x + 1)
3x – 5
x
x
6 One number is twice another number. If 30 is subtracted from both numbers, then one of the new numbers becomes four times that of the other new number. Find the numbers.
7 Find two consecutive even integers so that two-fifths of the smaller number exceeds two-elevenths of the larger number by 4.
8 One number is 3 times the other number. If 15 were to be added to both the numbers, then one of the new numbers becomes twice that of the other new number. Find the numbers.
Word Problems 1
After 30 years, Monty’s age will be 6 more than thrice his present age. What is Monty’s
2
Harsh’s mother is 30 years older than he. In 12 years, Harsh’s mother will be three times as
3
The length of a rectangular field is 7 less than twice its breadth. If the perimeter of the
4
Ravi deposited ₹25,000 in his bank account. He deposited the currency notes of ₹50, ₹100
5
The ratio of Naveen’s age to his father’s age is 2:5. After 10 years, the ratio of their ages is
present age?
old as Harsh. What are the present ages of Harsh and his mother? rectangle is 196 m. Find the area of the field.
and ₹500 in the ratio of 6:2:1. Find the number of notes of each denomination. 1:2. What was Naveen’s father’s age at the time of Naveen’s birth?
cost of the online ticket was ₹200 more than that of the offline ticket. Five-eighths of the online ticket’s cost is the same as three-fourths of the offline ticket’s cost. Let us find the cost of each type of ticket. Let the cost of the offline ticket be ‘y’.
Chapter 2 • Solving Equations in One Variables
23
1 y = 125 8 y = 1000 y + 200 = 1200
5 3 y + 200 = y 8 4
(
)
5 3 y + 125 = y 8 4 3 5 y− y= 125 4 8
Thus, the cost of the offline ticket is ₹1000 and the cost of online ticket is ₹1200. Solve for x. x + 8 3
x+3
=
2
Step 1: M ultiply the numerator of the LHS Step 2: S implify both sides of with denominator of the RHS. x+8 3
=
the equation.
Step 3: A pply transposition method to find the value of x.
2x + 16 = 3x + 9
x+3 2
16 – 9 = 3 x – 2 x x =7
2( x + 8) = 3( x + 3) Example 3
2x + 7 5x + 4 = 3 5 5(2x + 7) = 3(5x + 4) Solve for x:
Example 4
(2 x − 13) x − 3 x − 9 − = +1 5 11 5 ( x − 9) + 5 11(2 x − 13) − 5( x − 3) = 5 × 11 5
Solve:
10x + 35 = 15x + 12
22 x − 143 − 5 x + 15 = x −9+ 5 11
35 – 12 = 15x – 10x 23 = 5x x= Do It Together
17 x − 128 = x −4 11 17x − 128 = 11x − 44 6x = 84 ⇒ x = 14
23 5
Solve for m:
3x + 5 1 = 2x + 1 3
⇒ 17x − 11x = 128 − 44
3(3x + 5) = 1(2x + 1)
Do It Yourself 2B 1 Write YES if the correct solutions are given for the linear equations, else write NO. a x = 2 for
5x = 3 2x − 1
2 Solve. a d
24
5x − 4 7
=
8 9
6 9 = 3n + 1 5n + 3
b x = 6 for
b e
z + 5 6
−
(4x + 1) 3
7 5 = x +1 ( x − 1)
z +1 9
+
c x = 3 for
z + 3 = 4
(2 x − 1) 2
−
c (3 x − 7) 5
7x + 1 3
y − (4 − 3 y )
2 y − (3 − y )
6 =
=
=
1 4
5x + 6 4
3 Find the value of x using the cross-multiplication method. a d
3x − 8
= 6
5x
5(2 − x ) − 4(1 + x ) 2 − x
=
5 8
4 Answer the given questions. a Solve: b Solve:
5
3x − 1 2
+
4x + 6 3
b
5 7 = x x + 2
e
6 9 = 3x + 1 5x + 3
c
0.4x + 5 8
2.5 x + 4 15
2 3 43 + = 2 and find the value of y if = x y 6
2 x 4x − 5 3 x + 5 + = 11 and find the value of y if = 3 y 11 10
Solve the values of the variables to decode the postal code: 2abc5d. a
=
a −1
7a − 27
=
1 3
b
2b + 6 4
=
7b − 34 3
c
11c − 44 5
=
3c + 17 4
d
9d − 7
7d + 3
=
29 31
What is the postal code?
6
The numerator of a fraction is 3 less than the denominator. If the numerator and the denominator both are decreased by 2 then the simplest form of the fraction becomes
7
3 . What is the fraction? 4
Two complementary angles are in the ratio 1:2. When 15 is added to the smaller angle and 15 is subtracted from the bigger angle, the new ratio of the angles became 1:1. Find the angles.
Word Problem 1
The present ages of Abhijeet and Sahil are in the ratio 2:1. Four years ago, the ratio of their ages was 5:2. Find the present ages of Abhijeet and Sahil.
Application of Linear Equations Remember Lata and her friends sold hockey tickets. Jaya sold 54 tickets in total. If she earned a total amount of ₹58,800 and she sold two types of tickets, one for ₹1000 and the other for ₹1200, then how many tickets of each type did she sell? Linear equations help us in solving real-world problems. We need to set up an equation according to the condition given and solve it to find the value of the unknown. This method consists of two steps: 1
Translating the word problem into symbolic language.
2
Solving the equation
Let us solve a few real-life problems. Let the tickets sold for ₹1000 be x. Then the tickets sold for ₹1200 will be 54 – x.
Chapter 2 • Solving Equations in One Variables
25
According to the question, 1000x + 1200(54 − x) = 58800 200x = 64800 − 58800 x = 30
⇒ 1000x + 64800 − 1200x = 58800
⇒ 200x = 6000
⇒ 54 − x = 54 − 30 = 24
Thus, Jaya sells 30 tickets for ₹1000 and 24 tickets for ₹1200. Example 5
Suman’s present age is equal to one-fifth of her mother's age. Twenty-five years later, Suman’s age will be 4 years less than half the age of her mother. Find their present ages. y Let us assume the mother’s present age to be y years; Suman’s present age = 5 According to the condition, y + 25 = 5
y + 25 −4 2
Multiply both sides by 10, 2y + 250 = 5(y + 25) − 40 5y − 2y = 250 − 85
⇒ 2y + 250 = 5y + 125 − 40
⇒ 3y = 165
Suman’s present age =
⇒ y = 55
y 55 = = 11 years 5 11
Thus, Suman’s present age is 11 years and her mother is 55 years old.
Example 6
A positive number is 5 times another number. If 21 is added to both the numbers, then one of the new numbers becomes twice the other new number. What are the numbers? Let us assume the first number to be x; second number = 5x According to question, 5x + 21 = 2(x + 21) 5x + 21 = 2x + 42 3x = 21
⇒x=7
⇒ 5x − 2x = 42 − 21 ⇒ 5x = 35
Thus, the numbers are 7 and 35. Example 7
If the sum of three consecutive multiples of 9 is 108, then find the second multiple. Let us assume the smallest multiple of 9 to be 9x. Next multiple = 9(x + 1); Last multiple = 9(x + 2) So, 9x + 9(x + 1) + 9(x + 2) = 108 27x + 27 = 108
⇒ 27x = 81
⇒ x = 81 ÷ 27
⇒x=3
So, we can find, 9x = 27, 9(x + 1) = 36, 9(x + 2) = 45 Thus, the three consecutive multiples of 9 which add up to 108 are 27, 36, 45. Example 8
26
Tanya attempts a test in which there are 60 questions. For each correct answer x marks will be awarded and for each incorrect answer one-fourth of correct answer marks will be deducted and 0 marks will be awarded for un-attempted question. If Riya attempts 45 questions out of which 29 are correct, what is the value of x if Riya is awarded 200 marks?
Marks awarded for each correct answer = x Marks deducted for each incorrect answer = Total number of questions attempted = 45
x 4
Did You Know?
Total number of correct answers = 29
Knowledge of algebraic expression
Total number of incorrect answers = 45 – 29 = 16
days, as we are often estimating
Total marks scored = 200 x 200 So, (29 × x ) − 16 × 4 = 29x − 4x = 200 x=
can help us plan and schedule our and solving for unknown variables.
⇒ 25x = 200
200 ⇒x=8 25
Thus, the value of x is 8 marks. Do It Together
A car rental company charges ₹500 per day plus ₹12 per km. If the bill is ₹3320 for a day, then how many km were driven? Let the distance travelled be x km. Fixed charges for a day = __________________ Rate of the car for each km = __________________ Total bill amount = __________________
Do It Yourself 2C 1
Form the equation for the cases given. a Thrice x added to 5 is equal to the difference of half of x and 13. b The sum of four times y and 11 is equal to the sum of two times y and 56. c The sum of three consecutive even numbers is 144. d A train travels from point A to point B, a distance of 240 miles. On the return trip, the train’s speed is increased by 20 mph and takes 2 hours less.
2
The sum of four consecutive odd numbers is 352. What are the numbers?
3
The sum of three consecutive multiples of 11 is 429. What are the numbers?
4
A number is decreased by 10% and the new number obtained is increased by 25%, so the result is 81. Find the
5
The angles of a triangle are in the ratio 3:4:5. What is the difference between the largest angle and the smallest
6
Divide 6000 into two parts so that 15% of one number is equal to 35% of the other number.
number.
angle?
Chapter 2 • Solving Equations in One Variables
27
7
Puneet’s age is three times the age of his daughter. If the sum of their ages is 52 years, find the difference
8
The sum of two numbers is 45 and the numbers are in the ratio 2:1. What are the numbers?
9
The total cost of a table and a chair is ₹15,550. The cost of the table is ₹550 more than the cost of the chair. What
between their ages.
is the cost of the table?
10 Virat scores 20 more runs than twice the runs scored by Rohit. Together, their runs are four runs short of a triple century. What are the individual scores of Virat and Rohit, respectively?
11 Amit has a certain number of pencils that cost ₹4 per pencil. He again purchased 10 more than twice the number of pencils he had. If the total cost of all the pencils is ₹290, how many pencils did Amit have initially?
12 Animesh left one-half of his estate to his daughter, one-third of his estate to his son and donated the rest of his estate. If the donation was ₹5,00,000 then how much estate did, he give to his daughter and son?
13 One of the two digits of a two-digit number is 40% of the other digit. If you interchange the digits of this two-digit number and add the resulting number to the original number, you get 77. What is the original number?
14 The speed of the flow of a river is 3 km per hour. A boat goes upstream between two points in 5 hours while it covers the same distance going downstream in 3 hours. What is the speed of the boat in still water?
15 A sum of ₹770 is made up of denominations of ₹5 and ₹10. If the total number of notes is 72 then find the number of notes of each denomination.
16 Two cars simultaneously start from A and B in opposite directions and the distance between them after 4 hours is 60 km. What is the speed of each car?
's' km per h
('s' + 10) km per h A
60 km
B
300 km
Points to Remember • An equation is a mathematical statement with an equal symbol between two expressions. • Any operation done on one side of an equation must be done on the other side of the equation so that the equation remains true. • We use the transposition method to collect similar terms on one side. Change the signs of terms while using the transposition method and moving terms from one side to the other side. • While solving the word problems, identify the unknown terms, form an equation and solve the value of the unknown term. • The following rules can be applied while solving linear equations.
28
o
The same number can be added or subtracted from both sides of the equation.
o
The same number can be multiplied or divided on both sides of the equation.
o
A term can be transposed to the other side of the equation with its sign changed.
Math Lab Aim: Solve linear equations in one variable Material required: Chalk, stopwatch and equation cards Setting: In groups of 4 Method:
1
Prepare equation cards (unsolved equation on cards (should have linear equations)).
Mark the starting line and finishing line in the classroom or in an open area. There could be 2 different laps.
Each team lines up behind the starting line and the first person of each team is given an equation 3 card.
4
The student solves the equation and runs towards the finishing line with the correct solution.
Once they reach the finishing line, they hand the card to the next teammate. The next person 5 solves the next equation and continues the relay.
6
Use a stopwatch to calculate the total time taken by each team to complete the race.
7
The team which has the fastest completion time wins the relay race.
Chapter Checkup 1
Find the value of unknown value of x. a
2
3
4
b
c
Solve using the balancing method. a 4x + 14 = 56
b 3x + 4 = x + 18
c 5x – 7 = 2x +11
d 3x – 4 =1 – 2x
e 5x + 48 = 3(4x – 5)
f
11(x + 1) = 12(x – 1)
Solve using the transposition method. a 5(x + 3) = 3(1.5x + 18)
b 6(2x + 11) = 8(2x – 1)
c 8x – 7 – 3x = 6x – 2x – 3
d 10x – 5 – 7x = 5x + 15
e 5(x – 1) = 2(x + 8)
f
4x – 3 = (3x + 1) + (5x – 4)
Which of these is the correct solution of the equation? a 5x + 7 = 35 for x = 7
b 3(x – 2) = 4(x + 3) for x = 18
d 4(x +1) – 3(2x – 18) = 2x – 2 for x = 15
Chapter 2 • Solving Equations in One Variables
c 4x + 3 – 2(x + 1 ) = 8x for x = e 5(x + 9) – 2x = 90 for x = 15
1 6
29
5
6
7
Find the value of x using the cross-multiplication method. a
3x 9 = 2( x + 2) 10
b
4( x + 2) + 3 27 = 2 x + 3( x − 1) 17
c
d
4( x + 3) − 4 3x + 1 = 5 8
e
11(8 − 3) − 3 7( x − 3) + 2 = 7
f
b
3m + 2 3 = 9m + 4 5
c
5 z + 14 =3 2z
e
7 3 5w + = w − 14 2 2
f
3 b+7 5 4 = 2 4 b−4 5
9
23
=
11(2 x + 1) 11
4( x + 6) 26
= 6 x − 4 − 3x
Solve the equations. a
2 y − 1 17 = y + 3 12
d
2 5 1 − = 5a 3a 15
Answer the questions. F a If=
8
5 x + 11
9 C + 32, then find C when F = –40. 5
b Solve:
x+y 3 9 x + 4 4(2 x − 1) = . = and then find the value of y if xy 8 10 7
c Solve:
2 5 2x + 1 x − 2 3 + 2x = 4. − = and then find the value of m if + x m 10 6 15
Find the value of the unknown variables to decode the phone number. 8ab00c1d2e
a 9(x + 1) − 4(x + 2) = 6
b 11(x − 2) = 2(x + 1) + 3
d
e
2x + 7 1 = 3(7 x − 8) 4
11 x − 3(2 x − 5) 3
=
7 x + 15
c
2( x − 2) 1 = 3(3 x − 5) 3
4
Place the correct mathematical operations (+, –, ×, ÷) so that the equations are true. a 2x __ 3 = 7 for x = 2 for x = 2 c
2x __ 3(x __ 1)
15x __ 2(2x __ 4)
=
18 for x = 3 25
b 3(x __ 1) __ 4x = 46 for x = 7 d
3x __ 4(x __ 3) x __ 2(3x __ 7) = for x = 4 20 7
10 Two numbers are in the ratio 4:9 and the difference between the numbers is 250. What is the sum of the numbers? 11 If you subtract
1 1 1 then you get . What is the number? from a number and multiply the result by 2 3 15
12 Three consecutive even numbers add up to 696. What are the numbers? 13 The numerator of a fraction is less than its denominator by 8. If both the numerator and the denominator are increased by 17 then the fraction obtained is
5 in its simplest form. Find the original fraction. 6
Word Problems 1 The ages of Kushagra and Kush are in the ratio 5:6. Five years later the sum of their ages will be 43 years. What is the difference between the ages of Kushagra and Kush?
2 The present age of Shagun is one-fifth of her mother’s age. After twenty-five years, her age will be 4 less than half of her mother’s age. Find the difference between their present ages.
30
3 The perimeter of a rectangular swimming pool is 170 m. If the length of the pool is 4 m more than the breadth of the pool, what is the difference between the length and the breadth?
4 Sheetal has currency notes of denominations of ₹50, ₹100 and ₹500. The ratio of the number of notes is 12:8:7 respectively. If the total amount of money with Shetal is ₹24,500 how many notes of each denomination does she have?
5 There is a rectangular plot for a school. The length and breadth of the plot are in ratio
15:7. At the rate of ₹150 per metre, it will cost ₹1,32,000 to fence the plot. What are the dimensions of the plot?
6 Vikram took goats to the field. Half of the goats were grazing. Three-fourths of the
remaining goats were playing nearby. The rest of them, 15, were drinking water from the pond. How many goats were there in total?
7 The ratio of the speed of a boat to the stream is 7:2. The boat takes 6 hours more travelling upstream than downstream. What is the time taken by the boat for the entire journey?
8 The angles of a quadrilateral are in the ratio 7:17:19:29. What is the measure of the angles? 9 A book has 400 pages. Raj finishes writing 100 pages in x minutes. Raju takes twice the time taken by Raj to write the next pages. Vivek takes 2 hours more than half the time taken by
Raju to write the remaining pages. If the total time taken to write the book was 33 hours and 20 minutes then what is the value of x?
10 ₹x is to be divided among three friends A, B, and C. The share of A is two-fifths of the total money, the share of B is two-thirds of the remaining money and C’s share is ₹600. What is the value of x?
11 A father is 20 years older than his son. 12 years ago, the age of the father was six times that of his son. Find their present ages.
12 The age of the father is three times the age of his son. After five years, the age of the father will be 2
1 times the age of his son. Find the present ages of father and son. 2
Chapter 2 • Solving Equations in One Variables
31
3 Construction of 6 Quadrilaterals Let's Recall
Rhea drew a figure with different quadrilaterals. She asked her friend Gyan to guess the shape. Gyan was confused since all of them had 4 sides, 4 angles and 4 vertices. Let us identify the shapes using their properties. Properties of some special quadrilaterals A
Kite
Trapezium
Trapezium
Trapezium
Rhombus Parallelogram
Parallelogram
B
•
Opposite sides are equal and parallel
• Opposite angles are equal, and diagonals bisect each other
E D
C B
• All sides are equal
C
•
E
•
A
•
D
Rhombus
Opposite sides are parallel Opposite angles are equal
Diagonals bisect each other at right angles Kite
• Two pairs of adjacent sides are equal
B C
A D D
• Diagonals intersect at right angles
Trapezium • One pair of sides is parallel, and the other pair is non-parallel
C
• Diagonals in an isosceles trapezium are equal A
B
Letʼs Warm-up
Name the quadrilateral by reading the clues. 1 Exactly two pairs of adjacent sides are equal. _________________ 2 Exactly one pair of opposite sides are parallel. _________________ 3 All angles in a quadrilateral are right angles but only opposite sides are equal. _________________ 4 All sides are equal but all angles may not necessarily be equal. _________________ I scored _________ out of 4.
Construction of a Quadrilateral Real Life Connect
Rahul looked at building sketches and wondered how his architect aunt, Nilima, designed such amazing structures.
Nilima told him that she uses rulers and protractors to create shapes, which amazed Rahul. He was particularly eager to learn about making shapes, especially quadrilaterals.
Construction of Quadrilaterals To create quadrilaterals, we need a minimum number of elements. For instance, for a unique triangle, we need three measurements (at least one being a side). A quadrilateral requires at least five of its ten elements (four sides, four angles and two diagonals) to be given for a unique construction. You can construct a quadrilateral with the following sets of at least five elements. 1
Four sides and one diagonal
2
Three sides and two diagonals
3
Three angles and two included sides
mathematician, Brahmagupta,
4
Three sides and two included angles
quadrilaterals in his book
5
Four sides and one angle
Did You Know? In the 7th century, Indian explored properties of unique “Brāhma-sphuṭa-siddhānta.”
Let us discuss constructing quadrilaterals in each of the cases listed above one by one.
Construction with 4 Sides and 1 Diagonal A quadrilateral can be constructed when any 4 sides and one diagonal are given. Let us construct a quadrilateral PQRS in which PQ = 5.3 cm, QR = 4.2 cm, RS = 4.5 cm, SP = 3.7 cm and PR = 6.5 cm.
S
Draw a rough sketch of the quadrilateral PQRS with the marked measurements marked. We may divide quadrilateral PQRS into two triangles namely RSP and PRQ. 1 D raw a line segment PR = 6.5 cm.
2 D raw two arcs: one with center at P (5.3 cm radius) and another with center at R (4.2 cm radius), intersecting at Q.
3.7 cm
6.5 cm
5.3 cm
P
R
Chapter 6 • Construction of Quadrilaterals
P
6.5 cm
4.2 cm Q
Q
5.3 cm
6.5 cm
R
3 Join PQ and QR.
Q
P
4.5 cm
R
P
4.2 cm 6.5 cm
R
33
4 D raw two arcs on the opposite side of PR: one with center at P (3.7 cm radius) and another with center at R (4.5 cm radius), intersecting at S.
5 Join PS and RS. Q
Q
4.2 cm
5.3 cm
P
6.5 cm
5.3 cm P
R
4.2 cm
4.5 cm
3.7 cm S Example 1
R
6.5 cm
S m N
Construct a quadrilateral LMNO in which LM = 6.8 cm, MN = 4.5 cm, NO = 5.2 cm, OL = 3.9 cm and LN = 7.1 cm.
c O 5.2
3.9 cm
Draw a sketch of a quadrilateral LMNO with measurements roughly marked. We may divide the quadrilateral LMNO into two triangles namely ∆LMN and ∆LNO.
7.1 cm
L
6.8 cm M
1 Draw a line segment LN = 7.1 cm.
M
8 6.
3 Join LM and MN.
4 D raw two more arcs: one with center at L (3.9 cm radius) and another with center at N (5.2 cm radius), intersecting at O. 5 Join OL and ON. LMNO is the required quadrilateral.
4.5 cm
cm
2 D raw two arcs: one with center at L (6.8 cm radius) and another with center at N (4.5 cm radius), intersecting at M.
Do It Together
4.5 cm
L
N
86° 7.1 cm 2
3.9 cm
O
5.
cm
Construct a quadrilateral FEDC with given measures. Complete the rough figure. CD = 4.3 cm, DE = 3.6 cm, EF = 5.6 cm, FC= 4.7 cm, CE = 6 cm
Construction with 3 Sides and 2 Diagonals
S 3.5 cm
A quadrilateral can be constructed when any three sides and two diagonals are given.
4.9 cm
= 5.5 cm.
2 Draw two arcs: one with center
at P (4.9 cm radius) and another with center at Q (6.7 cm radius), intersecting at S.
P
S 4.9 cm
34
5.5 cm
R
P
5.5 cm
R
5.5 cm
3 Join PS and QS.
S
P
cm
cm
Draw a sketch of the quadrilateral PQRS with the measurements roughly marked. 1 D raw a line segment PQ
1
6.
7 6.
Let us construct a quadrilateral PQRS in which PQ = 5.5 cm, PS = 4.9 cm, RS = 3.5 cm, diagonal PR = 6.1 cm and diagonal QS = 6.7 cm.
P
6.7 cm
5.5 cm
R
Q
Q
4 D raw two arcs: one with center at P (6.1 cm radius) and another with center at S (3.5 cm radius), intersecting at R. S
5 Join RS and QR. PQRS is the required quadrilateral. S
R 6.7 cm
4.9 cm
7
Example 2
Q
P
6.
5.5 cm
1
cm
cm
P
3.6 cm
cm
8
5. D
Do It Together
3 cm
cm 5 cm
E
3.6 cm
8
5.
D
V 3 cm
cm
cm
6.2
V
A 5.1 cm
2 6.
Draw a sketch of the quadrilateral DAVE with the measurements roughly marked. 5.1 cm
Q
5.5 cm
Construct a quadrilateral DAVE in which DA = 3.6 cm, DE = 5 cm, EV = 3 cm, diagonal AE = 6.2 cm and diagonal DV = 5.8 cm. Find the length of side AV.
A
R
6.
4.9 cm
3.5 cm
E
5 cm
1 Draw a line segment DE = 5 cm. 2 D raw two arcs: one with center at D (5.8 cm radius) and another with center at E (3 cm radius), intersecting at V. 3 Join DV and EV.
4 D raw two arcs: one with center at D (3.6 cm radius) and another with center at E (6.2 cm radius), intersecting at A. 5 J oin DA and AE. DAVE is the required quadrilateral. On measuring, we get the length of AV as 5.1 cm.
Construct a quadrilateral LISA in which LI = 4.2 cm, SI = 5.5 cm, SA = 3.8 cm, diagonal IA = 6.7 cm and diagonal LS = 7.6 cm. Determine the length of the side LA.
D
Construction with 3 Angles and 2 Included Sides
120°
A quadrilateral can be constructed when any three angles and 2 included sides are given.
3 cm
Let us construct a quadrilateral ABCD in which AB = 4.9 cm, BC = 3 cm, ∠A = 60°, ∠B = 90° and ∠C = 120°. Draw a rough sketch of the quadrilateral ABCD and mark its measurements. Chapter 6 • Construction of Quadrilaterals
C
60° A
90°
4.9 cm
B
35
1 D raw a line segment AB = 4.9 cm.
2 Construct ∠XAB = 60° at A.
3 Construct ∠YBA = 90° at B. X
X
Y
90°
60° B
4.9 cm
4 U sing B as centre and a radius of 3 cm, draw an arc cutting BY at C. X
4.9 cm
B
A
X
60°
4.9 cm
D
Z
C
120°
3 cm
90°
B
4.9 cm
5 A t C, construct ∠ZCB = 120°, so that ZC and XA intersect at D. ABCD is the required quadrilateral.
Y
A
60°
B
Y C 3 cm
A
A
90° A
60°
4.9 cm
B
Remember! Keep the compass width consistent when constructing the 90° and 120°.
S Example 3
Construct a quadrilateral PQRS in which PQ = 3.9 cm, QR = 4.5 cm, ∠P = 75°, ∠Q = 100° and ∠R = 60°. Measure the length of PS. Draw a rough sketch of the quadrilateral PQRS with its measurements. 1 Draw a line segment PQ = 3.9 cm.
X Y
R 60°
S 3.3 cm
Z
4.5 cm
75°
P
36
3.9 cm
1
° 00
Q
P
R 60°
75° 100°
4.5 cm
3.9 cm Q
2 Construct ∠XPQ =75° at A. 3 Draw ∠YQP= 100° at B using a protractor. 4 U sing Q as the centre and a radius of 4.5 cm, draw an arc cutting QY at R. 5 A t R, construct ∠ZRQ = 60°, so that ZR and XP intersect at S. PQRS is the required quadrilateral. On measuring, we get the length of PS as 3.3 cm.
Do It Together
Construct a quadrilateral PUSH in which PU = 4.9 cm, SU = 5.5 cm, ∠P = 75°, ∠U = 100° and ∠S = 60°. Find the sum of the lengths of sides PH and SH.
Construction with 3 Sides and 2 Included Angles A quadrilateral can be constructed when any three sides and two included angles are given. Construct a quadrilateral ABCD in which AB = 5.4 cm, BC = 6.6 cm, CD = 3.3 cm, ∠B = 60° and ∠C =30°.
A D 5.4 cm
Draw a rough sketch of the quadrilateral ABCD with its measurements. 1 D raw a line segment BC = 6.6 cm.
2
Construct ∠XBC = 60° at B. X
60°
B
30°
6.6 cm
3.3 cm C
3 U sing B as centre and a radius of 5.4 cm, draw an arc to cut XB at A. X
5.4
cm
A
60° 6.6 cm
C
4 Construct ∠YCB = 30° at C. X
6.6 cm
B
5 U sing C as centre and a radius of 3.3 cm, draw an arc to cut YC at D.
6.6 cm
A
30°
C
Chapter 6 • Construction of Quadrilaterals
B
60°
cm
cm 5.4
cm 5.4
B
X
A
Y
C
6.6 cm
6 J oin AD. ABCD is the required quadrilateral.
X
A
60°
B
C
5.4
B
60°
Y
3.
Y
3
D 30°
6.6 cm
C
B
60°
30°
6.6 cm
cm
C
37
Example 4
Construct a quadrilateral PQRS in which PQ = 4.4 cm, QR = 5.7 cm, RS = 2.5 cm, ∠Q = 60° and ∠R = 100°. Find the length of PS. Draw a rough sketch of the quadrilateral PQRS with its measurements.
3 Using Q as centre and a radius of 4.4 cm, draw an arc to cut XQ at P.
m
S
4 Construct ∠YRQ = 100° at R, using the protractor.
2.5 cm
5 U sing R as the centre and having a radius of 2.5 cm, draw an arc to cut YR at S.
R
6 J oin PS. PQRS is the required quadrilateral. On measuring, we get the length of SP as 4.1 cm.
4.4 cm 60°
Q Do It Together
60° 100° 2.5 cm Q 5.7 cm R
2 Construct ∠XQR = 60° at Q. 4.1 c
P
S
4.4 cm
1 Draw a line segment QR = 5.7 cm.
Y
X
P
100°
5.7 cm
Construct a quadrilateral JKLM in which JK = 5.7 cm, KL = 4.4 cm, LM = 2.8 cm, ∠K = 60° and ∠L = 120°. Find the measure of ∠J.
Construction with 4 Sides and 1 Angle A quadrilateral can be constructed when four sides and one angle are given. Construct a quadrilateral PQRS in which PQ = 4.7 cm, QR = 3.6 cm, RS = 5.6 cm, SP = 2.7 cm and ∠Q = 60°.
S
3.6 cm 2.7 cm
60°
Draw a rough sketch of the quadrilateral PQRS with its measurements. 1 D raw a line segment PQ = 4.7 cm.
2 Construct an angle of 60° at Q. X
R
5.6 cm
P
4.7 cm
3 W ith the centre Q and radius as equal to 3.6 cm, draw an arc to cut XQ at R. X
R 3.6 cm
P
38
4.7 cm
60° Q
P
4.7 cm
Q
Q
60° P
4.7 cm
Q
4 D raw an arc with center P (2.7 cm radius) and another arc from center R (5.6 cm radius), intersecting the first at S.
5 J oin SP and SR. PQRS is the required quadrilateral.
X
X
R
m 6c
5.
3.6 cm S
S
60° P
Q
4.7 cm
2.7 cm
Draw a rough sketch of the quadrilateral PQRS with its measurements. D
X
5c
m
P
5.
1
m
cm
3.5 c
A Do It Together
75° 4.7 cm
4.7 cm D 5.5 cm
1 Draw a line segment AB = 4.7 cm.
Q 5 cm
C
3.5 cm 75° A 4.7 cm B
2 Construct an angle of 75° at B. 3 W ith the centre B and the radius equal to 3.5 cm, draw an arc to cut XB at C.
C 5.5 cm
3.6 cm 60°
Construct a quadrilateral ABCD in which AB = 4.7 cm, BC = 3.5 cm, CD = 5 cm, AD = 5.5 cm and ∠B = 75° Find the measure of diagonal AC.
Example 5
R
4 D raw an arc with center A (5.5 cm radius) and another arc from center C (5 cm radius), intersecting the first at D. 5 Join AD and CD. ABCD is the required quadrilateral.
B
On measuring, we get the length of diagonal AC as 5.1 cm.
Construct a quadrilateral EFGH in which EF = 4.7 cm, FG = 3.8 cm, GH = 4 cm, HE = 7 cm and ∠F = 90° Find the sum of the lengths of both diagonals.
Chapter 6 • Construction of Quadrilaterals
39
Do It Yourself 6A 1 Construct quadrilaterals where the lengths of 4 sides and 1 diagonal are given. a Quadrilateral LMNO: LM = 5.6 cm, MN = 4.2 cm, NO = 7.8 cm, LO = 6.5 cm, MO = 8.1 cm.
b Quadrilateral XYZW: XY = 6.3 cm, YZ = 4.7 cm, ZW = 5.2 cm, WX = 7.4 cm, WY = 8.6 cm.
2 Construct quadrilateral GLOW with the given measures. Measure the unknown side. a GL = 5.6 cm, OW = 3.4 cm, GW = 4.8 cm, GO = 6.7 cm, LW = 7.3 cm
b GL= 5.2 cm, LO = 4.3 cm, WG = 3.8 cm, GO = 6.1 cm, WL = 7.2 cm
3 Construct the following quadrilaterals when three angles and two sides are given. Find the measure of both of its diagonals.
a ABCD: AB = 7 cm, AD = 4 cm, ∠A = 75°, ∠B = 110° and ∠D = 95°.
b WXYZ: WX = 8 cm, XY = 6 cm, ∠W = 120°, ∠X = 85° and ∠Y = 75°.
4 Construct quadrilaterals ABCD with the given measures. a AB = 3.5 cm, BC = 5.4 cm, CA = 4.9 cm, ∠B = 125° and ∠C = 80°
b AB = 4.2 cm, BC = 5.3 cm, CA = 3.6 cm, ∠B = 135° and ∠C = 60°
5 Construct the quadrilaterals given below, where 4 sides and one of the angles are known. a RSTU: RS = 2.9 cm, ST = 3.6 cm, TU = 4 cm, RU = 6 cm, ∠S = 90°.
b JKLM: JK = 3.9 cm, KL, 3.1 cm, LM = 3.6 cm, MJ = 3.3 cm, ∠ J = 60°.
Word Problem 1
You are creating a poster for your school’s geometry fair and want to include a unique quadrilateral shape. You know that you want to use a quadrilateral with four sides
measuring 9 cm, 6 cm, 12 cm and 7 cm, and you want one of the angles between two
consecutive sides to be 60 degrees. How can you use your ruler and compass to accurately draw this quadrilateral for your poster? Draw two quadrilaterals that fit this description.
Constructing Special Quadrilaterals Some quadrilaterals are special and have some of their measures equal. Some examples of special quadrilaterals are squares, rectangles, rhombus, trapezium, kite and parallelogram. The properties of special quadrilaterals help in constructing them. So, we do not need 5 measures to construct special quadrilaterals.
Constructing a Square We know that a square has all its sides equal, all angles measure 90°, its diagonals are equal in length and bisect each other at right angles. A square can be constructed when 1 of its diagonals or 1 of its sides are given.
40
Let us construct a square with the length of its diagonal as 4.6 cm.
It is known that diagonals of a square are equal in length and bisect each other at right angles. This property can help us construct the square.
P
Let’s say, the square is PQRS and the diagonals bisect at O.
2.3 cm
Q O 4.6 cm
Given:
R
S
PR = QS = 4.6 cm (diagonals are equal)
X
PO = OR = QO = OS = 2.3 cm (Diagonals bisect each other)
Q 2.3 cm
1 Draw a line segment PR = 4.6 cm. 2 Draw the right bisector XY of PR, intersecting PR at O. 3 With the centre O and radius equal to 1 × 4.6, i.e., 2.3 cm, draw arcs on either side 2 of PR, cutting XY at Q and S.
O P 4.6 cm
R
S
4 Join PQ, QR, RS and SP. PQRS is the required square.
Y
Think and Tell Is there any other measure, other than the diagonal, that can help us construct a square?
Constructing a Rhombus A rhombus has lengths of all its sides equal, its opposite angles are equal and the diagonals bisect each other at right angles. Let us construct a rhombus whose diagonals are of lengths 5 cm and 7 cm.
It is known that the diagonals of a rhombus bisect each other at the right angle. This property can help us construct the rhombus. Let’s say, the rhombus is PQRS and the diagonals bisect at O.
Q 2.5 cm
PR = 7 cm, PO = OR = 3.5 cm (Diagonals bisect each other)
QS = 5 cm, QO = OS = 2.5 cm (Diagonals bisect each other)
cm O
cm
5
3. P
S X
1 Draw a line segment PR = 7 cm. 2 Draw XY, its perpendicular bisector and let it intersect PR at O. 3 U sing O as centre and a radius of 2.5 cm, draw two arcs on the opposite sides of PR to intersect XY at Q and S.
R
2.5
Given:
5
3.
cm
Q O 2.5 cm R 7 cm
P
4 Join PQ, QR, RS and SP. PQRS is the required rhombus.
S
Constructing a Rectangle
Y
We know that a rectangle has its opposite sides equal, all angles measure 90° and its diagonals are equal in length and bisect each other. Let us construct a rectangle PQRS, given that side PQ = 6.2 cm and the diagonal PR = 7.8 cm. We know that the opposite sides are equal and all the interior angles in the rectangle are right angles. This property can help us construct the rectangle. Let’s say, the rectangle is PQRS and PR is its diagonal. Chapter 6 • Construction of Quadrilaterals
S
R
7.8 P
cm
6.2 cm
Q
41
Given:
PR = 7.8 cm (Length of diagonal)
X
PQ = RS = 6.2 cm (Opposite sides of a rectangle are equal) ∠Q = 90° (All angles are right angles)
1 Draw a line segment PQ = 6.2 cm.
S
R
2 Draw XQ ⊥ PQ at point Q. 3 Using P as the centre and a radius of 7.8 cm, draw an arc cutting QX at R.
8
7.
4 D raw two arcs: one with center R and a 6.2 cm radius, and another with center P and a radius equal to QR, ensuring they intersect at point S.
P
5 Join SP and SR. PQRS is the required rectangle.
Let us construct a trapezium EFGH in which EF ॥ GH, HE = 3.7 cm, FE = 4.5 cm, GH = 6.7 cm and ∠H = 60° cm.
90° Q
6.2 cm
Constructing a Trapezium We know that a trapezium has one pair of parallel sides and in an isosceles trapezium, the non-parallel sides are equal and the diagonals are equal.
cm
E
4.5 cm
F
3.7 cm 60°
H
G
6.7 cm
We know that one pair of opposite sides are parallel in a trapezium. This property can help us construct the trapezium. Given: EF ॥ GH; HE = 3.7 cm, FE = 4.5 cm, GH = 6.7 cm.
1 Draw a line segment GH = 6.7 cm. E
X
4.5 cm
F
Y
3.7 cm H
60°
2 Construct an angle of 60° at H. 3 Using H as centre and radius of 3.7 cm, draw an arc to cut XH at E. 4 Construct XY parallel to HG, passing through E.
6.7 cm
G
5 Using E as centre and radius of 4.5 cm, draw an arc to cut EY at F. 6 Join GF. EFGH is the required trapezium.
Constructing a Parallelogram
In a parallelogram, the opposite sides are equal and parallel, the opposite angles are equal and the diagonals bisect each other.
Let us construct a parallelogram one of whose sides is 6 cm and whose diagonals are 7 cm and 8 cm. D
3.5
It is known that diagonals of a parallelogram bisect each other. This property can help us construct the parallelogram. Let’s say, the parallelogram is ABCD and the diagonals bisect at O.
cm
AC = 8 cm, AO = OC = 4 cm (Diagonals bisect each other)
BD = 7 cm, BO = OD = 3.5 cm (Diagonals bisect each other)
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A
O
6 cm
cm
AB = 6 cm
cm
3.5
4
Given:
cm
4
B
C
1 Draw a line segment AB = 6 cm.
C
D
2 D raw two arcs: one with center A and a 3.5 cm radius, and another with center B and a 4 cm radius, intersecting at O.
O
3 Join OA and OB.
4 E xtend AO to meet C, making OC equal to AO and extend BO to meet D, making OD equal to BO. 5 Join AD, BC and CD. ABCD is the required parallelogram.
3.5 cm
4 cm A
B
6 cm
Constructing a Kite In a kite, two pairs of adjacent sides are equal and the diagonals intersect at right angles. Let us construct a kite in which the measure of two adjacent sides are 5 cm and 3 cm with the length of the diagonal passing through the common vertex of these sides is 4 cm. It is known that the two pair of the adjacent sides have equal lengths in a kite. This property can help us construct the kite. Let’s say, WXYZ is a kite and WY is its diagonal.
X
5 cm
5 cm 4 cm
W 3 cm
Give: WY = 4 cm (Length of diagonal); WX = XY = 5 cm (adjacent sides are equal); WZ = YZ = 3 cm (Adjacent sides are equal)
Y 3 cm
Z
1 Draw a line segment WY = 4 cm.
X
2 D raw two arcs: one with center at W (3 cm radius) and another with center at Y (3 cm radius), intersecting at Z.
5 cm
3 Join WZ and YZ. 4 D raw two more arcs: one with center at W (5 cm radius) and another with center at Y (5 cm radius), intersecting at X.
5 cm 4 cm
W 3 cm
5 Join WX and YX. WXYZ is the required kite.
Y 3 cm
Z Example 6
Construct a parallelogram PQRS in which PQ = 3 cm, QR = 4 cm and diagonal PR = 6 cm. It is known that the opposite sides of a parallelogram are equal. This property can help us construct a parallelogram. Let’s say, the parallelogram is PQRS and PR is its diagonal. Given: PQ = RS = 3 cm, RQ = SP = 4 cm, PR = 6 cm.
Do It Together
Construct a parallelogram with its adjacent sides measuring 3 cm and 4 cm, respectively and the included angle being 60°.
S
R
6 P
cm
3 cm
4 cm
Q
Complete the given rough diagram of the parallelogram before starting the construction.
Error Alert! You may or may not able to construct a quadrilateral with any random set of measures of elements.
Chapter 6 • Construction of Quadrilaterals
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Rough Diagram
Construction
Do It Yourself 6B 1 Construct a square whose each side measures 5 cm. 2 Construct a rhombus ABCD whose side is 5 cm and an angle is 120°. 3 Construct a rectangle whose adjacent sides are 5.6 cm and 4.5 cm. 4 Construct a parallelogram with one side as 4.9 cm and diagonals as 5.4 cm and 8 cm. 5 Construct a parallelogram PQRS in which PQ = 4.6 cm, QR = 6.7 cm and ∠Q = 45°. 6 Construct a trapezium ABCD in which AB ॥ CD, AB = 5.8 cm, CD = 4.5 cm, ∠A = 67° and AD = 3.6 cm. What is the measure of the largest angle in the quadrilateral?
Word Problem 1
You are making a greeting card, and you want to create a square-shaped window in the
front cover. The diagonals of this window are 8 cm. You have a ruler and a compass. What steps will you follow to accurately draw a square on the card’s cover?
Points to Remember •
A quadrilateral can be constructed if the measures of 5 out of its 10 elements are given.
•
Diagonals of parallelograms, rectangles, rhombuses and squares bisect each other.
• A quadrilateral can be constructed when the measure of the 4 sides &1 diagonal, 3 sides &2 diagonals, 3 sides &2 included angles, 3 angles &2 included sides or 4 sides &1 angle are given. • You need the knowledge of fewer elements to construct special quadrilateral as their properties help you determine other unknown elements. •
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The sum of the interior angles of a quadrilateral is 360°.
Math Lab Quadrilateral Pattern Creation Materials needed: Drawing paper, rulers, compasses, pencils, protractors, coloured pencils/ markers (optional) Instructions: 1 Start by reviewing the basic construction techniques for drawing quadrilaterals using a ruler and a compass. 2 Ask each student to brainstorm and plan a design or pattern that they want to create using quadrilaterals. They should consider the type of quadrilaterals they want to include (e.g., squares, rectangles, parallelograms, and so on) and the measurements for each side and angle. 3 Allow students to begin constructing their chosen patterns on their drawing paper. They should use their rulers, compasses and protractors. 4 Encourage students to get creative with their patterns. They can experiment with different colours and arrangements of the quadrilaterals to make their designs visually appealing. 5 Assess each student’s pattern on the basis of creativity, precision in construction and the ability to explain their design choices.
Chapter Checkup 1 Construct quadrilaterals with the given lengths of the 4 sides and 1 diagonal. a Quadrilateral UVWX: UV = 3.9 cm, VW = 5.1 cm, WX = 4.5 cm, XU = 6.8 cm, VX = 7.2 cm. b Quadrilateral STUV: ST = 4.5 cm, TU = 5.7 cm, UV = 6.2 cm, VS = 7.3 cm, SV = 8.9 cm. c Quadrilateral KLMN: KL = 6.1 cm, LM = 3.5 cm, MN = 5.8 cm, KN = 4.2 cm, LN = 7.4 cm.
2 Construct a quadrilateral PINE with the given lengths of 3 sides and 2 diagonals. Find the measure of the unknown side.
a PI = 6.9 cm, EN = 3.2 cm, IN = 5.5 cm, PN = 4.7 cm, IE = 7.6 cm. b PI = 7.4 cm, NE = 3.2 cm, PE= 5 cm, IE = 6.8 cm, PN = 4.7 cm c PI = 6.5 cm, IN = 4.2 cm, PE = 3.9 cm, PN = 7.1 cm, IE = 8.3 cm.
3 Construct the following quadrilaterals when the measure of the 3 angles and 2 included sides are given. Find the measure of both of its diagonals.
a Quadrilateral EFGH: EF = 6 cm, FG = 5 cm, ∠E = 60°, ∠F = 45°, and ∠G = 120° b Quadrilateral IJKL: IJ = 4.9 cm, IL = 7 cm, ∠I = 70°, ∠J = 90° and ∠L = 115° c Quadrilateral MNOP: MN = 6.5 cm, MP = 4.5 cm, ∠M = 100°, ∠N = 90° = and ∠P = 135°
Chapter 6 • Construction of Quadrilaterals
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4 Construct quadrilaterals WXYZ with the given measures. a WX = 4.5 cm, XY = 6.4 cm, YZ = 3.9 cm, ∠X = 150° and ∠Y = 60° b WX = 4.6 cm, XY = 5.5 cm, YZ = 6.6 cm, ∠X = 80° and ∠Y = 100° c WX = 3.2 cm, XY= 5.7 cm, YZ = 4.4 cm, ∠X = 135° and ∠Y = 60°
5 Construct the quadrilaterals given below where 4 sides and one of the angles are known. a Quadrilateral ABCD: AB = 4.6 cm, BC = 4.1, CD = 5.6 cm, AD = 5.2 cm and ∠B = 70° b Quadrilateral PQRS: PQ = 3.9 cm, QR = 3.3 cm, RS = 6 cm, PS = 5 cm and ∠Q = 80° c Quadrilateral WXYZ: WX = 7.6 cm, XY= 6.8 cm, YZ = 5.2 cm, ZW = 4.6 cm and ∠Y = 120°
6 Construct a square whose each diagonal measures 5.8 cm. 7 Construct a rhombus PQRS in which: a PQ = 4 cm and SQ = 6 cm
b QS = 6.6 cm and QR = 3.7 cm
8 Construct a rectangle ABCD when it is given that: a One side measures 4 cm and the diagonal has a length of 6.6 cm. b The diagonals measure 5.6 cm and the angle between them is 60°. c Its adjacent sides measure 3.7 cm and 2.9 cm, respectively.
9 Construct a parallelogram ABCD, given that BD = 4.6 cm, AC = 6 cm and the angle included between the diagonals is 60°.
10 Construct a trapezium EFGH in which EF ॥ GH, EF = 4.5 cm, BC = 3.7 cm, AD = 5 cm and ∠F = 40°. Can we construct more than one trapezium in this case?
11 Construct a parallelogram ABCD in which AC = 5.5 cm, AB = 3.6 cm and the altitude of AM from A to CD is 2.6 cm.
Word Problem ou are working on a craft project and need to create a rhombus-shaped outline 1 Y with a diagonal of 6 cm in length and the side of the length 4.2 cm. How can you use your ruler and compass to draw the rhombus accurately?
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