IES Llanes. Sección bilingüe
Dpto. de Matemáticas
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Index of contents:
Plane Geometry……………………………………………………………………………………...........1 Similarity.............................................................................................................12 The right triangle.............................................................................................14 Polyhedra..............................................................................................................17 Solids of revolution...........................................................................................20 Statistics and probability..............................................................................22 Integer numbers...............................................................................................26 Rational numbers...............................................................................................29 Proportionality……………………………………………………………………………………………...31 Algebra…………………………………….........................................................................33
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3
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CIRCULAR SHAPES: Circumference: is a curved line in which every point is at the same distance from a fixed point called the centre.
Circle: is the zone inside a circumference. Lines of a circumference and number Pi:
Circular slices:
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Area of PlaneShapes Triangle Area = ½b × h b = base h = vertical height
Square Area = a2 a = length of side
Rectangle Area = w × h w = width h = height
Parallelogram Area = b × h b = base h = vertical height
Trapezoid (US) Trapezium (UK) Area = ½(a+b) × h h = vertical height
Circle Area = πr2 Circumference=2πr r = radius
Ellipse Area = πab
Sector Area = ½r2θ r = radius θ = angle in radians
Angles: Units: degrees ( º ) o A circle is divided into 360 equal degrees, so, 1 circle = 360º o A right angle=90º o Degrees can be divided into minutes: 1º = 60’ o Minutes can be divided into seconds: 1’ = 60’’ Another unit is the radian; 1 circle = 2 rad 6.28 rad o 1 radian 57º Angles in the circumference:
inscribed
half-inscribed
interior
exterior
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EXERCISES 1. Write the correct word for each definition: a. Lines that never meet are…………….....lines b. A triangle with three equal sides is a ………..….......triangle c. A quadrilateral with four equal sides and two pairs of equal angles is a …….............. d. The perpendicular bisectors of a triangle meet on a point called………….............. e. The heights of a triangle meet on a point called………................... f. A ...................... angle measures more than 180º but less than 360º g. Lines that meet on one point are………….........…..lines h. A triangle with two equal sides is an ………..…............triangle i. A quadrilateral with four equal angles and two pairs of equal sides is a ……............... j. The medians of a triangle meet on a point called ………….......... k. The bisectors of a triangle meet on a point called ………............
2. Classify these triangles taking into account their sides and their angles: a. Two angles of 35º and 55º b. Two angles of 35º each one c. Two angles of 55º and 65º d. Two angles of 45º each one e. Two angles of 30º and 50º 3. Find the measures of all the unknown angles in these polygons: a. A rhombus with an angle of 130º b. An isosceles trapezium with an angle of 110º c. A rhombus with an angle of 130º d. An isosceles trapezium with an angle of 110º e. A right triangle with an angle of 40º f. An isosceles triangle with the different angle of 120º 7
4. Classify these quadrilateral: a. Four equal sides and for equal angles. b. Only opposite sides are parallel. c. Four eual angles and two pairs of equal sides. d. Four equal sides and two pairs of equal angles. 5. Solve these exercises about area and perimeter: 5.1 Get the diagonal, the perimeter and the รกrea of this rectangle
5.2Get the perimeter and the รกrea of this right trapezium:
5.3 Calculate the perimeter and the รกrea of this polygon and write its name:
5.4Calculate the perimeter and the รกrea of these shapes:
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6. Mark goes riding his bike around a track of radius 50 m. How far does he go in 1 lap? In 5 laps? In 3 and ½ laps? 7. Answer these questions: How many diameters does a circumference have? How many centres? How many radius? 8. A thread with a length of 50 cm can be rolled around a reel (carrete) 10 times. What is the diameter of the reel? 9. What is the length of an arc with a radius and angle of: 4 cm and 20º 5 km and 150º 12 m and 30º 10.James and Tom want to share a pizza with radius 30 cm. James will eat 3/5 of the pizza. Which part (fraction) of the pizza will be for Tom? How many square cm will each one eat? 11.Convert these angles into single units: 20º 30’ 45º 20’ 30’’ 120º 120’’ 12.Convert these angles into complex units: 7200’’ 1354’ 125’ 13.Calculate: 26º 30’ 15’’ – (15º 40’ 25’’) 3 · (35º 35’ 30’’) (121º 37’ 25’’):5 17 h 15 min – (14 h 30 min) 7· (4h 20min 50s)
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Similarity 1. Are similar two triangles with these sides?
a 6 cm,
b 9 cm, c 5 cm.
a' 12 cm, b' 18 cm, c' 10 cm
2. Complete the measures (lengths) of the sides in these triangles to be similar: 3 cm6 cm 2 cm4 cm10 cm
3. What is the height of the tree? Use these data: height and shadow of the statue: 3 m and 4 m. respectively; shadow of the tree: 6 m
4. Draw a segment with a length of 5 cm and divide it into 7 parts. 5. A square has an area of 4 cm2 . What is the area of another square with a side: a)double? b) half? 6. Find similar triangles among these right-angled ones: 63º
41º
27º
45º
49º
47º
A
C
D
E
F 7. The length of a road onB a map with a scale of 1:500 000 is 8 cm. What is its real length?
8. The distance between Sevilla and Madrid is about 540 km. What is this distance on a map with a scale of 1:150 0001 9. Calculate the distance between Tribeca and City Hall
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11. There are four sets of similar triangles below. Can you work out the lengths of the sides marked with a question mark
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Similar polygons have the same shape but different size.
a b c d The lengths of their sides satisfy the relationship: a ' b' c ? d ' Their corresponding angles are equal: A = A’, B = B’ , C = C’ , D = D’ d d’
a
c b
a’
c’ b’
Similar triangles: two triangles are similar if: 3 angles of one triangle are the same as 3 triangles of the other, or 3 pairs of corresponding sides are in the same ratio (proportional), or An angle of one triangle is the same as the angle of the other triangle and the sides containig these angles are in the same ratio.
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The right triangle 1. Can you construct a triangle with these lengths? If it is possible, what kind of triangle is it? (right triangle, acute triangle, obtuse triangle) a. 3 m, 4m, 7m b. 4 cm, 5 cm, 6 cm c. 3m, 5m, 7m d. 6 cm, 8 cm, 10 cm 2. What is the diagonal of a rectangle with sides of 3 m and 7 m? 3. What is the apothem of a hexagon with sides of 6 cm? Solve these Pythagorean problems: 4. To get from point A to point B you must avoid walking through a pond(charco). To avoid the pond, you must walk 34 metres south and 41metres east. If you walk directly from A to B, how many metres will you walk? Round to the nearest dm. 5. A baseball diamond is a square with sides of 90 feet. What is the shortest distance, to the nearest tenth of a foot, between first base and third base?
6. A suitcase measures 24 inches long and 18 inches high. What is the diagonal length of the suitcase to the nearest tenth?
7. Two joggers run 8 miles north and then 5 miles west. What is the shortest distance, to the nearest tenth of a mile, they must travel to return to their starting point?
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8. Find the height of an isosceles triangle with base of 48 cm and equal sides of 26 cm. 9. Find the area of a triangle with the following sides: a. 20 cm, 26 cm, 26 cm b. 2 mm, 2mmm (right triangle) c. 20 m, 20 m, 24 m d. 8 cm of hypotenuse 10. What is the height of a equilateral triangle with sides of 10 dm? 11. Calculate c in the figure
12. Calculate a and bin each triangle: a)
b)
13. What is the length of the diagonal AB?
14.Calculate the area of this roof:
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A right triangle (or right-angled triangle, formerly called a rectangled triangle) has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs or catheti (singular: cathetus) of the triangle.
Pythagorean theorem:”Thesquare on the hypotenuse is equal to the sum of the squares on the legs”. If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the equation
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Solid geometry: Polyhedra and solids of revolution 1. Write the names of these polyhedra:
2. Draw the following polyhedra and mark the main elements on them: e. A regular triangular prism. f. A regular hexagonal prism. g. A square pyramid. 3. Draw the net (plane development) for the shapes in exercise 2, and identify the elements. 4. Can you recognize the shapes with these nets?
5. The edges of a cuboid measure 7m, 4m, and 4m; is it a regular or an irregular prism? What is its surface area? And its volume? 6. Get the surface area and the volume of the shapes below:
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The dihedral angle is the angle between two planes.
A polyhedron is a solid made of flat surfaces. Each surface is a polygon. Regular polyhedra:
Tetrahedron Cube
OctahedronDodecahedron
Prisms:
Icosahedron
Pyramids
Each polyhedron has sides called faces, edges which connect the faces, and vertices or corners which connect the edges.
Surface Area: to get the area of a polyhedrondraw its net and add the areas of all the polygons. Volume of prisms and pyramids: the volume of prisms and pyramids can be found using the formulas:
V prism base' s area height
V pyramid
base' s area height 3 17
SOLIDS OF REVOLUTION 1. Write the name of these solids:
2. Draw the net for a cylinder with a radius of 1.5 cm and a height of 2.5 cm, and mark the measures on it. What is its surface area? And its volume? 3. Draw the net for a cone with a radius of 2 cm and a slant height (generatriz) of 3.5 cm. What is its surface area? And its volume? 4. Get the surface area and the volume of the following solids: h. A cone with a diameter of 6 cm and a height of 4 cm. i. A cylinder with a height of 4 m and a radius of 50 dm. j. A sphere with radius of 10 cm. 5. A rectangle with sides of 7 cm and 5 cm rotates around its shortest side. What kind of solid do you get? Calculate its surface area and volume. 6. Write the name of the solid you get when these shapes rotate around the following axes: k. A right triangle around one of the legs. l. A semicircle around its diameter. m. A square around one of its sides. n. A right trapezium around its height. 7. A painter gets ÂŁ 1000 to paint a cylindrical tank with a height of 4 m and a diameter of 4m. How much will he get to paint a spherical tank with radius of 2 m? 8. Calculate the volume of the space between two cylinders, one inside the other, both with height of 5 m and radii of 4 m and 2 m. 9. Match the descriptions below with an answer given in the list on the right. Round all answers to hundredths. Use 3.14 for pi. 1. 267.95 c cm o. Volume of a cone with height of 9 cm and radius of 7 cm p. Height of a cone with volume of 132 cubic cm and radius of 3 cm 2. 1 3. 113.04 c ft q. Volume of a cone with diameter 16 cm and height of 4 cm 4. 461.58 c cm r. Volume of a cylinder with radius of 5 cm and height of 3 cm 5. 0 s. Number of vertices on a cone 6. 235.50 c cm t. Number of vertices on a cylinder 7. 14.01 cm u. Volume of a sphere with a radius of 3 ft 18
 Solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis) that lies on the same plane.  The main solids of revolution are: cylinder, cone and sphere.
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STATISTICS AND PROBABILITY
1.
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2. Use the pie chart to answer the questions.
2 11 taxi car
15
walk bike
3
bus
7
How many people walk to school? What percentage? What is the most popular method of transport? What is the least popular method of transport? What is the total number of pupils? If the bus breaks down and they have to walk, how many people walk to school? How many people use petrol to get to school? What percentage?
3.Mean and range A group of children were given examinations in five different subjects. This table shows their percentage scores in each examination.
NAME
MATHS
ENGLISH
HISTORY
SCIENCE
GEOGRAPHY
Brian
67
59
72
66
71
Susan
53
61
55
68
63
Pamela
66
57
64
61
67
John
72
75
59
75
59
Peter
69
81
60
58
67
Jill
63
69
74
56
63
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a. What is each child’s range of scores in the five exams? b. What is each child’s mean score in the five exams? c. Who has the highest mean? d. Who has the lowest mean? e. Which two children received their highest result in their history exam? f. Which two children received their lowest result in their English exam? g. What is the mean score in English? h. What is the mean score in geography? i. In which subject was the range of scores smallest? j.
In which subject was the range of scores biggest?
Probability 4. Two boxes of sweets contain different numbers ofhard- and soft-centred sweets. Box 1 has 8 hard-centred sweets and 10 with soft centres. Box 2 has 6 hard-centred sweets and12 with soft centres. Kate only likes hard-centred sweets. She can pick a sweet at random from either box. Which boxshould she pick from? Why? Kate is given a third box of sweets with5 hard-centred sweets and 6 with sofcentres. Which box should Kate choose from now? Why?
4. A newsagent delivers (reparte) these papers, one to each house. Sun 250 Times 120Mirror 300 Mail 100Telegraph 200 Express 80 What is the probability that a house picked at random has: a. the Times? b. the Mail or the Express? c. neither the Sun nor the Mirror? 22
Population: población Sample: muestra Data: característica que se quiere estudiar To sort the data:organizar los datos
Continuous (time, length ,..) Quantitati ve ( you can measure it ) Types of data Discrete ( number of brothers,..) Qualitativ e
Organisation o Tally charts o Frequency tables Graphs o Bar charts o Pie charts o Pictograms o Line graphs
Averages and ranges The mean is equal to the total divided by the number of things The median is the middle number The mode is the number that appears the most The range is the difference between the largest and smallest numbers in a group
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Probabilityis a way of measuring the chance or likelihood (posibilidad) of a particular outcome(resultado) on a scale from 0 to 1, with the lowest probability at zero (impossible) and the highest probability at 1(certain).
As well as words, we can use fractions or decimals to show the probability of something happening. Impossible is zero and certain is one. A fraction probability line is shown below
Probability can be represented as a fraction, decimal or percentage. The probability of a particular outcome is:
number of events favourable to the outcome total number of possible events This result is known as Laplace’s rule
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INTEGER NUMBERS Intergalactic Temperature Quest Your Space Commander sends you on a quest around the universe to collect temperatures for research purposes. You have the use of a spaceship:
…and a thermometer:
First find the difference between the hottest and coldest temperatures you collected for each planet. Copy the table into your book. The first one has is done for you.
Planet
Hottest recorded
Coldest recorded
temperature
temperature
Kark
10°C
-5°C
Hikl
126°C
37°C
Topa
54°C
12°C
Uni-7
-13°C
-179°C
Jolarg
54°C
-68°C
Vortan
104°C
-97°C
Intop
-95°C
-214°C
Yurg
-27°C
-163°C
Difference
15°C
Those last two planets were really cold! Now that you are back in the lab have a nice hot chocolate and work out these three puzzles: 25
What is the mean average of the difference column? (Use a calculator). What is the mode average of the hottest temperature column? What is the median average of the coldest temperature column?
1. Order these temperatures on the thermometer:16°C, 37°C, -33°C, -6°C 25°C, 2°C, 25°C, 1°C, 14°C
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2. Tell me an integer to describe each situation: a. 5 degrees below zero b. a loss of 7 pounds c. a gain of 10 yards d. 3 feet below sea level e. 2 degrees above zero 3. Tell me the opposite of each integer: 3, -5, -151, 1420, -1 4. a. Write 4 consecutive even integers beginning with each of the following integers: -26, 20 b. Find two consecutive even integers such that twice the smaller is 26 less than three times the larger. 5. The record high temperature in Nowhere, USA is 98 degrees and the record low is -23 degrees. What is the temparature variation between the record high and the record low? 6. What number are you on if you: a. Start at 2 and take away 4 b. Start at -7 and add on 3 c. Start at 5, take away 8 and add on 1 d. Start at -3m take away 3 and add on 4
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7. Calculate:
a. 3 7
b. 3 7
c. 5 7
d. 5 7
e. 3 1
g. 7 2
h. 7 2
i. 1 3 1
8. The highest elevation in North America is Mt. McKinley, which is 20,320 feet above sea level. The lowest elevation is Death Valley, which is 282 feet below sea level. What is the distance from the top of Mt. McKinley to the bottom of Death
Valley? 9. Jenny has $2. She earns $5, spends $10, earns $4, then spends $3. How many dollars does she have or owe?(To owe= deber) 10. The temperature in Anchorage, Alaska was 8°F in the morning and dropped to 5°F in the evening. What is the difference between these temperatures?(to drop
= caer) 11. A submarine was situated 800 feet below sea level. If it ascends 250 feet, what is its new position? Use integer numbers to solve this problem. 12.Roman Civilization began in 509 B.C. and ended in 476 A.D. How long did Roman Civilization last?
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RATIONAL NUMBERS 1. Write each improper fraction as a mixed number in the simplest form: 14 5
5 12
55 12
86 16
8 6
33 9
2. Matt worked for 43 hours and earned (ganó) 678.31 dollars. How much does he earn per hour?(round to the nearest cent) 3. What is the distance Jesse has covered after 6 hours, if he was travelling at a rate of 49.4 miles per hour? 4. The cost for a field trip (viaje escolar) is £99.9 and 31 students have to pay for it. How much does each student need to pay?(round to the nearest cent) 5. In 6 hours Tom travelled 633.9 miles. Calculate the speed (velocidad) at which Tom is travelling. 6. A pack of 4 marbles (canicas) weighs 0.75 lbs. What is the weight of one marble? 7. If you travel at a speed of 70.2 miles per hour, what distance would you cover in 5.2 hours? 8. Tom scored 28 out of 70 in the spelling test. What is his percentage?(Round to nearest tenths) 9. In Matt school 52% of the students are boys. The total number of students in the school is 8000. Find the number of boys. How many girls are there? 10.In a town there were 2500 people two years ago. The population increased by 5%. What is the population of the town now? 11.A basketball team won in 8 games, lost in 9 games. What percent did the team win? What percentdidthey lose? 12.Kate saved £ 5394. She spent 10% of it on her dress and 21% on games for her sister and brother. How much did she spend on games and how much money does she have now?(Round to nearest pences) 13.83 kids went to a field trip, which is 62% of the total number of kids in the first course. Find the total number of kids in the first course? 14.Find the increased or decreased percentage: a. 64 bags sold on Thursday, 86 bags sold on Friday. b. 70 computers sold on Tuesday, 14 computers sold on Wednesday c. 30 dollars spent last week, 16 dollars spent this week. d. 16 cents spent on Friday, 89 cents spent on Saturday. 28
15.Write each mixed number as an improper fraction in the simplest form: 6
1 11
1
13 14
5
2 4
3
14 16
2
3 6
16.At the geologists convention 1/4 of the people are students, 1/4 are professors and 1/6 are industry representatives. The rest are employees (empleados) of the convention center. If there are 348 people at the convention, how many are employees of the center? 17.Nineteen people have signed up for the bowling tournament at Snookie's Bowling Alley and Arcade. This is about 3/4 of the maximum number of people allowed (permitido) in the tournament. How many more can still sign up? -
18. Which is cheaper, the red jumper or the blue one?
19.1/2 of the auditorium is filled (estálleno) with students and the rest are teachers and parents. Of the students, 1/7 is girls. What fraction of auditorium is filled with girl students? What fraction is filled with boy students? 20.Jane and Stephen drive from Edimburgh to London. After driving 3/8 of the journey, they have to drive 250 more miles. What is the distance between Edimburgh and London? 21.A shop received some tomatoes that day. Of them, they sold 1/5 of tomatoes and 1/9 of tomatoes were spoiled (estropeados). There are still 225 tomatoes with them. How many tomatoes did they receive? 22.3/7 of distance to grandma house can be covered in 15 minutes. How long will it take to reach grandma’s house? 23.5/9 of the auditorium is filled by students. If the number of others is 270, what is the total population in the auditorium? 24.How many half-gallon bottles are required to fill 84 2/7 gallons of water? 25.Dad and Kelly painted 7/18 of the house and mum painted 9/15 of the house. Who painted more - dad and Kelly or mum? Did they finish the work? If no, howmuchisleft?(¿cuánto queda?)
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PROPORTIONALITY 1.
Decide if the relationship between the two variables in each case is direct proportion, inverse proportion, or neither (ninguna de las dos): (a) the length and the width of a rectangle with a constantperimeter. (b) the mark you get in an exam and the amount of effort you put in, (c) the area of a square and the length of its sides, (d) the average speed and the time taken for a journey of constant length, (e) the circumference of a circle and its diameter
2. Solve the following problems A. To make tins of purple paint you need to mix blue and red. 1. 2. 3. 4.
If one in every four is blue what proportion are red. If there are 2 tins of blue how many red tins would be needed? If you have 20 tins of blue paint how many red tins would you need? Mary made 32 tins. How many tins of blue paint did she use?
B. A box of sweets contains chocolates and toffees. 1. 2. 3. 4.
If one in every five sweets is a toffee, what proportion are chocolates? There are 10 toffees. How many chocolates? If there are 20 sweets altogether how many are chocolates? Mary only likes toffees and Tim loves chocolate. If Mary eats all the toffees from a box containing 40 sweets. Howmany chocolates are leftfor Tim?
C. A farmer has enough cattle feed (piensoparaganado) to feed 64 cows for 2 days. 1. How long would the same food last for 32 cows? 2. And for 16 cows?
D. The lions are fed 10kg of meat a day. 1. If the father lion eats 4kg and the mother eats 4 kg how much is left for the cub (cachorro)? 2. What proportion does the cub eat? 3. What proportion does the mother eat? 4. On Saturdays the lions are given 20kg so the keeper doesn’t need to work on Sundays. How much meat will the cub have? 30
E. It takes 96 hours for 3 workers to clear (limpiar) a field. 1. How long would it take for 6 workers? 2. How many workers would be required to do the job in 9 days?
F . At the tennis club members are grouped in to adult and junior. 1. If one in every 6 is a junior what proportion are adult members? 2. In one section of the club there are 4 juniors. How many adults are there in that section? 3. In another section there are 36 members. a. Howmany are juniors? b. Howmany are adults? G. Barry is making a fruit cake which contains raisins, currants and cherries. 1. 2. 3.
If one in every 10 pieces of fruit is a currant and two in every five is a cherry what proportion are raisins? He decides that he will make some more cakes for the school summer fair. If he has 30 raisins how many currants and cherries will he need? Martha has made 40 cakes for the summer fair. How many of each of the following does she need a. Raisins? b. Cherries? c. Currants? H. A car travelling at 45 km/h takes 33 minutes for a journey. How long does a car travellingat 55 km/h take for the same journey?
I.
A bag of D & D sweets contain just yellow and orange sweets. For every 2 yellow sweets, there are 6 orange sweets. Completethetablebelow: Yellow Orange Sweets
4 6
6
12 24
40
31
1. 2. 3. 4. 5.
What is the ratio of orange to yellow sweets? If you have 8 yellow sweets, how many orange sweets will you have? There are 32 sweets in the medium sized bag. How many will be yellow? In the super fat size there are 40 sweets. Whatproportion are orange? You look in the sweet bowl and count out 16 yellow sweets. How many sweets are in the bowl?
A proportion is an equation that states that two ratios are equal, such as
. In a true proportion, the product of the means equals the product of the extremes If two quantities are in DIRECT proportion, when one quantity is doubled, so is the other, i.e. they both increase in the same proportion. If two quantities are in DIRECT proportion, their quotient is a constant, i.e. if you divideone by the other you always get the same result If two quantities are in INVERSE proportion to each other, then if one gets doubled, the other gets halved.If two quantities are in INVERSE proportion, their product is a constant, i.e. if you multiply them together you always get the same result.
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ALGEBRAIC EXPRESSIONS 1. Translate these sentences into algebraic language: a. Suppose you are “n” years old now. i. After 1 year, you will be ............... years old. ii. After 2 years, you will be ............ years old. iii. 3 years ago you were................ years old. iv. After x years, you will be ............... years old. b. John is three times as tall as Paul. If Paul is zm tall, what is John's height? c. Two more than a number d. Five less than three times a number e. Seven times a number, increased by 4 f. Jane is y years old. What is the present age of Jane's aunt if her aunt is four times as old as Jane will be 2 years from now? EQUATIONS AND SYSTEMS 1. Write the steps of the substitution method for solving systems of equations. 2. Write an equation to solve this problems: a. A number decreased by 11 equals 14. What is this number? b. The sum of two consecutive integers is 15. Findthenumbers. c. María spent £15 at a restaurant for dinner for herself and a friend. The bill came to £11.75 and tax was £0.83. How much tip (propina) did she leave? d. Twelve times a number is 204. What is this number? e. In January of the year 2000, my husband John was eleven times as old as my son William. In January of 2012, he will be three times as old as my son. Howoldwasmy son in January of 2000? f. Suppose one painter can paint the entire house in twelve hours, and the second painter takes eight hours. How long would it take the two painters together to paint the house? g. Working together, Bill and Tom painted a fence (valla) in 8 hours. Last year, Tom painted the fence by himself. The year before, Bill painted it by himself, but took 12 hours less than Tom took. How long did Bill and Tom take, when each was painting alone? 33
h. The product of two consecutive negative even integers is 24. Findthenumbers 2. Solve the following problems by writing an equation or a system of equations: a. A box of oranges and a box of apples cost 14 dollars. The box of apples cost 8 more than the box of oranges. Find the cost of the box of oranges and the box of apples b. 8 pen boxes and 6 pencil boxes cost $164. 5 pencil boxes cost $34 more than 8 pencil boxes. Find the price of a pen box and a pencil box c. A fraction is equal to 7/9 when both the numerator and denominator is increased by 3. The fraction becomes 2/3 when numerator is decreased by 2 and denominator decreased by 3. Findthefraction d. 9 packs of coffee and 3 packs of tea cost $87. 3 pack of coffee and 9 packs of tea cost $45. Find the price of one pack of coffee and one pack of tea. e. The ages of two cousins are in the ratio 7:22 now. 4 years before the ratio was 1:4. Howold are theynow? f. The three sides of a right triangle form three consecutive even numbers. Find the lengths of the three sides. g. Find the dimensions of a rectangle with an area of 10 square feet if its length is 8 feet more than 2 times its width. h. A ladder (escalera de mano) is resting against a wall. The top of the ladder touches the wall at a height of 15 feet. Find the distance from the wall to the bottom of the ladder if the length of the ladder is one foot more than twice its distance from the wall.
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Equations: An equation is a statement that two numbers or expressions are equal. Equations are useful for relating variables and numbers. Many word problems can easily be written down as equations with a little practice. Unknown: incógnita, cantidad desconocida Solution: to the equation is the number that makes the equation true when we replace the variable with its value. Strategies for Solving Word Problems with Variables Often, word problems appear confusing, and it is difficult to know where to begin. Here are some steps that will make solving word problems easier: 1. Read the problem. 2. Determine what is known and what needs to be found (what is unknown). 3. Try a few numbers to get a general idea of what the solution could be. 4. Write an equation. 5. Solve the equation. 6. Check your solution--does it satisfy the equation? Does it make sense in the context of the problem? (e.g. A length should not be negative.)
Simultaneous Equations Simultaneous equations are a set of equations which have more than one value which has to be found There are two methods of solving simultaneous equations: Substitution: 1. Isolate one of the variables ( ‘x’ ) on one side of one of the equations: 2. Substitute for the isolated variable in the other equation: We will then have one equation in one unknown, which we can solve Elimination or addition: 1. Make one pair of coefficients negatives of one another, 2. Add the equations vertically, and that unknown will cancel. We will then have one equation in one unknown, which we can
solve.
Systems can be: compatible (determined or undetermined) or incompatible 35
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