Acknowledgements
Academic Authors: Muskan Panjwani, Evgenii Todorov
Creative Directors: Alena Sizintseva
Book Production: Larisa Pavlenko, Anastasia Voitovich
All products and brand names used in this book are trademarks, registered trademarks or trade names of their respective owners.
© Uolo EdTech Private Limited First edition 2026
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Book Title: DinoLab Math Smartbook 7
ISBN: 978-93-89789-96-6
Published by: Uolo EdTech Private Limited
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DinoLab is an AI-powered self-learning platform that helps children learn Mathematics and other subjects step by step, at their own pace. Students can practise and revise every topic digitally and through printed Smartbook.
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1 2 3
1. INTEGERS
ADDITION AND SUBTRACTION OF INTEGERS
1. Write the numbers in ascending order (from smallest to largest): −12, 7, 0, −3, 5, −8. What is the sum of the smallest and the largest?
2. For each statement, write T if the statement is true and F if the statement is false:
a) −5 > −11
b) |−6| > |4|
c) 0 is less than every positive integer
d) −1 < 0
3. Insert the correct sign (>, <, =):
a) −3 2
b) | − 7 | | − 4|
c) −10 − 15 d) 0 | − 5 |
4. Label the points: −5, −3, 2, 4, 5 on a number line below:
5. Look at the number line below:
Connect each letter to its corresponding number with a line.
6. Use a number line to find the distance between two numbers.
a) distance between 0 and 3 is
b) distance between 5 and 2 is
c) distance between −5 and −1 is
d) distance between −3 and 4 is
e) distance between 3 and −2 is
7. a) On a number line, label a point halfway between numbers −4 and 6.
b) On a number line, label all integers strictly between numbers −3 and 3.
c) On a number line, label the midpoint between numbers −5 and 1.
8. Find:
|4−2| =
9. Find all integers x such that: a) |2| = b) |− 9| =
|7−9| =
|–5 + 8| = a) |x| = 2
b) |x| = 0
10. Use the absolute value to find the distance between numbers.
a) The distance between −4 and 5 is
b) The distance between −7 and −1 is
a) −12 + 7 =
b) 5 + (−9) =
11. Add the integers.
−3 + (−8) = d) 14 + (−14) =
12. Add all numbers together.
−5 + 3 + (−7) + 2 + 10
13. Fill the blank spaces with integers to make each equation correct.
14. Subtract the integers
a) 5 − (−3)
15. Rewrite each subtraction as addition and then solve.
b) −10 − 8
16. Fill the blank spaces with integers to make each equation correct.
17. The temperature at 6 a.m. was −3 ℃. By noon, it increased by 9 ℃. What was the temperature at noon?
18. A submarine is at a depth of 450 metres below sea level. It rises 120 metres, then dives 200 metres. What is its final depth?
a) − 4 = −9b) 6 – = 11c) − (−7) = 2
19. Ravi’s bank account has a balance of ₹1500. He withdraws ₹1200. What is his balance now?
20. In a game, a player gains or loses points as follows:
• +8 points for each correct answer
• −5 points for each incorrect answer
After answering 12 questions, the player’s total score is 5 points. How many answers were correct?
MULTIPLICATION AND DIVISION OF INTEGERS
21. Multiply the integers.
a) (−12) × 8 =
(−7) × (−6) =
15 × (−4) =
(−5) × 0 =
22. Fill the blank spaces with integers to make each equation correct.
a) (−3) × = 21b) × (−8) = −64c) (−9) × 0
23. A box weighs 2 kg lighter than its reference weight. How much lighter is a stack of 5 such boxes com pared to the reference weight?
24. The product of four integers is positive. Exactly three of these integers are negative. Is this possible? Explain your reasoning.
25. Tick (mark with ✓) all divisions that give an integer answer.
a) 75 ÷ (−8)
b) −56 ÷ 7
c) 48 ÷ 9 d) 39 ÷ (−3) e) −81 ÷ 9
26. Divide the integers.
a) 84 ÷ (−6) = b) −96 ÷ 12 =
с) −100 ÷ (−25) = d) −45 ÷ 9 =
27. Fill the blank spaces with integers to make each equation correct.
a) ÷ (−5) = −9
b) −48 ÷ = 6c) ÷ 4 = 0
28. A diver is exploring an underwater cave. He swims downward at a speed of 15 metres per minute for 6 minutes, then upward at a speed of 12 metres per minute for 8 minutes, and finally downward again at a speed of 10 metres per minute for 5 minutes. What is his final position relative to the starting point (in metres)?
29. A lift descends 4 floors every 8 seconds. How many floors does it descend in 1 minute?
30. A trader earns ₹250 for each sale but loses ₹180 for each refund. He makes 6 sales and 3 refunds. What is his total profit or loss?
31. Tick which property is shown?
(−5) + (−7) = (−7) + (−5)
Options:
a) Commutative property of addition
b) Associative property of addition
c) Distributive property
32. Tick which property is used?
(−3) × (12 + 5) = (−3) × 12 + (−3) × 5
Options:
a) Associative property of multiplication
b) Distributive property
c) Commutative property of multiplication
33. Tick the correct property name for each equation:
a) a + (b + c) = (a + b) + c
Associative
Commutative
Distributive
b) a × b = b × a
Associative
Commutative
Distributive
c) a × (b + c) = a × b + a × c
Associative
Commutative
Distributive
34. Rearrange the numbers using properties to make the calculation easier: (−2) + 8 + (−5) + (−1). Explain which properties you used.
35. Simplify using the distributive property: (−4) × (6 − 3)
36. Fill the blank spaces.
a) The additive inverse of 7 is .
b) The additive inverse of −11 is .
c) The additive inverse of 0 is .
b) The additive inverse of 25 is .
37. Fill the blanks with integers to make each equation true.
a) 5 + =0 b) + (−9) = 0c) 0 + = 0
38. Write the multiplicative inverse of each nonzero integer as a fraction.
a) The multiplicative inverse of 4 is .
b) The multiplicative inverse of −3 is .
c) The multiplicative inverse of 1 is .
b) The multiplicative inverse of −10 is .
39. For each statement, write T if the statement is true and F if the statement is false:
a) The additive inverse of x is always −x.
b) The number 0 has a multiplicative inverse.
c) The multiplicative inverse of −1 is −1.
40. Solve each expression.
a) (−12) + 15 × (−2) ÷ 3b) (6 − (−4)) × (−3)
c) −52 + 3 × (−4)
41. Insert parentheses to make the statement true:
42. Simplify: (−3) × ((−2) + 5) − 6 ÷ (−3)
43. Fill the blank with numbers to make the equation true:
−8 + 6 × −2 = 4 + 12 × (−3) = 0
44. A student wrote: −8−(−5) = −8−5 = −13. Spot and correct the mistake. Write the correct answer.
45. Start at 0. Move according to these steps: +4, −7, +5, −3, +6. Where do you end up?
46. Complete a 3×3 magic square with integers from −4 to 4 (use each integer only once) so that each row, each column, and each diagonal sums to 0.
47. The sum of three consecutive integers is −9. What are the integers?
2. FRACTIONS AND DECIMALS
OPERATIONS ON FRACTIONS
Unless otherwise stated, write all fractions in their simplest form. !
1. For each picture, write the fraction that is shaded.
a) fraction:
b) fraction:
2. Shade the correct fraction of each shape:
a) Shade 3 4 of a rectangle
b) Shade 2 5 of a circle
c) fraction: c) Shade 5 8 of a square
3. Write the fraction for each case:
a) A bar is divided into 6 equal parts. 4 of them are colored:
b) A pizza is cut into 10 equal slices. 7 of them are eaten:
c) A class has total of 20 students. 15 are wearing glasses:
= 5 1 8
4. Write each improper fraction as a mixed number.
5. Write each mixed number as an improper fraction.
6. Find positive integers indicated in each case.
a) Find a such that 2 7 9 = a 9
b) Find a and b such that 58 15 = a + b 15
c) Find m such that m + 3 8 = 83 8
7. Reduce each fraction to its simplest form. a) 3 12 18 = 2 3 с) 16 24 = f) 54 81 = i) 81 108 = b) 45
= g) 63 84 =
8. Fill in the blanks with numbers to make equivalent fractions. a) 3 5 = 12 20 d) 9 = 3 4 b) 12 = 5 6 e) 18 = 2 3 c) 7 = 21 24 f) 11 = 33 45
9. Match equivalent fractions across the four columns. Draw lines to connect all fractions that are equal.
10. Write each fraction with denominator 60.
3 4 = f) 11 15 =
Unless otherwise stated, always use the least common denominator (LCD) when rewriting fractions.
11. Rewrite each pair of fractions with a least common denominator.
12. Find a common denominator for each set of fractions and rewrite them.
13. Compare the fractions. Write <, >, or = between them.
14. Compare the fractions. Write <, >, or = between them.
15. Arrange the fractions in increasing order.
16. Add or subtract the fractions.
17. Add or subtract the fractions.
3 4 + 5 6 =
18. A tank is 2 8 full of water. Another is 3 10 of its capacity is added. Then 1 4 of the total capacity is drained. What fraction of the tank’s capacity is filled now?
19. Find the value of each expression.
a) 3 5 + 7 10 –2 15
b) 4 9 + 5 6 –7 18
c) 7 8 –5 12 + 1 6
d) 5 14 + 3 7 –2 21
20. Simplify and find the value.
21. Multiply the fractions and write the answer in simplest form.
a) 2 1 3 × 3 4
22. Multiply the mixed numbers and write the answer in simplest form.
b) 1 1 2 × 2 2 5
c) 3 3 8 × 4 7
d) 5 2 3 × 1 3 5
23. A recipe uses 3 4 cup of sugar to make 600 grams of cookies. How much sugar is needed to make 400 grams of cookies? What fraction of the tank’s capacity is filled now?
24. Divide the fractions and write the answer in simplest form.
25. A ribbon is 3 4 metre long. How many pieces can you cut from the ribbon if each piece needs to be 1 8 metre?
26. Divide the mixed numbers and write the answer in simplest form. a) 2 1 3 ÷ 3 4 b) 1 1 2 ÷ 2 2 5 c) 3 3 8 ÷ 4 7 d) 5 2 3 ÷ 1 3 5
27. A book has 240 pages. You have read 7 12 of the book. How many pages have you read?
28. 2 9 of a number is 18. Find the number.
29. A shop increases the price of a toy by 1 4 of its original price. If the original price was $32, what is the new price?
30. Compare and order the decimals from smallest to largest
32. Write each fraction as a decimal a) 0.5 = d) 0.75 = b) 0.25 = e) 0.6 = c) 0.125 = f) 0.2 =
31. Write each decimal as a fraction in simplest form a) 3.5 + 2.4 = b) 7.8 – 3.2 = c) 4.75 + 5.25 = d) 10 – 6.35 =
33. Add or subtract a) 12.8 – 4.65 = b) 3.15 + 7.45 = c) 8.08 – 2.39 = d) 5.25 + 6.75 =
34. Add or subtract
35. Find the value
a) 0.75 + 0.85 + 1.25
b) 5.8 − 2.35 + 4.2
c) 6.05 + 3.95 − 2.1
d) 7.7 − 3.15 − 2.25
e) 15.25 + 7.6 − 12.45
f) 8.08 − 3.5 + 6.92
g) 20 − 7.35 − 4.65
h) 11.11 + 9.99 − 10
a) 3.5 × 2
Multiply
b) 6.25 × 4
c) 7.5 × 0.8
d) 0.75 × 0.6
e) 5.4 × 1.25
f) 12.5 × 0.08
g) 4.2 × 1.5
h) 3.14 × 2.5
37. Divide
a) 7.5 ÷ 5 = b) 8.4 ÷ 4 =
c) 6.6 ÷ 3 =
e) 18.9 ÷ 3 = d) 9.9 ÷ 6 = f) 7.35 ÷ 5 =
38. Divide
a) 16.5 ÷ 3
c) 19.6 ÷ 4 b) 22.4 ÷ 7 d) 33.75 ÷ 9
a) 6.4 ÷ 0.8
39. Divide decimals
c) 8.25 ÷ 0.5
e) 3.75 ÷ 0.025
g) 6.3 ÷ 0.009 b) 7.5 ÷ 1.5 d) 9.6 ÷ 1.2 f) 7.92 ÷ 0.04
40. A rope is 7.5 metres long. It is cut into 5 equal pieces. How long is each piece?
41. A bag of apples weighs 4.8 kilograms. If each apple weighs 0.24 kilograms, how many apples are in the bag?
42. A car travels 182.4 km in 3.2 hours. Find its average speed in km/h.
43. A piece of cloth costs $5.75 per metre. How much will 8.4 metres of cloth cost?
3. DATA HANDLING
MEAN, MEDIAN AND MODE
1. The table contains savanna animals weight and height. Study the table and answer the questions.
a) What is the height of a giraffe? cm.
b) Choose the animal with the smallest height. Write its height: cm.
c) Which animal is 140 cm tall?
d) Which animal has the highest weight?
e) Choose the animal that weighs 4500 kg. Write its height: cm.
2. The table shows the area and the largest depth of the world’s oceans. Study the table and answer the questions.
a) Which ocean has the largest area?
b) What is the depth of the Pacific Ocean? m.
c) Arrange the oceans by depth in ascending order. Write all the depths in order: m
d) Arrange the oceans by area in descending order. Which ocean comes first?
e) Which ocean has the second largest area?
3. Tick (✓) the correct formula for the arithmetic mean.
Number of observations
Sum of all observations
Sum of all observations
Number of observations
Mean = Number of observations × Sum of observations
Mean = Mean = Mean =
Sum of all observations 2
4. Find the sum of the first 5 even numbers, starting from 2.
5. Find the arithmetic mean of the first 5 even numbers, starting from 2.
6. The sum of four numbers is 41. Find the arithmetic mean of these numbers.
7. The mean of ten numbers is 12.2. Find the sum of these numbers.
8. The mean of two numbers is 30. If one number is 25, what is the other number?
9. The mean of four numbers is 50. If three of the numbers are 40, 41, and 42, what is the fourth number?
10. Find the mean of the following set of numbers: 7, 7, 7, 7, 10, 10.
11. The numbers 10, 20, and 30 appear with the following frequencies: 10 appears 2 times, 20 appears 5 times, 30 appears 3 times. What is the mean of these numbers?
12. Ayesha practiced playing basketball and took 30 shots in total. She made 12 successful two-point shots and 8 successful three-point shots.
a) What is the total number of points Ayesha scored from her successful two-point and three-point shots?
b) What is the mean number of points Ayesha scored per successful shot?
c) What is the arithmetic mean number of points Ayesha scored per shot, including both successful and unsuccessful shots?
13. The mean of 5 numbers is 50. If we add one more number, the mean increases by 3. Find the added number.
14. The mean of 6 numbers is 40. One of the numbers, which was 35, was removed. Find the new mean of the remaining numbers.
15. Study the table with data about football matches in one school. Answer the questions.
a) What is the total number of spectators across all four matches?
b) What is the mean number of spectators per match?
c) What is the total number of players across all four matches?
d) What is the mean number of players per match?
e) What is the total number of people (players, referees, and spectators) across all four matches?
f) What is the mean total number of people (players, referees, and spectators) per match?
16. Match the terms with their correct definitions.
mean mode median range
a) The middle value in a set of data when arranged in ascending or descending order.
b) The difference between the largest and smallest values in a data set.
c) The sum of all values in a data set divided by the total number of values.
d) The value that appears most frequently in a data set.
17. Find the mode in this data: 2, 3, 3, 3, 4, 4, 6, 6, 8.
18. Find all modes in this data: 2, 2, 3, 5, 5, 7, 8, 8.
19. Arrange the following numbers in increasing order and find the mode: 35, 25, 15, 20, 35, 20, 12, 35, 30.
20. Arrange the following numbers in increasing order and find all the modes: 35, 25, 15, 20, 30, 20, 12, 35, 30.
21. The table shows how many books each of five friends read in each of three seasons. Study the table and answer the questions.
a) Find the mode of the number of books read by the friends in autumn.
b) Find the mode of the number of books read by the friends in winter.
c) Find all modes of the number of books read by the friends in spring.
22. Find the median in this data: 3, 3, 3, 4, 5, 6, 6.
23. Arrange the following numbers in increasing order and find the median: 35, 25, 15, 20, 35, 20, 12, 35, 30.
24. Find the median in this data: 2, 3, 3, 3, 4, 5, 6, 6.
25. Arrange the following numbers in increasing order and find the median: 25, 15, 20, 35, 20, 12, 35, 30.
26. The median of the given data is 18.5. What is the value of x? The data: 20, 15, 30, x.
27. The table shows the number of children of each age at a party. Study the table and answer the questions.
a) How many children were 12 years old?
b) What is the total number of children at the party?
c) What is the sum of ages of all children?
d) What is the mean (average) age of the children at the party?
e) Find the mode of the ages of the children at the party.
f) Find the median age of the children at the party.
g) Find the range of the ages of the children at the party.
ORGANISATION AND REPRESENTATION OF DATA
28. The chart shows how many years different animals live. Study the chart and answer the questions.
a) How many years do cats live, according to the chart?
b) How many years does a wolf live?
c) Which animal lives the longest?
d) What is the difference in years between a hen and a dog?
e) Which animal lives fewer years, a dog or a rabbit?
29. The chart shows the different marks Raj received in mathematics throughout the year. Study the chart and answer the questions.
a) How many Cs did Raj receive?
b) Did Raj receive more As than Bs?
c) How many marks did Raj receive in total?
30. The chart shows the height of the most famous towers in the world: the Leaning Tower of Pisa, the Spassky Tower, the Big Ben, and the Tower of Belen in Lisbon. Study the chart and answer the questions.
height, m
a) What is the height of the Spassky Tower?
b) Which tower is the tallest?
c) Which tower has the smallest height?
d) Which tower is approximately three times higher than the Tower of Belen?
T. of PisaSpassky T.Big BenT. of Belen
31. The chart shows the amount of rainfall (in mm) in Mumbai for six months. Study the chart and answer the questions.
a) How much rainfall was there in May?
b) Which month had the lowest rainfall?
c) Which month had more than 60 mm of rainfall?
d) What is the total amount of rainfall from March to May?
32. The chart shows the speed records (in km/h) of different animals: falcon — 300 km/h, ostrich — 50 km/h, cheetah — 150 km/h, gazelle — 100 km/h. Draw and colour the column marked ”cheetah” to the correct height.
33. At the biology lesson, the children measured the length of insects in centimetres. Mosquito length is 0.9 cm, beetle length is 2.1 cm, bee length is 1.5 cm, and ant length is 0.6 cm. Draw the missing columns using the given data.
34. The table shows how many gadgets (tablets, phones and laptops) were sold over four days. Complete the chart using the data provided in the table.
4. INTRODUCTION TO PROBABILITY
UNDERSTANDING PROBABILITY
1. Write the chance of the given events happening, using the terms: sure, impossible, unlikely, likely, and equally likely.
a) A rock talking in human language.
b) The sun setting in the evening.
c) Tossing a coin and getting tails.
d) A randomly picked clover turning out to be a four-leaf clover.
e) Finding your notebook where you left it.
2. Consider tossing a fair die with six faces, each showing a number from 1 to 6. Write the chance of the given events happening, using the terms: sure, impossible, unlikely, likely, and equally likely.
a) Getting a number less than two.
b) Getting a number greater than six.
c) Getting a number greater than three.
d) Getting a number greater than two.
e) Getting a number less than seven.
3. Consider an aquarium with 10 fish. Their colors are shown in the picture.
Write the chance of the given events happening, using the terms: sure, impossible, unlikely, likely, and equally likely.
A randomly picked fish turned out to be:
a) green
b) not green
c) black
d) not black
e) blue or yellow
4. Rakesh has a collections of 20 crystals, shown in the picture:
Fill in the numbers:
a) The chance of selecting red crystal is out of 20.
b) We can write the probability of selecting a red crystal as:
20 = 2
c) The probability of selecting a yellow crystal is:
20 =
d) The probability of selecting a blue crystal is:
5. A letter is picked at random from a word DINOLAB. What is the probability of picking a vowel?
6. Fill in the numbers.
Disha invited guests to her birthday party and baked three kinds of cakes for them. There were 3 strawberry cupcakes, 4 blueberry cupcakes and 5 vanilla cupcakes. In total, Disha made cupcakes.
a) The probability of selecting a strawberry cupcake is:
b) The probability of selecting a blueberry cupcake is:
c) The probability of selecting a vanilla cupcake is:
7. Fill in the numbers.
There are 10 keys on the table: 3 golden keys, 4 silver keys and 3 bronze keys.
The probability of picking a key that is not made of silver is:
8. Fill in the numbers.
Malati has a box of colorful geometric shapes for a math activity. In the box, there are 6 triangles, 5 squares, and 11 circles. In total, Malati has shapes in her box.
The probability of selecting a triangle or a square is:
9. Fill in the numbers.
The table shows the results of rolling a dice 100 times.
a) The experimental probability of rolling 1 is:
b) The theoretical probability of rolling 1 is:
d) The theoretical probability of rolling an odd number is: = 100
c) The experimental probability of rolling an odd number is:
10. Fill in the numbers.
Shyam has a collection of marbles: 5 red marbles, 7 green marbles, and 8 blue marbles. He conducted a series of experiments. He placed all his marbles in a hat and randomly drew one marble 50 times. The table shows the results of the experiments.
a) The experimental probability of drawing a red marble is:
b) The theoretical probability of drawing a red marble is:
c) The experimental probability of drawing a blue marble is:
d) The theoretical probability of drawing a blue marble is:
5. SIMPLE EQUATIONS
LINEAR EQUATIONS IN ONE VARIABLE
1. Match each equation to its root.
3x + 4 = 16
4x – 3 = 2x + 7 x – 7 = 3 5x − 2 = 13
5 x + 2 = 1 2 x + 5 2x – 2 = 10
4 x + 2 = 7
= 6 x = 4 x = 10 x = 20 x = 5 x = 4 x = 11 x = 3 x = –30 x + 5 = 9
(x – 3) = x + 5
a)
2. Match the equation and its description.
Twice a number decreased by 4 is 10
Seven more than a number is 15
Half of a number is equal to 6
Three times a number increased by 8 equals 20
b)
One third of a number increased by 5 is 11
Three quarters of a number minus 2 equals 7
Five sixths of a number plus 4 is equal to 9
Seven tenths of a number decreased by 3 equals 8
Twice a number minus one-fifth of that number is equal to 200
Seven-tenths of a number added to 150 equals 850 c)
Three-fourths of a number plus 12 equals 500
Half of a number decreased by 25 is equal to 150
a) x – 8 = –5
3. Solve the equations.
1) x = –5 + 8 2) x = 3
b) x + 2 3 = 7 3
c) x –5 4 + 2 = 3 2
d) x + 3 5 –7 10 = 2 –1 2
4. Solve the equations with integer coefficients.
a) 3x − 5 = 13
1) 3x = 13 + 5
3) x = 6
b) 2x = 10
c) 4x + 7 = −9
d) 5x − 18 = −3
2) 3x = 18 divide both sides by 3
5. Solve the equations.
a) 2x – 1 = 2
1) 2x = 2 + 1 2) x=3 divide both sides by 2
3) x = 3 2
b) 3x + 1 = 5
c) 4x – 1 = 5
d) 5x + 2 = 8
6. Solve the equations with negative coefficients.
a) –2x + 4 = 1
1) –2x= –1 –4
2) 2x = –5 change the sign
3) 2x = 5 divide both sides by 2 4) x = 5
b) –3x – 5 = 1
c) –4x + 2 = 5
d) –5x – 3 = –1
7. Solve the equations with fractional coefficients.
a) 3 2 x + 3 = 7
1) 3 2 x =7 – 3
2) 3 2 x = 4 multiply both sides by 2
3) 3x = 8 divide both sides by 3
4) x = 8 3
b) 2 3 x – 1 = 2
c) 3 4 x + 2 = 5 2
5 6 x – 2 = 1 2
8. Solve the equations with braces.
a) 2(x + 3) + 2 = 10
1) 2(x+3)=10-2
2) 2(x + 3) = 8 divide both sides by 2
3) x + 3 = 4
4) x = 4 – 3
5) x = 1
c) 4(x + 2) – 3= 18
b) 3(x – 5) – 5 = 6
d) 5(x – 3) + 15 = 2
9. Solve the equations with x on both sides.
a) 2x + 3 = x + 7
1) gather x on one side: 2x + 3 – x = 7
2) x + 3 = 7
3) x = 7 – 3
4) x = 4
b) 5x – 4 = 3x + 10
c) 7x + 2 = 5x – 8
d) 4x – 7 = 2x + 5
a) x 5 = 3 4
10. Use the cross-multiplication property to solve the given equations.
1) use the cross-multiplication property: x × 4 = 3 × 5
2) 4x = 15 divide both sides by 4
3) x = 15 4
b) 7 x = 14 5
c) x + 2 3 = 5 6
d) 8 x – 1 = 4 9
11. Use all your previous skills to solve the given equations.
a) 1 2 x + 3 = 1 4 x + 5
b) –3(x – 2) = 2x + 5
c) 2 3 x – 4 = –1 6 x + 2
d) –4(x + 1) + 3 = 1 2 x – 5
12. A shop sells mangoes at 50 rupees per kilogram. Priya bought some mangoes and paid a total of 350 rupees. How many kilograms did she buy?
Choose the equation that can be used to solve the problem:
a) 50 + x = 350
b) 50x = 350
c) x 50 = 350
d) 350x = 50
13. Ravi bought 12 pencils and paid a total of 96 rupees. Each pencil costs the same. What is the cost of one pencil?
Choose the equation that can be used to solve the problem:
a) 12 + x = 96
b) 12x = 96
c) x 12 = 96
d) 96x = 12
14. A water tank already contains 15 litres of water. Water is being added at a rate of 3 litres per minute. After how many minutes will the tank have 27 litres of water?
Choose the equation that can be used to solve the problem:
a) 15 + 3x = 27
b) 15x + 3 = 27
c) 3 x + 15 = 27
d) 27x = 15 + 3
15. The temperature in Shimla in the morning was 12°C. It drops by 2.5°C every hour. After how many hours will the temperature be –3°C?
Choose the equation that can be used to solve the problem:
a) 12 – 2.5x = –3
b) 12 + 2.5x = –3
c) –2.5x = 12 – 3
d) 2.5 – x = –3
16. The sum of two numbers is 45. If one number is twice the other, find the numbers.
Solve the problem by writing and solving the equation:
1) Let the smaller number be x. Then the larger number will be 2x.
2) As per the given condition: x + 2x = 45
3) 3x = 45 divide both sides by 3
4) ...
5) ... Answer:
17. The sum of two numbers is 51. One number is 3 more than twice the other. Find the numbers.
Solve the problem by writing and solving the equation:
18. Two numbers differ by 7. One half of the larger number plus 3 equals the smaller number. Find the numbers.
Solve the problem by writing and solving the equation:
19. The alloy consists of five parts nickel and four parts iron. How much of each element is present in an alloy weighing 1440 grams?
Solve the problem by writing and solving the equation:
20. The alloy consists of two parts gold, one part platinum, and ten parts silver. How much of each metal is contained in an alloy weighing 910 grams?
Solve the problem by writing and solving the equation:
21. A cleaning solution consists of seven parts vinegar, five parts baking soda and four parts water. If the total mass of the solution is 1280 grams, how much of each substance is present?
Solve the problem by writing and solving the equation:
22. Anita, Ramesh, and Priya together read 72 pages of a book. Priya read 5 times as many pages as Anita, and Ramesh read twice as many pages as Anita. How many pages did Priya read?
Solve the problem by writing and solving the equation:
6. ANGLE PAIRS AND PARALLEL LINES
ANGLE PAIRS
Unless stated otherwise, any angle read with a protractor must be recorded to the nearest 1°.
1. Four separate angles are shown. Measure each with a protractor and record A, B, C, D.
2. The angles x and y are shown.
a) Measure x.
b) Construct a 30° angle sharing the vertex and one ray with x.
c) Compute x + 30°
d) Construct a 45° angle from y (same base ray) and compute y + 45°.
3. Each figure shows an angle with a student’s recorded measure. For each,decide if the record is correct. If not, write the correct measure.
4. Find complementary (comp.) and supplementary (supp.) angles of.
a) 34°: comp. = , sup. =
b) 67°: comp. = , sup. =
c) 102°: comp. = , sup. =
d) 145°: comp. = , sup. =
e) 89°: comp. = , sup. =
(if a complement is not possible, write “no”)
5. Classify pairs around one vertex on the figure. For each requested pair, state complementary, supplementary, or neither.
a) AOB and BOC:
b) AOB and AOD:
c) BOC and COD:
d) AOC and COD:
e) List all complementary pairs among { AOB, BOC, COD, DOA}:
f) List all supplementary pairs among the same set:
a) A is 30° greater than its complement. Find: A, comp.
b) B is 18° less than its supplement. Find: B, supp.
c) Two complementary angles C and D are in the ratio 2 : 3 Find: C, D
d) Two supplementary angles E and F differ by 56°. Find: E, F
e) Angles G = 2x + 7° and H = 5x − 1° are complementary. Find: x, G, H
All images in the following problems are approximate. Unless stated otherwise, use the information from the problem text rather than a protractor to find the size of a given angle.
7. Angles AOB and BOC form a linear pair.
a) Given that AOB = 2x + 18° and BOC = 4x – 12°, find x.
b) Find: AOB = and BOC =
c) Find the complement of the smaller of the two angles.
d) A ray OD bisects BOC. Find the measures of the two angles formed at OD:
8. A right angle AOB split by a ray OC.
a) Suppose AOC = 2y + 40° and COB = y – 4°, find y.
b) Find AOC = and COB .
c) Are AOC and COB complementary or supplementary?
d) If the ray OC is rotated so that AOC decreases by 12°, find the new measures.
of AOC = and COB = .
9. Let the three adjacent angles x, y and z lie on the same side of line ℓ with a common vertex O.
Assume x + y + z = 180°.
y
a) Given that x = 3y + 3° and z = 4y + 1°, find x, y, and z.
b) Find the supplements and complements of angles x, y, and z
(if a complement is not possible, write “no”):
x: comp. = , supp. = ;
y: comp. = , supp. = ;
z: comp. = , supp. = ;
See the figure below.
a) AOC and COB from a linear pair. If AOC = 64°, find COB = .
b) BOD and AOD form a linear pair. If BOD = 38°, find DOA = .
c) Do COD and DOA form a linear pair?
Underline the correct answer: Yes/No
d) Do COD and DOA form al inear pair?
Underlinethecorrectanswer: Yes/No
e) List all distinct linear pairs determined by the rays OA, OB, OC, OD.
11. In each part, A and B form a linear pair. Find x and both measures:
a) A = 4x + 12°, B = 2x + 48°
b) A = 3x + 15°, B = 5x + 45°
c) A = 7x + 14°, B = 2x + 22°
d) A = 9x – 9°, B = 3x + 15°
12. Angles A, B, C lie on the same side of line ℓ with common vertex O and A + B + C = 180°.
a) If A = 2x + 6°, B = 3x + 24°, C = x + 30°, find x.
a) Find: A = , B = , C = . b) State which of them is the largest: .
13. Rays OA and OB point at the opposite directions. Ray OP lies between them.
a) Given that AOP = 3x + 6° and POB = 96°, find x .
b) Find: AOP = and POB = .
c) Which of the two is acute? Which is obtuse?
Acute: , obtuse: .
14. Check and correct the student’s records using the protractor. In each mini-figure, the two marked angles share a side and their other sides are opposite rays.
a) For (i): Is the student’s B correct?
Underlinethecorrectanswer: Yes/No.
If not, write the correct value:
b) For (ii): Is the student’s B correct?
Underlinethecorrectanswer: Yes/No. If not, write the correct value:
c) For (iii): Is the student’s B correct?
Underlinethecorrectanswer: Yes/No.
If not, write the correct value:
d) For (iv): Is the student’s B correct?
Underlinethecorrectanswer: Yes/No.
If not, write the correct value:
15. Lines ℓ1 and ℓ2 intersect at O. The four angles are labeled A, B, C and D in clockwise order starting from the top-right region.
a) Assuming that A = 3x + 16° and C = 5x – 8°, find x
b) Find: A= , B = , C = and D = .
c) List all pairs of vertical angles:
d) A ray OE bisects B. Find the measures of the two angles formed at OE:
B A
16. Angles B and C are adjacent and form a linear pair.
a) If B = 2y + 12° and C = y + 24°, find y.
b) Find: A= , B = , C = and D = .
c) Which of the angles are acute? Which are obtuse?
Acute: , obtuse: .
17. Compute from one given angle A = 68°.
a) Find: B = , C = , D = .
b) Find the sum of the two acute angles at O.
c) Find A + B = .
d) By how many degrees does the obtuse angle exceed the acute. angle at this intersection? By .
18. Around a point: five rays from O cut the full turn into consecutive angles A, B, C, D, E (see figure).
Let A = 2k + 20°, B = k + 16°, C = 3k + 4°, D = 2k + 10°.
The full turn is 360°, and E is 10° greater than A.
a) Find k.
b) Find: A = , B = , C = , D = and E = .
c) Which angle is the largest? Which are acute?
Largest: Acute:
d) Find C + E = .
19. The angle AOB is shown:
a) Measure AOB = .
b) Construct a 30° angle sharing vertex O and the base ray OA.
Compute AOB + 30° = .
c) Construct a 45° angle sharing the same base ray OA
Compute AOB + 45° =
(d) Classify each resulting sum as acute, right, obtuse, or straight:
AOB + 30° is
AOB + 45° is
20. See the figure below.
a) Measure AOB = , BOC = .
b) Compute AOB + BOC = .
c) Compute | AOB – BOC| = .
(d) Compute 180° − ( AOB + BOC) = .
State whether this angle is acute or obtuse.
Underline the correct answer: Acute/Obtuse.
21. Three angles are defined by P = 2x + 14°, Q = 3x + 16°, R = x + 20°.
a) If P + Q = 120°, find x, then find P and Q.
b) Compute P + R = .
c) For what value of x is Q + R = 180°?
d) For x from part (a), is P < Q?
Underline the correct answer: Yes/No
22. The rays on one side of line AD form consecutive angles AOB, BOC, COD with a common vertex O.
Let us assume following: AOB = 3y + 12°, BOC = 2y – 6°, COD = 4y + 3°, AOB + BOC + COD = 180°.
a) Find y.
b) Find AOB = , BOC = and COD = .
c) Which angle is the largest?
PARALLEL LINES
23. Two parallel lines ℓ1 ǁ ℓ2 are cut by transversal t.
For each named pair, state the relationship: corresponding, alternate interior, alternate exterior, same-side interior, or none.
a) A and E:
b) B and F:
c) C and G:
d) D and H:
e) B and E:
f) A and G:
g) C and E:
h) D and F:
24. Same figure as in problem 23. List all pairs of each type (do not repeat pairs; order within a pair does not matter).
a) All corresponding pairs:
b) All alternate interior pairs:
c) All alternate exterior pairs:
d) All same-side interior pairs:
25. Two parallel lines ℓ1 ǁ ℓ2 are cut by transversal t.
a) If A = 4x + 8° and E = 2x + 38°, find x
b) Find A = and E = .
c) Find B = .
d) Find F = and H = .
26.Same setup as above. Assume C and E are alternate interior. Two parallel lines ℓ1 ǁ ℓ2 are cut by transversal t.
a) Given that C = 3x + 10° and E = 2x + 40°, find x.
b) Find C = and E = .
c) Find D = and H = .
C
27. Same setup. Assume B and E are sameside interior. Two parallel lines ℓ1 ǁ ℓ2 are cut by transversal t.
a) Given that B = 6x + 12° and E = 3x + 6°, find x.
b) Find B = and E = .
c) Find A = and G = .
C
28. Three parallel lines ℓ1 ǁ ℓ2 ǁ ℓ3 are cut by a single transversal t.
At the top intersection, the marked angle A equals 38°.
a) Find B = , C = , D = .
b) Assuming E corresponds to A, find E = and the linear-pair angle next to it: F = .
c) Find the two acute angles at the bottom intersection: J = , L = .
29. A trapezoid ABCD has bases AB ǁ CD (see figure). The leg BC is a transversal of the bases.
a) If ABC = 3x + 24° and BCD = 6x + 12°, find x (use same-side interior along BC).
b) Find ABC = and BCD = .
c) State which one is acute and which is obtuse.
Acute: Obtuse:
d) Find the exterior angle adjacent to ABC along line CD: =
30. See the figure below.
a) Are m and n parallel?
E = 138°
Underline the correct answer: Yes/No
A = 148°
b) Are m and n parallel?
B = 75° F = 105°
Underline the correct answer: Yes/No
31. Correct the student’s records for each minifigure, assuming that ℓ₁ ǁ ℓ₂ and the value of angle B is correct.
a) Is the second measure correct? Yes/No
The correct value of C = .
= 70° C = 120°
a) Is the second measure correct? Yes/No
The correct value of C = . B = 96°
= 86°
a) Is the second measure correct? Yes/No
The correct value of C = .
7. TRIANGLES AND THEIR PROPERTIES
UNDERSTANDING TRIANGLES AND THEIR PROPERTIES
1. For each set of angles or sides of the triangle ABC, underline its correct type by sides and by angles?
a) A = 60°, B = 60°, C = 60° by sides:
equilateral / isosceles / scalene by angles: acute / right / obtuse
b) A = 30°, B = 60°, C = 90° by sides:
equilateral / isosceles / scalene by angles: acute / right / obtuse
c) A = 70°, B = 70°, C = 40° by sides:
equilateral / isosceles / scalene by angles: acute / right / obtuse
d) A = 100°, B = 40°, C = 40° by sides:
equilateral / isosceles / scalene by angles: acute / right / obtuse
e) Sides: 7, 24, 25 by sides: equilateral / isosceles / scalene by angles: acute / right / obtuse
f) Sides: 9, 10, 12 by sides:
equilateral / isosceles / scalene by angles: acute / right / obtuse
g) Sides: 13, 14, 15 by sides: equilateral / isosceles / scalene by angles: acute / right / obtuse
h) Sides: 9, 10, 12 by sides:
equilateral / isosceles / scalene by angles: acute / right / obtuse
2. For each set of angles, underline the correct type of the triangle by angles and, where possible, by sides (if impossible, underline “impossible”).
a) 35°, 55°, 90° by sides:
equilateral / isosceles / scalene by angles:
acute / right / obtuse / impossible
b) 50°, 50°, 80° by sides:
equilateral / isosceles / scalene by angles:
acute / right / obtuse / impossible
c) 20°, 30°, 130° by sides:
equilateral / isosceles / scalene by angles:
acute / right / obtuse / impossible
d) 60°, 60°, 60° by sides:
equilateral / isosceles / scalene by angles:
acute / right / obtuse / impossible
e) 45°, 45°, 90° by sides:
equilateral / isosceles / scalene by angles:
acute / right / obtuse / impossible
3. Write the correct term for each description: median, centroid, altitude, orthocenter.
a) A segment from a vertex to the midpoint of the opposite side:
b) The common point of the three medians:
c) A line through a vertex perpendicular to the opposite side (or its extension):
d) The common point of the three altitudes:
e) The point that divides each median in the ratio 2:1 (from the vertex):
f) In a right triangle, it coincides with the rightangled vertex (name the point):
4. In ABC, D is the midpoint of BC, E is the midpoint of AC, and the medians AD and BE intersect at G.
a) If AD = 21 cm, find: AG = and GD =
b) If AG = 12 cm, find: GD = and AD =
c) If BE = 30 cm, find: BG = and GE =
5. For each figure: RIGHT-ANGLED TRIANGLES
• State where the orthocenter H is located (inside / at a vertex / outside).
• State which type of triangle it is (acute / right / obtuse).
a) orthocenter H is located:
type of triangle:
b) orthocenter H is located:
type of triangle:
c) orthocenter H is located:
type of triangle:
6. In ABC, D, E, F is the midpoint of BC, CA, AB respectively, and the medians intersect at G.
a) Given AG = 10 cm and GE = 7 cm, find: AD = and BE = .
b) Given AD = 27 cm and BG = 18 cm, find: AG = , GD = , BE = .
a) In ABC with AB =AC, if A = 36°, find B and C.
b) In ABC with AB = AC, if B = 52°, find A and C.
c) In ABC with BC = CA, if B = 38°, find A and C.
8. Each set forms a triangle. Find the missing angle.
a) A = 62°, B = 47°, C = .
b) A = 35°, B = , C = 85°.
c) A = , B = 73° , C = 62°.
d) Angles are in the ratio A : B : C = 3 : 5 : 7. Find each angle.
e) Angles are in the ratio A : B : C = 2 : 2 : 1. Find each angle.
f) A = x + 14°, B = 2x – 9°. Find C.
9. In ABC, side AC is extended at C
a) If A = 45° and B = 63°, find the exterior angle at C and the interior C.
b) If the exterior angle at C equals 118° and A = 50°, find B and C.
c) If A = 2x°, B = (x + 28)°, the exterior angle at C equals 136°, find x and all angles of the triangle.
10. Which of the pictures correctly shows an exterior angle at C. Underline the correct option(s).
Correct option(s): pic. 1 / pic. 2 / pic. 3
11. In ABC, AB = AC. Side BA is extended at A to form an exterior angle.
a) If the exterior angle at A equals 132°, find: B = and C = .
b) If B = С = (y + 7)° and the exterior angle at A equals 118°, find all angles.
c) If A = (2k + 6)° and the exterior angle at A equals (5k + 6)°, find k and all angles.
12. Mark each statement as true or false (circle T/F), and correct the false ones.
a) An exterior angle equals the sum of the two adjacent interior angles. T/F
Correct:
b) If an exterior angle is 120°, then the opposite interior angles must be 60°each.
T/F
Correct:
c) In any triangle, each exterior angle is greater than any interior angle of the triangle. T/F
Correct:
d) In an isosceles triangle, the exterior angle at the vertex equals the sum of the base angles. T/F
Correct:
e) If two exterior angles are equal, then the adjacent interior angles are equal.
T/F
Correct:
f) If A = B, then the exterior angles at A and B are equal. T/F
Correct:
13. In the figure ABC and BCD share side BC.
a) If ABC = (2x + 10)° and DCB = (x + 25)°, express ABC in terms of x: ABC = .
b) If ACB= 70° and the exterior angle at C for BCD is 128°, find: BCD = .
14. Consecutive exterior angles are formed at B and C by extending both sides at each vertex.
a) Given A = (x + 20)° and the exterior at B equals 3x – 10°, and the exterior angles at B and C are equal, find x and B.
b) If the exterior angles at B and C are equal, the exterior at C equals y + 46° and A = 64°, find A and C.
15. In ABC, both AC and AB are extended.
a) If the exterior angle at C is 114°, A = 48° and AB = AC, find: B = and C = .
b) If the exterior angle at A is x + 70° and C = 2x − 16°, and AB = AC, find x and all angles.
16. a) In ABC with AB = AC and A = 48°. Find the exterior angles at B and C:
b) In ABC with BC = CA, the exterior angle at A equals 138°. Find all interior angles.
17. Can a triangle with given sides be formed? Circle Yes / No for each case.
a) 5, 7, 11: Yes / No
b) 6, 9, 14: Yes / No
c) 8, 8, 16: Yes / No
d) 10, 12, 21: Yes / No
18. For each pair (a, b), write all integer values of c such that a triangle with sides a, b, c exists.
a) a = 7, b = 12, values of c:
b) a = 9, b = 15, values of c:
c) a = 11, b = 18, values of c:
19. a) Right triangle with legs 6 cm and 8 cm. The hypotenuse is cm.
b) Right triangle with legs 9 cm and 12 cm. The hypotenuse is cm.
20. a) Right triangle with hypotenuse 13 cm and one leg 5 cm. Find the other leg.
b) Right triangle with hypotenuse 25 cm and one leg 7 cm. Find the other leg.
21. Classify triangle by its angles. Hint: Use the Pythagoras theorem:
• If the square of the largest side equals the
• sum of the squares of the two smaller sides, the triangle is right.
• If it is greater, the triangle is obtuse.
• If it is smaller, the triangle is acute.
a) Sides: 7, 24, 25. Type:
b) Sides: 5, 6, 7. Type:
c) Sides: 8, 15, 18. Type:
22.
a) Rectangle ABCD has sides 9 cm and 12 cm. Find its diagonal.
b) Rectangle SPQR has sides 8 cm and diagonal 17 cm. Find its other side.
23. A ladder of length 13 m stands with its foot 5 m from the wall. Find the height of the top of the ladder above the ground.
24. In right triangle ABC with the right angle at B, the perimetre is 36 cm and AB = 9 cm. Find BC and AC.
CONGRUENCE
25. In the figures, two sides and the included angle are marked as equal.
Write the correspondence of vertices and circle the correct congruence criterion.
A , B , C
Criterion: SSS / SAS / ASA / AAS
26. Two angles and the included side are equal.
Write the correspondence of vertices and circle the correct criterion.
A , B , C
Criterion: SSS / SAS / ASA / AAS
27. Do the triples determine congruent triangles (by SSS)? Circle Yes / No.
a) AB = 7 cm, BC = 9 cm, AC = 12 cm and DE = 7 cm, EF = 9 cm, DF = 12 cm
Yes / No
b) AB = 6 cm, BC = 10 cm, AC = 13 cm and DE = 6 cm, EF = 10 cm, DF = 12 cm
Yes / No
28. Two angles and a non-included side are equal.
Write the correspondence of vertices and circle the correct criterion.
A , B , C
Criterion: SSS / SAS / ASA / AAS
29. Looking at the marks, identify which criterion ensures congruence.
Criterion: SSS / SAS / ASA / AAS
30. Triangles are congruent as marked.
State which sides/angles are equal and find the marked part (with “?”).
Answer:
31. In two congruent triangles ABC and DEF, AB corresponds to DE, and BC corresponds to EF.
If AB = 5 cm, DE = 15 cm and DF = 21 cm then AC equals cm.
32. In ABC, AC = BC, and CD is the angle bisector of C meeting AB at D. If AB = 16 cm and AD = 7 cm, find DB.
33. In right triangle ABC with C = 90°, altitude CH is drawn to the hypotenuse.
a) Write all three similar triangles.
b) If AB = 15 cm and AC = 9 cm, find BC = cm.
c) If CH = 12 cm and AC = 15 cm, find AB = cm.
34. In the figure, ABC and ADE share A and have DE ǁ BC.
If AB = 8 cm and AD = 4 cm, find AE when AC = 10 cm.
35. Write all values of x such that the triangles are similar.
a) ABC with sides 6, 8, 10 and PQR with sides 9, 12, x.
b) XYZ with sides 7, 14, x and DEF with sides 5, 10, 15.
36. In two similar triangles, the ratio of similarity is 3 : 5.
a) If the smaller triangle has perimetre 30 cm, find the perimeter of the larger one: cm
(b) If the larger triangle has area 250 cm2, find the area of the smaller one: cm2
8. RATIO AND PROPORTION
RATIO AND PROPORTION
1. Write each ratio in three different forms (fraction, with “:”, and in words). The first item is done for you.
a) 4 apples to 6 apples: 4 6 = 4 : 6 = “4 to 6”.
b) 6 apples to 9 apples: = : = .
c) 15 pencils to 10 pencils: = : = .
d) 24 students to 36 students: = : = .
2. In each case, write the required ratio in simplest form:
a) There are 20 boys and 30 girls in a school.
Ratio of boys to girls: :
b) There are 48 students in total, among them 18 study music.
Ratio of students who study music to all students: :
c) A garden has apple trees, 24 orange trees and 36 banana trees. Write:
i. ratio of apple trees to banana trees: :
ii. ratio of banana trees to total trees: :
3. Express each ratio in simplest form:
a) 12 : 18 :
b) 35 : 49 :
c) 64 : 80 : d) 150 : 225 :
4. Convert to the same units first, then write the ratio in three different forms
a) 3 m to 250 cm.
= : =
b) 4 kg to 750 g.
= : =
c) 2.5 L to 1500 mL.
= : =
d) 45 minutes to 2 hours.
= : =
5. Are the following pairs of ratios equivalent? Circle Yes / No
a) 8 : 12 and 10 : 15 Yes / No
b) 21 : 35 and 9 : 15 Yes / No
c) 45 : 60 and 9 : 12 Yes / No
d) 16 : 24 and 20 : 30 Yes / No
6. Simplify each ratio to its lowest terms:
a) 84 : 126 :
b) 144 : 180 :
c) 315 : 525 :
d) 128 : 226 :
7. Simplify the following three–term ratios to its lowest terms:
a) 18 : 27 : 45 : :
b) 72 : 108 : 180 : :
c) 150 : 225 : 300 : :
8. Convert to the same units, then simplify the ratio:
a) 2 m 50 cm to 150 cm
Simplest form: :
b) 3 kg 600 g to 1 kg 200 g
Simplest form: :
c) 1.5 L to 2500 mL
Simplest form: :
d) 90 minutes to 2 hours 15 minutes
Simplest form: :
9. A recipe uses flour, sugar and butter in the ratio
3 : 2 : 1. If 450 g of flour are used, how much sugar and butter are needed?
10. Compare the ratios. For each pair, write one symbol in the box: >, < or =.
a) 7 : 9 5 : 6
b) 16 : 24 3 : 5
c) 14 : 21 20 : 30 d) 9 : 20 11 : 25
11. Circle the correct answer. Two baskets contain red and blue beads. Basket A has the ratio red : blue equal to 24 : 18. Basket B has 35 : 28.
Which basket has the greater ratio? A / B
12. Order the following ratios from least to greatest. Write the ranks 1–4 in the boxes next to each ratio (1 = smallest).
a) 5 : 12 rank
b) 7 : 15 rank
c) 9 : 16 rank
d) 11 : 18 rank
13. Divide each quantity into the given ratio. Write each part.
a) Divide 180 into 2 : 4. Parts: ,
b) Divide 350 into 3 : 4. Parts: ,
c) Divide 560 into 5 : 9. Parts: ,
14. A prize of $720 is shared among three friends A, B and C in the ratio 2 : 3 : 7. How much does each receive?
A: , B: , C: .
15. The sides of a rectangle are divided in the ratio 3 : 5. If the perimetre of the rectangle is 64 cm, find the lengths of the sides.
Side lengths: cm, cm.
16. If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C.
Answer: : :
17. If P : Q = 3 : 7 and Q : R = 2 : 9, find P : Q : R.
Answer: : :
18. Three numbers are in the ratio 2 : 3 : 5. If their sum is 200, find the numbers.
Numbers: , ,
19. The monthly incomes of A and B are in the ratio 7 : 9. If A earns $14,000, what is the income of B?
Income of B:
20. The angles of a triangle are in the ratio 2 : 3 : 4. Find the three angles.
Angles: °, °, °
21. Check if the following ratios form a proportion. Circle Yes / No
a) 7 : 9 and 21 : 27 Yes / No
b) 25 : 35 and 15 : 20 Yes / No
c) 14 : 18 and 21 : 28 Yes / No
22. Find the value of x:
a) 6 : 9 = 10 : x x =
b) 15 : x = 25 : 40 x =
c) x : 36 = 5 : 12 x =
23. The sides of two similar triangles are proportional. If one triangle has sides measuring 9 cm, 12 cm and 15 cm, and the side of another triangle that corresponds to the 9 cm side measures 18 cm, find the remaining sides of the second triangle. Find the remaining sides of the second triangle.
Sides: cm, cm.
24. On a map, 1 cm represents 5 km. If two towns are 7.2 cm apart on the map, find the actual distance between them.
Answer: km
Answer:
25. Find the number that must be added to both terms of the ratio 5 : 7 so that it becomes 3 : 4.
Answer:
26. Find the number that must be subtracted from both terms of the ratio 33 : 53 so that it becomes 3 : 5.
Answer:
27. Find the number which must be added to both terms of the ratios 3 : 5 and 7 : 11 so that the new ratios are equal.
Answer:
28. Find the mean proportional between 9 and 25.
29. The areas of two similar rectangles are in the ratio 9 : 16. If the smaller rectangle has side 12 cm, find the corresponding side of the larger rectangle.
Answer: cm
Answer:
30. Numbers a, b, c are said to be in continued proportion if a : b = b : c. If a = 4, b = 6, find c.
Answer: ₹
31. The cost of 4 notebooks is ₹56. Find the cost of 7 notebooks.
Answer: ₹
32. 2.5 kg of sugar cost ₹150. Find the cost of 4 kg of sugar.
Answer:
33. A machine produces 360 parts in 6 hours at a constant rate. How many parts will it produce in 10 hours at the same rate?
34. 12 workers can finish a job in 8 days, working at the same rate. How many days will 6 workers take to finish the same job?
Answer: days
! Reminder. For speed, distance and time we use the formulas: S = D T , D = S × T, T = D S
Where S = speed, D = distance, T = time.
35. A car travels 180 km in 3 hours at a constant speed. How far will it travel in 5 hours at the same speed?
Answer: km
36. A cyclist covers 84 km in 4 hours at a constant speed. Find the speed in km/h.
Answer: km / h
37. A car travels at 54 km/h. Find its speed in m/s.
(Use 1 km/h = 5 18 m/s).
Answer: m / s
38. A runner completes a distance of 1500 m in 5 minutes. Find his speed in km/h.
Answer: km / h
39. A car travels 60 km at 40 km/h and then 90 km at 60 km/h. Find the average speed for the whole journey.
Answer: km / h
40. A cyclist travels for 2 hours at 12 km/h and then for 3 hours at 8 km/h. Find the average speed.
Answer: km / h
41. A bus goes 60 km at 30 km/h, then 60 km at 40 km/h, and finally 75 km at 25 km/h. Find its average speed for the whole journey.
Answer: km / h
42. Two cars start from two towns 300 km apart and travel towards each other. Their speeds are 50 km/h and 70 km/h. How many hours will it take them to meet?
Answer: hours
43. A train 120 m long passes a man walking at 6 km/h in the same direction in 12 seconds. Find the speed of the train in km/h. (Take 1 m/s = 3.6 km/h).
Answer: km / h
44. Two cyclists start at the same time from the same point in the same direction. One moves at 18 km/h and the other at 12 km/h. How long will it take for the faster cyclist to be 24 km ahead of the other?
Answer: hours
45. A train leaves City A at 7:30 am and reaches City B at 10:00 am. The distance between A and B is 150 km. Find the average speed of the train.
Answer: km / h
Answer:
46. A bus leaves Town X at 6:00 am and travels at 40 km/h. Another bus leaves Town Y at 7:30 am and travels at 60 km/h towards Town X. The towns are 300 km apart. At what time will the two buses meet?
47. A car gives a mileage of 20 km per litre at a constant rate. How many litres of fuel are needed to travel 260 km?
Answer: L
48. 12 workers can finish a job in 10 days, with all workers identical and working the same number of hours per day. How many days will 15 workers take to finish the same job under the same conditions?
Answer: days
49. To cover a fixed distance, a bus takes 4 hours at 60 km/h. How long will it take at 80 km/h to cover the same distance? Give your answer in hours.
Answer: hours
! Reminder. For a fixed amount of gas at constant temperature:
P1 × V1 = P2 × V2 or P1 P2 = V1 V2
Where P = pressure and V = volume.
50. A gas has pressure 120 kPa and volume 30 L. If the pressure changes to 80 kPa (temperature constant), find the new volume.
Answer: L
51. State whether each situation shows Direct or Inverse proportion. Then solve.
a) For 24 students the canteen prepares 24 cups of juice (same number of cups per student). How many cups are needed for 30 students?
Circle Direct / Inverse
Answer:
b) A runner covers a fixed 12 km route. At 8 km / h he takes 1.5 hours. How long will he take at 10 km/h? (Give your answer in hours as a decimal.)
Circle Direct / Inverse
Answer: hours
52. To paint a wall of area 200 m2 Michael needs 3 L of paint. If paint and number of coats are the same, how many litres are needed for 350 m2?
Answer: L
53. A team of 12 identical workers can complete 720 units of work in 10 days, working the same number of hours per day. If the total work increases by 25% and the team size increases to 15 workers, how many days will be needed now?
Answer: days
DATA H ANDLIN G
9. PERCENTAGE, PROFIT AND LOSS, SIMPLE INTEREST
PERCENTAGE AND ITS APPLICATIONS
1. The pie chart shows the favorite fruits of a group of students.
a) What fraction of the students like Mango?
Answer:
b) What percentage of the students like Orange?
Answer:
c) How many students like Banana, if there are 200 students in all?
Answer:
2. The pie chart shows the means of transport used
by students.
a) What is the ratio of students going by bus to those going by bicycle?
Answer:
b) What percent of students go on foot?
Answer:
c) If 300 students are surveyed, how many go by car?
3. The pie chart shows marks obtained by a student in four subjects.
Which subject has the highest marks?
Answer:
b) What percent of the total marks is obtained in English?
Answer:
c) If the total marks are 400, find marks obtained in Science.
Answer:
4. Convert the following fractions into percentages.
5. Convert the following percentages into fractions in lowest form.
a) 45% b) 125%
6. Convert the following fractions into percentages.
a) 0.07 %b) 1.5 %
7. Express each percentage as a decimal.
a) 8% b) 150%
8. Write each number as a percentage of the other.
a) 15 and 60
15 is % of 60;
60 is % of 15
b) 40 and 80
40 is % of 80;
80 is % of 40
9. Find the number.
a) 12 is 30% of what number?
Answer:
b) 48 is 120% of what number?
Answer:
10. Compare the following and arrange them in ascending order by converting each into a percentage: 0.45, 2 5 , 38%.
Answer: % < % < %
(Write all values in percentages)
11. In a class of 50 students, 30 are girls and 20 are boys.
a) Write the ratio of girls to boys. Answer: :
b) What percent of the class are girls? Answer: %
12. Out of 100 books in a library, 45 are story books and the rest are textbooks.
a) Write the ratio of story books to textbooks. Answer: :
b) What percent of the books are textbooks? Answer: %
13. The ratio of boys to girls in a team is 2 : 3. If there are 30 students in all.
a) How many are boys? Answer:
b) What percent of the team are girls? Answer: %
14. In a survey, 70% of the people liked tea and 30% liked coffee.
a) Write the ratio of tea-lovers to coffee-lovers. Answer: :
b) If 200 people were surveyed, how many liked tea? Answer:
15. Find:
a) 45% of 200:
b) 12% of 150:
16. Express each number as a percentage of the other.
a) 40 of 160: %b) 90 of 150: %
17. Find the number.
a) 18 is 30% of what number?
Answer:
b) 42 is 70% of what number?
Answer:
18. Convert the following into percentages of a whole.
a) 250 m out of 1 km
Answer: %
b) 45 s out of 3 min?
Answer: %
19. A tank contains 800 L of water. 120 L is taken out.
a) What percent of the water is removed?
Answer: %
b) What percent of water remains?
Answer: %
20. A school has 600 students. 40% are in class 7, 25% in class 8, and the rest in class 9.
a) How many students are in class 7? Answer: .
b) How many students are in class 8? Answer: .
c) What percent of students are in class 9? Answer: %.
21. A book cost ₹200 last year. This year its price is ₹240.
a) By how many rupees did the price increase? Answer: .
b) What percent is this increase on the original price?
Answer: %
22. The population of a town decreased from 50,000 to 45,000.
a) Find the decrease in population. Answer: .
b) Find the percent decrease.
Answer: %
23. A shopkeeper increased the price of a shirt from ₹400 to ₹500.
a) What percent was the increase?
Answer: %.
b) If the shirt is again reduced back to ₹400,what percent is the decrease from ₹500?
Answer: %.
24. Compare:
a) Is 72 more or less than 80? By what percent?
Answer:
b) Is 96 more or less than 80? By what percent?
Answer:
25. The salary of a worker was first increased by 20% and then decreased by 10%.
a) Find the net percent change in the salary.
Answer: %
b) If the original salary was ₹10,000, find the new salary.
Answer:
26. A toy was bought for ₹150 and sold for ₹180.
a) Find the profit. Answer: .
b) Find the profit percent.
Answer: %
27. A book was bought for ₹250 and sold for ₹200.
a) Find the loss. Answer:
b) Find the loss percent.
Answer: %
28. A pen was sold for ₹96 at a profit of 20%.
a) Find the cost price of the pen.
Answer:
29. A table was sold for ₹1,320 at a loss of 12%.
a) Find the cost price of the table.
Answer:
30. A machine was bought for ₹12,000 and sold at a loss of 15%.
a) Find the selling price of the machine
Answer:
31. A shirt has a marked price (MP) of ₹1,500. A discount of 10% on MP is offered.
a) Find the discount amount. Answer:
b) Find the selling price (SP).
Answer:
32. A jacket has CP = ₹800 and MP = ₹1,200. It is sold after a discount of 20% on MP.
a) Find the selling price. Answer:
b) Find the profit percent on CP. Answer:
Answer:
33. After a discount of 30% on MP, an article is sold for ₹840. Find the marked price.
34. A store announces successive discounts: first 20% on MP, then a further 10% on the reduced price.
a) Find the single equivalent discount percent.
Answer: %
b) If MP = ₹2,000, find the final selling price.
Answer:
35. A trader says: “Selling at ₹176 gives me the same rupee profit as the rupee loss when I sell at ₹144”.
a) Find the cost price (CP)
Answer:
b) Find both the profit percent at ₹176 and the loss percent at ₹144.
Answer: % and %
36. A shopkeeper bought 40 T-shirts at the same cost price (per piece) and sold all at the same price per piece. His total loss was equal to the cost price of 8 T-shirts.
a) If the cost price per T-shirt was ₹150, find the selling price per T-shirt.
Answer:
b) Find his loss percent on the whole transaction.
Answer: %
! Reminder. For simple interest we use:
SI = P × R × T 100 , where P = Principal, R = Rate (per year), T = Time (in years),
SI = Simple Interest
Answer: ₹
37. Find the simple interest on a sum of ₹5,000 at 8% per annum for 2 years.
38. A sum of ₹3,600 gives ₹972 as simple interest in 3 years. Find the rate of interest.
Answer: %
39. For what time will ₹2,500 produce a simple interest of ₹600 at 12% per annum?
Answer: years
10. RATIONAL NUMBERS
1. Write a rational number and reduce it to its standard form. INTRODUCING RATIONAL NUMBERS
a) whose numerator is (−3) × 4 and denominator is 6 × 3
Standard form:
b) whose numerator is (−5) × (−6) and denominator is 3 × 20
Standard form:
c) whose numerator is 10× (−7) and denominator is (−14) × 11
Standard form:
d) whose numerator is 7 × 15 and denominator is 35 × (−2)
Standard form:
2. Label the numbers on a number line.
3. Bring the denominator to 24 and write the equivalent fraction in standard form.
4. Complete with equivalent fractions (denominators are given):
5. Reduce each to standard form.
28 −98 =
6. Make the denominator positive and simplify completely.
7. Convert each mixed number to an improper fraction, then reduce to standard form.
8. Write the additive inverse of each number (use standard form).
9. Write the additive inverse and then verify the sum is 0.
10. Write the multiplicative inverse and verify the product is 1.
a) Number: −7 5 Inverse: , Check:
b) Number: − 12 −11 Inverse: , Check:
11. Order the numbers from least to greatest. Show the LCM used for denominators.
order:
12. Order the numbers from least to greatest. Show the LCM used for denominators.
13. Fill the blank so that the equality holds and the result is in standard form:
14. Check whether the pairs are equivalent by using ad = bc. Circle Yes / No.
a) 8 12 and 10 15
b) −9 14 and 27 −42
c) 21 28 and 9 12
Yes / No
Yes / No
Yes / No
Final:
15. Reduce −216 −378 to standard form.
Answer:
16. Given that a b is equivalent to 3 8 with a, b Z, list all integer pairs (a, b) with 0 < b ≤ 40.
Yes / No
17. Insert >, <, or = by converting to a common denominator:
Find: m =
18. Decide if the statement is true. Circle Yes / No. “If two negative fractions have the same denominators, the one with the larger absolute numerator is smaller”.
19. Let x = m 24 with m Z. Place x so that it is exactly halfway between − 5 6 and − 1 4 on the number line.
Answer:
20. Using the midpoint method repeatedly, write the first three rationals strictly between 1 6 and 1 3 produced by.
21. Write the two fractions whose denominator is 40 that are closest to the midpoint between numbers − 5 8 and − 1 4 one just below and one just above it. (Their numerators will be consecutive.)
Answer: ,
22. Using denominator 36, write the first three (smallest) fractions strictly between 1 9 and 1 3.
Answer: , ,
Answer:
23. Between − 7 10 and − 1 2 , choose the unique fraction with denominator 60 that is exactly in the middle (by distance).
24. Are the following pairs multiplicative inverses? Circle Yes / No. a) − 5 3 and − 3 5 Yes / No Yes / No Yes / No b) −7 8 and 8 7 c) 2 −9 and −9 2
25. Compute and reduce to standard form.
26. Find the sum by bringing to LCM denominators 13
Answer: 27. Subtract and simplify: −
28. Find x so that the equation holds (unique solution).
29. Solve and reduce:
30. Multiply and reduce:
−7 9 × 15 14 = ;
31. Divide (as multiplication by reciprocal) and reduce:
32. Evaluate.
h) (3 7 + 2 7) × 5 6 = g) (5 8 × 12 25 ) + 1 10 = i) 13 15 − (2 5 × 3 4) = j) (1 2 × 3 5) + (1 2 × 4 5) = k) (7 9 − 1 3) ÷ 2 3 = l) ( 4 11 + 3 22 ) − 5 22 =
Use this space for calculations:
33. Find area of a rectangle with sides 11 12 m and 3 5 m.
Answer (in standard form): m2
34. Find x: 5 8 × x = − 15 32 Answer: x =
Answer:
35. A tank initially has 3 5 of its capacity filled. Then 1 6 of the full capacity is added. What fraction of the capacity is filled now?
Answer:
36. A rope 11 12m long is cut into two parts in the ratio 2 : 1. Find the length of the longer part (in metres).
37. The temperature at midnight was − 7 ° 4 C. By noon, it increased by 11 ° 8 C. What is the temperature at noon?
Answer: ° C
Answer:
38. A price is decreased by 3 10 and then increased by 1 5 of the new price. Find the overall multiplicative factor relative to the original price.
39. Reduce and compare. 84 −126 and −2 3 .
Are they equal? Yes / No
40. A quantity is scaled by 7 6 and then by 12 7 . What single rational factor describes the overall change?
Answer (in standard form):
DATA H ANDLIN G 11. CONSTRUCTION OF TRIANGLES
CONSTRUCTION TRIANGLES
1. Given a segment AB and a point P on AB, construct the perpendicular to AB at P.
2. Given that ∠XOY = 40°, construct the bisector of ∠XOY.
3. Circle Yes / No.
a) The bisector of an angle divides it into two equal angles
b) The sum of angles around a point is 360°
c) The three angle bisectors of any triangle meet at one point.
Yes / No
Yes / No
Yes / No
4. Fill in the blanks:
a) A triangle can be uniquely constructed if we know at least elements;
b) At least one of the known elements must be a .
5. Which data sets are sufficient to construct a unique triangle? Circle Yes / No.
a) Three sides
b) Two sides and the included angle
c) Three angles
d) One angle and two opposite sides
e) It is possible to construct a triangle with sides 3 cm, 3 cm, and 7 cm
Yes / No
Yes / No Yes / No
Yes / No
Yes / No
6. Construct a triangle with sides 5 cm, 6 cm, and 7 cm.
Classify the triangle by its sides:
7. Construct an equilateral triangle △ABC with side 5 cm and find its angles.
8. The perimeter of a triangle △ABC is 18 cm and the sides are in the ratio AB : BC : AC = 2 : 3 : 4. First find the three side lengths, then construct the triangle.
9. A student constructs △ABC with AB = 7 cm, AC = 4 cm, and ∠A = 45°.
Which criterion is used here?
10. Construct an isosceles triangle △ABC with two equal sides AB = AC and included angle 120°.
Classify the triangle by its angles:
11. Construct a triangle with given side BC, ∠B = 45°, and ∠C = 60°.
Measure ∠A:
12. While constructing △PQR with ∠P = 40°, ∠Q = 70°, and side PQ = 6 cm, a student constructs ∠R = 70° instead of ∠Q. Will this result in an error when constructing a triangle?
Yes / No
13. Construct a triangle with given side AB, ∠A = 50°, and ∠B = 80°.
Which side is the longest?
14. Given that legs ML and MN are in ratio ML : MN = 4 : 3, construct a right-angled triangle with right angle at M and given leg ML = 8.
State the length of the hypotenuse:
15. Triangles △ABC and △PQR are right-angled. △ABC has legs 3 and 4, while △PQR has legs 12 and 9. Find the ratio of their areas.
SABC : SPQR = :
16. Are the following data sets are sufficient to construct a unique triangle? Circle Yes / No.
a) AB = 6 cm, BC = 7 cm, CA = 5 cm
b) AB = 5 cm, BC = 6 cm, ∠B = 120 °
c) ∠A = 40 ° , ∠B = 60 ° , AB = 5 cm
d) ∠A = 60 ° , ∠B = 70 ° , ∠C = 50 °
Yes / No Yes / No Yes / No Yes / No
17. Assertion: A triangle is uniquely determined if its three angles are given. Reason: The sum of the three angles of a triangle is 180°.
Choose:
a) Both true and Reason explains Assertion
b) Both true but Reason does not explain Assertion
c) Assertion false, Reason true
d) oth false
Answer:
Yes / No
18. A student is given BC = 6 cm, ∠B = 40°, and ∠C = 50°. Is this data sufficient to construct △ABC uniquely?
DATA H ANDLIN G 12. PERIMETER AND AREA
SQUARES, RECTANGLES, TRIANGLES AND PARALLELOGRAMS
1. A square has a side of 8 cm. Find its perimeter and area.
2. Find the area of a square that has a perimeter of 44 cm.
3. The area of the rectangle is 180 cm² and one side is 12 cm long. What is the perimeter of this rectangle?
4. The figure is shaped like an ‘L’ and consists of two rectangles joined together, with an area of 64 cm². One rectangle has a length of 10 cm and a width of 4 cm; the other has a width of 6 cm. Find the perimeter of the whole figure.
5. The meander is one of the oldest and most popular motifs in art and architecture. It consists of repeating fragments shown in the illustration. Imagine an artist who has decided to decorate a vase with a meander pattern, with lines and spaces measuring 1 cm in width. If the artist needs 1 g of paint per 1cm², how much paint will be needed to colour in the entire fragment shown in the illustration?
6. A rectangle measures 18 cm × 12 cm. From each corner, a square of side 3 cm is cut out. Find the area of the remaining figure.
7. A square with side 20 cm is divided by a diagonal into two triangles. From one of the triangles, a square of side 6 cm is cut out along the diagonal. Find the area of the remaining figure.
8. A rectangle of 14 cm × 10 cm has a smaller rectangle of 6 cm × 4 cm cut out from the inside, so that strips remain on all four sides. Find the area of the remaining figure.
9. A rectangle has dimensions 24 cm × 16 cm. A square of side 8 cm is cut out from the middle of the top side, forming a U-shaped figure. Find the area and perimeter of the new figure.
10. A staircase-shaped figure is made of three rectangles: the bottom one 12 cm × 4 cm, the middle one 8 cm × 4 cm, and the top one 4 cm × 4 cm. Find the perimeter of the figure.
11. In parallelogram ABCD, the height h1 to side AB is 9 cm, and the height h2 to side AD is 12 cm. The perimeter is 70 cm. Find the area of ABCD.
12. In parallelogram ABCD with AB parallel to CD, point P is taken on AB. Through P a line is drawn parallel to AD, meeting BC at Q. The small parallelogram APQD has area equal to three-fifths of the area of ABCD. If AB = 24 cm, find AP.
13. In a parallelogram, the altitudes to two adjacent sides differ by 4 cm, and the lengths of these two sides differ by 3 cm. The longer side corresponds to the smaller altitude. Find the lengths of the sides of the parallelogram.
14. In parallelogram ABCD (AB parallel to CD), point P is taken on side AD and point Q on side BC so that PQ is parallel to AB. The line PQ divides the parallelogram into two smaller parallelograms: APQD and PQCB. If AP = PD, find the ratio of the areas of APQD and PQCB.
15. The area of a triangle is 132 cm², and its base is 12 cm. Find the corresponding height.
16. The raja’s land is rectangular in shape, as shown in the picture. Part of it is covered by a forest (shaded in grey), and the rest is a lake. Which is larger in area: the forest or the lake?
17. Two triangles have the same height of 10 cm. The bases are 12 cm and 8 cm. Compare their areas.
18. A triangle with base 12 cm and corresponding height 8 cm is divided into two smaller triangles by its median. Find the area of each smaller triangle.
19. On side AB of triangle ABC, point D is marked so that AD = 4 cm and BD = 7 cm. The area of triangle ACD is 12 cm². Find the area of triangle BCD.
20. A triangle of area 48 cm² is divided by a line parallel to the base into two smaller figures of equal height. If the smaller FG is 6 cm, find the base of the larger one.
21. Triangle XYZ has base YZ = 18 cm. The altitude from X to YZ is 10 cm. Point M lies on the altitude such that YM = 6 cm. Through M draw a line parallel to YZ, cutting XY at A and XZ at B. Find the area of triangle XAB.
22. Point E is marked on the CD side of square ABCD, and point F is marked at the intersection of segment BE and diagonal AC, as shown in the figure. If the area of triangle AEF is 32 cm², what is the area of triangle BCF?
In the following problems, we will take π as (22/7).
CIRCLES
23. The radius of the circle is 7 cm. The diametres BC and DE are perpendicular to each other. Calculate the area and perimeter of the shaded figure.
WORD PROBLEMS ON PERIMETER AND AREA
24. As shown in the figure, four quarters of a circle were cut from a square pie with a side length of 21 cm. Two of the quarters had a radius of 7 cm and the other two had a radius of 14 cm. What is the area of the remaining part of the pie?
25. A man is walking along a blue, rectangular route with sides measuring 14 and 28 m. Next to him, a dog runs along a red route consisting of six semicircles, each with a diametre of 14 m. How much longer is the dog’s route than the man’s?
26. Adhya and Aarav are running along the routes shown in the diagram. The long horizontal segments of these routes are identical, measuring 28 m each. However, Adhya runs along short vertical segments measuring 14 m, while Aarav runs along semicircular segments with a radius of 14 m. How much longer is Aarav’s route than Adhya’s?
Aarav
Adhya
DATA H ANDLIN G 13. ALGEBRAIC EXPRESSIONS
GENERATING RULES IN FORMULAS AND PATTERNS
1. Continue each pattern by writing the next three terms.
a) 2, 5, 8, 11, , , . c) 7, 10, 13, 16, , , .
b) 9, 12, 15, 18, , , . d) 4, 1, −2, −5, , , .
2. The nth term of a sequence is given. Find the 21st term.
a) an = 2n + 1
b) an = 10n − 1
c) an = 3n − 4
c) n = 50 Answer: a21 = Answer: a21 = Answer: a21 = Answer: Answer: Answer:
3. Find the sum of the first n natural numbers using 1 + 2 + 3 + ... + n = n(n + 1) 2
a) n = 10 b) n = 20
4. Find the perimeter of regular polygon of side a.
a) Triangle: c) Pentagon:
b) Square:
d) Hexagon:
5. Two rectangles have sides (l, b) and (l + 2, b − 1).
Find (l + 2)(b − 1) − lb and write the result in standard form.
Answer:
6. Compare perimeters: which is greater? Write each perimeter.
A: equilateral triangle, side 3a ;
B: square, side 2a .
Circle which is greater? A / B
7. List the terms.
a) 7x − 3y + 5
b) −4a + 2ab − b
(c) 3p2q − 2q + 9
Terms:
Terms:
Terms:
a) 2.5ab
b) −7a
c) − ab 3
8. For each expression, state the numerical coefficient of a (pure number) and the algebraic coefficient of a (remaining factor).
Numerical: , Numerical: , Numerical: , Algebraic:
9. Identify all constant terms.
a) 5x − 3y + 9
b) −2m + 3mn − n2
c) 7 − k
Constant:
Constant: Constant: Algebraic: Algebraic:
10. Complete the factor tree for 12ab.
a) 15ab:
11. Complete the factor trees.
b) 18a2b:
c) 24abc:
12. Form the expression.
a) Sum of x and twice y:
b) Three-fourths added to one-third z:
c) One-fourth of n subtracted from three n:
d) t decreased by five:
Expression:
Expression:
Expression:
Expression:
13. Write the expression.
a) Perimeter of rectangle (l, b). P =
b) Area of square of side a. S =
c) Perimeter of equilateral triangle of side a. P =
14. Based on the text of the problem, write an expression.
a) Arjun has ₹m, he gets ₹50 more.
Expression:
b) Priya spends ₹x on books and ₹y on a pen.Expression:
c) A ribbon of length r cm is cut into two parts: one is r/3 cm. Express the other. Expression:
15. Regular n-gon of side a: find its perimeter in terms of a, n.
Expression:
16. From each set, list like terms.
a) 3x, −2x, 5y, x, y
(b) ab, −3ab, a2b, b2, 4ab
17. Circle Like / Unlike.
a) 7p and −3p
Like / Unlike b) 2a2 and 2ab
Like / Unlike
Like terms:
Like terms:
c) xy and yx
Like / Unlike
18. Group the like terms.
a) 5x − 2y + x + 3y − 4
b) ab + 2a − 3ab + b − a
a) 7x
b) 3y − 5
Answer:
Answer:
19. Classify each algebraic expression based on the number of terms.
c) a + b + c
d) p − q + r − s
20. Find the odd one: x + y, a + b, p + q + r.
The odd one:
21. Identify all terms and classify.
a) 2m − 3n + 5
b) −a + b
c) 7 Terms: Terms: Terms: Type: Type: Type:
22. Add (horizontal).
a) (3x − 2y) + (x + 5y)
b) (a + 2b + 3) + (2a − b − 1)
Answer: Answer:
23. Add (column).
a) 5p − 2q + 1 and −3p + 4q − 6
Answer:
b) 2m + n and m − 3n + 7
24. Add and simplify.
a) x + y + 2x − 3y + 4
b) ab − 2a + 3ab + b − a
25. Add three expressions.
x − 2, 3x + 5, −2x + 1
Answer: Answer: Answer: Answer:
26. Rohan has ₹x, Meera has ₹2x + 50. Find total.
Answer:
27. Rectangle (l, b) and square (a, a). Find sum of perimeters.
Answer:
28. Subtract (horizontal).
a) (5x − 3y) − (2x + y)
b) (a + 4b − 7) − (2a − b + 1)
Answer: Answer:
29. Subtract (column).
a) 4p + q − 6 minus 2p − 3q + 1
Answer:
b) m − 2n minus (3m + n − 5)
Answer:
Answer:
30. What should be subtracted from x + 3y to get 2x − y?
Answer:
31. What should be subtracted from 5a − 2b + 7 to get 2a + 3b − 1?
Answer:
32. Shop sells ice creams: earned ₹x + 40 on Day 1, earned ₹x − 10 on Day 2. How much more on Day 1?
Answer:
33. Ribbon r cm, cut (r/4 + 3) cm. Remaining?
34. Remove brackets and write the result in standard form.
a) [ 2x − { x − (3 − y) } ]
Result:
b) { 3a − [ 2a − (b − 1) ] }
Result:
35. Simplify.
a) x − (2x − 3) + 4
b) − { 2y − [ 3y − (1 − y) ] }
Answer:
Answer:
36. Simplify and evaluate.
x + 2x − 3 at x = −2
b) 2a − b + (a − 2b) at a = 3, b = −1
Answer: Answer:
c) 3m − 2n − (m + n) at m = −4, n = 5Answer:
37. Evaluate.
2p − q at p = 7, q = 3
b) a2 − b at a = −3, b = 4
Answer:
xy − 2x at x = −2, y = −5
Answer: Answer:
a)
a)
c)
a) Rectangle (l, b) and square (a, a). Find (2l + 2b)−4a and write the result in standard form.
Answer: Answer:
b) Square side (k + 2). Area (do not expand).
39. Mixed. Write the result in the standard form: (x − 3y + 1) + (2x + y − 4) − (x − 2y − 5)
Answer: .
40. Classify.
a) 9t
b) u − 7
c) p + q + r
d) a − b + c − d
41. Circle Like / Unlike.
a) 3xy and −5yx
Like / Unlike
b) 2a2 and 2a
Like / Unlike
42. Add and simplify.
a) (4x − 3y + 2) + (x + y − 5)
b) (m − 2n) − (n − 3m + 1)
Answer: Answer:
Answer:
43. Find the perimeter of right triangle with legs b, h and hypotenuse c.
44. Evaluate.
a) 2x + 3 at x = 5
b) a2 − 2a at a = −2
(c) 3p − q at p = −1, q = −4
Answer: Answer: Answer:
14. EXPONENTS AND POWERS
EXPONENTS
1. Fill in the blanks:
a) 1 lakh = units;
b) 1 crore = units;
c) 10, 00, 000 = lakhs.
2. Match the following:
a) 100 hundreds b) 1000 tens c) 10 lakhs
1) 10000 units 2) 1000000 units
3) 10000 units
3. Write the following in usual form:
b) 3 crore = units;
c) 2 crore 50 lakh = units.
4. Compute:
a) 5 lakh = units; 101 = ; 102 = ; 103 = .
5. Circle Yes / No: Is 100000 equal to 105?
Yes / No
6. Write as repeated multiplication:
a) 25 = ;
7. Identify base and power:
a) 57 base: , power:
b) (−3)4 base: , power:
c) 109 base: , power:
8. Compute:
9. Find the square or cube:
a) Square of 11 = ;
10. Compare using >, < or =:
a) 25 52
= ; a1 = ;
b) 73 = ; b) Square of 12 = ; b) 33 24 c) 104 = . c) Cube of 7 = . c) 102 210
11. Fill in the gaps:
1n = ;
. 0n = (n > 0).
Answer:
12. Evaluate:
17. Decide if the result is positive or negative: −32 = ; −24 = ;
13. Complete the table:
14. Circle True / False.
a) 23 = 32 a) (−7)12 True / False Positive / Negative (−3)2 = ;
16. Compute:
= ; b) 42 = 24
= 125
True / False
15. A bacteria doubles every hour. Initially there were 8 bacteria. How many will there be after 2 hours?
/ Negative
18. Circle Yes / No: Is (−3)5 negative? Yes / No
19. Compare using >, < or =:
a) 74 54 b) (−2)6 (−2)7
20. Write in prime factor form:
a) 216 =
b) 768 =
c) 1500 =
a) 540 = b) 490 =
21. Express in prime factor form:
22. Find HCF and LCM of:
a) 23 × 32 × 5 and 22 × 33
b) 24 × 72 and 25 × 3 × 7
, LCM: .
, LCM: .
23. There are 216 students. Express the number in prime factor form and find how many groups of 9 can be formed.
Prime factorization of 216 = Number of groups:
24. Write all factors of 12 as pairs of divisors (e.g. (1, 12)).
Answer:
Answer:
25. Do the same for 36. What happens in the middle?
26. Simplify:
23 × 25 = , 54 × 52 = .
27. Simplify:
× 7)3 = .
28. Compute:
29. Simplify:
30. Simplify:
31. Simplify: (3 × 5)2 = ;
= ;
32. Simplify:
= .
33. Find x and t:
10x = 1000 ;
Answer: x = ; t = .
34. Given that 6p × 63 = 362, find p.
Answer: p = .
35. Find n: x2 × xn = x7.
Answer: n = .
36. Write as repeated multiplication:
37. Compute:
38. Find reciprocals:
39. Write in standard form:
a) 45000 = ;
40. Write in usual form:
b) 8000000 = .
a) 3.6 × 105 = ;
b) 7.2 × 107 = .
41. Compare using >, < or =:
a) 3.8 × 107 7.2 × 106
b) 5.01 × 108 4.9 × 109
Answer:
42. Find the number of digits in 8.3 × 106.
43. Multiply and write in standard form (k × 10n): (2.5 × 103) × (4 × 102) =
(Remember to write the coefficient k so that 1 ≤ k < 10.)
44. The population of one country is 1.25 × 109.
Express it in usual form:
45. A bacteria triples every 2 hours. Initially there are 9 bacteria. How many after 6 hours?
Answer:
46. A game doubles the score each round. If a player starts with 5, express the score after n rounds as a power.
Answer:
47. Given that 1 KB = 210 bytes, 1 MB = 210 KB, 1 GB = 210 MB and 1 TB = 210 GB, express 1 GB and 1 TB in bytes using powers of 2.
Answer: 1 GB = bytes, 1 TB = bytes.
15. SYMMETRY
UNDERSTANDING SYMMETRY
1. Circle the figures with 1 axes of symmetry.
2. Circle the figures with 2 axes of symmetry.
3. Circle the figures with 3 axes of symmetry.
4. Circle the figures with 4 axes of symmetry.
5. Identify and draw the lines of symmetry in the given figures.
6. Which of the lines is the axis of symmetry of the figures?
7. Relative to which of the three straight lines are the figures symmetrical in the drawing?
8. Which of the drawings shows the correct case of central symmetry.
9.Draw a shape symmetrical to the point O.
10. Complete the given figures such that they are symmetrical about the given mirror line of symmetry.
11. Find the angle of rotation.
12. Draw the axes of symmetry and find the angle of rotation, if it is possible.
16. VISUALISING SOLID SHAPES
3D SHAPES AND 2D SHAPES
1. Name the shapes.
2. Find and separate 3D and 2D shapes. circle
3. Cross out the cones and circle the pyramids.
4. Name the shapes that these objects look like. circle
5. Find a shape that has the specified properties out of 3 possible ones.
a) faces: 6 edges: 12 vertices: 8
b) faces: 5 edges: 8 vertices: 5
c) faces: 8 edges: 18 vertices: 12
d) faces: 4 edges: 6 vertices: 4
e) faces: 7 edges: 15 vertices: 10
6. For each 3D shape, shade the correct net.
c)
d) 1. 2. 3.
e) 1. 2. 3.
1. SIMPLE EQUATIONS
Page 1
1. Ascending order: −12, −8, −3, 0, 5, 7.
Sum of smallest and largest: −5.
2. T, T, T, T.
3. a) −3 < 2; b) |−7| > |−4|; c) −10 > −15; d) 0 < |−5|;
4. 0 –3–5 1245
Page 2
5. A B C D E –5 –2 2 3 4
6. a) 3; b) 3; c) 4; d) 7; e) 5.
7. a) 1; b) −2, −1, 0, 1, 2; c) −2.
Page 3
8. a) 2; b) 9; c) 2; d) |7−9| = 2; e) |−5+8| = 3.
9. a) x = ±2; b) x = 0.
Page 4
10. a) |−4 − 5| = 9; (b) |−7 − (−1)| = 6.
11. a) −5; b) −4; c) −11; (d) 0.
12. 3
13. a) 6; b) −6; c) −12. 14. a) 25; b) 25; c) 0.
Page 5
15. a) 8; b) −18.
16. a) −5; b) −5; c) −5.
17. 6°C.
18. 530 m below sea level.
Page 6
19. ₹300.
20. 5.
21. a) −96; b) 42; c) −60; d) 0.
Page 7
22. a) −7; b) 8; c) 0.
23. 10 kg lighter.
24. Not possible.
25. b), d), e).
Page 8
26. a) −14; b) −8; c) 4; d) −5.
27. a) 45; b) −8; c) 0.
28. −44 m (below start).
Page 9
29. 30 floors.
30. ₹960 profit.
31. a) Commutative property of addition.
32. b) Distributive property.
Page 10
33. a) Associative, b) Commutative, c) Distributive.
34. 0.
35. −12.
Page 11
36. a) −7; b) 11; c) 0; d) −25.
37. a) −5; b) 9; c) 0.
38. a) 1 4 ; b) – 1 4 ; c) 1; d) – 1 10
39. a) T; b) F; c) T.
Page 12
40. a) −22; b) −36; c) −70.
41. (−8 + 6) × (−2) = 4.
42. −7.
43. 36.
44. −3
Page 13
45. 5. 46.
47. −4, −3, −2.
2. FRACTIONS AND DECIMALS
Page 14
1. a) 2 6 = 1 3 ; b) 3 8 ; c) 5 12 ;
2. Answers vary (student shading; check the target fraction of the area).
Page 15
3. a) 4 6 = 2 3 ; b) 7 10 ; c) 15 20 = 3 4 .
4. a) 5 1 4 ;
5. a) 19 5 ; b) 43 6 ; c) 43 8 ; d) 8 3 ; e) 9 6 = 3 2 ; f) 47 10
Page 16
6. a) a = 2601; b) a = 68, b = 3; c) m = 8
Page 18
10. a) 30 60 ; b) 40 60 ; c) 45 60 ; d) 42 60 ; e) 45 60 ; f)
44 60 11. a) 9 12 and 10 12 ; b) 8 12 and 7 12 ; c) 25 40 and 12 40 ; d) 28 36 and 3 3 ;
12. a) 6 12 , 8 12 , 9 12 ; b) 3 5 , 5 8 , 7 10 c) 576 1008 , 560 1008 , 378 1008
Page 19
13. a) <; b) >; c) >; d) <; e) <.
14. a) >; b) <; c) >; d) >; e) >.
7. a) 12 18 = 2 3 ; b) 45 60 = 3 4 ; c) 16 24 = 2 3 ; d) 28 42 = 2 3 ; e) 40 50 = 4 5 ; f) 55 81 ; g) 63
Page 17
8. b) 10; c) 8; d) 12; e) 12; 15
=
;
15. a) 3 8 < 1 2 < 2 3 < 5 6 b) 5 9 < 4 7 < 3 5 < 7 10
Page 20 16. a) 5 7 ; b) 1 9 ; c) 9 11 ; d) 2 5 ; 17. a) 1 7 12 ; b) 1 12 ; c) 37 40 ; d) – 5 36 ; 18. 9 20
Page 21
19. a) 7 6 ; b) 8 9 ; c) 5 8 d) 29 42
Page 22
20. a) 2 3 ; b) 19 20 ; c) 3 16 ;
21. a) 12 35 ; b) 21 44 ; c) 1 6 ; d) 3 10 ;
Page 23
22. a) 1 3 4 ; b) 3 3 5 ; c) 1 13 14 ; d) 9 1 15
Page 24
23. 1 2 cup
24. a) 1 1 20 ; b) 1 29 48 ; c) 1 23 27 ; d) 128 135 ;
25. 6 pieces
Page 25
26. a) 3 1 9 ; b) 4 19 24 ; c) 5 29 32 ; d) 3 13 24 ;
Page 26
27. 140 pages 28. 81 29. $40
30. a) 0.405 < 0.45 < 0.5; b) 0.8 < 0.805 < 0.82; c) 0.12 < 0.125 < 0.13.
Page 27
31. a) 1 2 ; b) 1 4 ; c) 1 8 ; d) 3 4 ; e) 3 5 ; f) 1 5 ;
32. a) 0.75 b) 0.7; c) 0.45; d) 0.44; e) 0.4; f) 0.125.
33. a) 5.9; b) 4.6; c) 12; (d) 3.65.
34. a) 8.15; b) 10.8; c) 5.69; d) 12
Page 28
35. a) 2.85; b) 7.65; c) 7.9; d) 2.3; e) 10.2; f) 11.48; g) 6; h) 11.10
Page 29
36. a) 7; b) 25 c) 6; d) 0.45; e) 7; f) 1; g) 6.3; h) 7.9
Page 30
37. a) 1.5; b) 2.15; c) 2.2; d) 1.65; e) 6.3; f) 1.47;
38. a) 5.5; b) 3.228571; c) 4.9; d) 3.75;
Page 31
39. a) 8.25; b) 5; c) 16.5; d) 8; e) 150; f) 197.5; g) 700
Page 32
40. 1.5m
41. 20 apples
42. 57 km/h
43. $48.30
3. DATA HANDLING
Page 33
1. a) 600; b) 70; c) Zebra; d) Elephant; e) 200
Page 34
2. a) Pacific; b) 11022; c) 5527, 7725, 8442, 11022; d) Pacific; e) Atlantic
Page 35
3. (b) 4. 30 5. 6 6. 10.25
Page 36 7. 122 8. 35
9. 77
10. 8 11. 21
Page 37
12. a) 48; b) 2.4; c) 1.6
13. 68
14. 41
Page 38
15. a) 528; b) 132; c) 73; d) 18.25; e) 612 f) 153
Page 39
16. c d a b 17. 3
18. 2, 5, 8
19. 12, 15, 20, 20, 25, 30, 35, 35, 35; mode:
35
Page 40
20. 12, 15, 20, 20, 25, 30, 30, 35, 35; modes: 20, 30, 35
35
21. a) 2; b) 7; c) 2, 3
Page 41
22. 4
23. 12, 15, 20, 20, 25, 30, 35, 35, 35; median: 25
24. 3.5
25. 12, 15, 20, 20, 25, 30, 35, 35; median: 22.5
Page 42
26. 17
27. a) 3; b) 15; c) 165; d) 11; e) 13; f) 12 g) 7 years
Page 43
28. a) 25; b) 17; c) cat; d) 6; e) rabbit
Page 44
29. a) 44; b) No; c) 204
Page 45
30. a) 71 m, b) Big Ben; c) Tower of Belen; d) Big Ben
Page 46
31. a) 60 mm; b) February; c) June; d) 132mm
4. INTRODUCTION TO PROBABILITY
Page 50
1. a) Impossible; b) Sure; c) Equally likely; d) Unlikely; e) Likely
2. a) Unlikely; b) Impossible; c) Equally likely; d) Likely; e) Sure
Page 51
3. a) Unlikely; b) Likely; c) Impossible; d) Sure; e) Equally likely
Page 52
4. a) 8; b) 8 20 = 2 5 ; c) 5 20 = 1 4 ; d) 7 20
Page 53
5. 3 7
6. 12 in total; a) 3 12 = 1 4 ; b) 4 12 = 1 3 ; c) 5 12
Page 54
7. 6 10 = 3 5
8. 22; 11 22 = 1 2
Page 55
9. a) 20 100 = 1 5 ; b) 1 6 ; c) 47 100 ; d) 1 2
Page 56
10. a) 16 50 = 8 25 ; b) 5 20 = 1 4 ; c) 19 50 ; d) 8 20 = 2 5
5. SIMPLE EQUATIONS
Page 57 1. Page 58 2.
Twice a number decreased by 4 is 10
Seven more than a number is 15
Half of a number is equal to 6
Three times a number increased by 8 equals 20
One third of a number increased by 5 is 11
Three quarters of a number minus 2 equals 7
Five sixths of a number plus 4 is equal to 9
Seven tenths of a number decreased by 3 equals 8 b)
Twice a number minus one-fifth of that number is equal to 200
Three-fourths of a number plus 12 equals 500
Seven-tenths of a number added to 150 equals 850 c)
Half of a number decreased by 25 is equal to 150
Page 59
3. a) x = 3; b) x = 5 3 ; c) x = 3 4 ; d) x = 8 5
Page 60
4. a) x = 6; b) x = 5; c) x = –4; d) x = 3
Page 61
5. a) x = 3 2 ; b) x = 4 3 ; c) x = 3 2 ; d) x = 6 5
Page 62
6. a) x = 3 2 ; b) x = –2; c) x = – 3 4 ; d) x = – 2 5
Page 63
7. a) x = 8 3 ; b) x = 9 2 ; c) x = 2 3 ; d) x = 3
Page 64
8. a) x = 1; b) x = 26 3 ; c) x = 13 4 ; d) x = 2 5
Page 66
9. a) x = 4; b) x = 7; c) x =–5; d) x = 6
Page 67
10. a) x = 15 4 ; b) x = 5 2 ; c) x = 1 2 ; d) x = 19
Page 68
11. a) x = 8; b) x = 1 5 ; c) x = 36 5 ; d) x = 8 9
Page 70
12. b) 13. b) 14. a)
Page 71
15. a) 16. 15 and 30
Page 72
17. 16 and 35 18. 20 and 13
Page 73
19. Nickel: 800 g, Iron: 640 g
20. Gold: 140 g, Platinum: 70 g, Silver: 700g
Page 874
21. Vinegar: 560 g, Baking soda: 400 g, Water: 320 g
22. 45
6. ANGLE PAIRS AND PARALLEL LINES
Page 75
1. A = 28°, B = 68°, C = 118°, D = 156°.
Page 76
2. x = 62°, x + 30° = 92°, y = 43°, y + 45° = 88°
Page 77
3. E = 35°, F = 90°, G = 140°, H = 25°.
4. a) 34°: comp. = 56°, sup. = 146°.
b) 68°: comp. = 22°, sup. = 112°.
c) 102°: comp. = no, sup. = 78°.
d) 145°: comp. = no, sup. = 35°.
e) 89°: comp. = 1°, sup. = 91°.
Page 78
5. a) AOB and BOC: complementary.
b) AOB and AOD: supplementary.
c) BOC and COD: neither.
d) AOC and COD: neither.
e) Complementary pairs: AOB and BOC; AOB and COD.
f) Supplementary pair: AOB and DOA.
Page 79
6. a) A = 60°, comp. = 30°.
b) B = 81°, supp. = 99°.
c) C = 36°, D = 56°.
d) E = 118°, F = 62°.
e) x = 12, G = 31°, H = 59°.
Page 80
7. a) x = 29°.
b) AOB = 86°, BOC = 104°.
c) comp. = 14°.
d) BOD = COD = 38°.
Page 81
8. a) y = 18°.
b) AOC = 86°, BOC = 140°.
c) Complementary.
d) AOC = 64°, BOC = 26°.
Page 82
9. a) x = 69°, y = 22°, and z = 85°.
b) x: comp. = 21°, supp. = 111°.
y: comp. = 68°, supp. = 158°.
z: comp. = 1°, supp. = 91°.
Page 83
10. a) COB = 116°; b) DOA = 142°; c) No;
d) No; e) AOC and COB; BOD and DOA.
Page 84
11. a) x = 20; A = 92°, B = 88°.
b) x = 15; A = 60°, B = 120°.
c) x = 16; A = 126°, B = 56°.
d) x = 14.5; A = 121.5°, B = 58.5°.
Page 85
12. a) x = 20.
b) A = 46°, B = 86°, C = 50°.
c) B.
Page 86
13. a) x = 26.
b) AOP = 84°, POB = 96°.
c) Acute: AOP; obtuse: POB.
Page 87
14. a) No; correct B = 135°.
b) No; correct B = 110°.
c) Correct; B = 84°.
d) Correct; B = 152°.
Page 88
15. a) x = 12.
b) A = 52°, B = 128°, C = 52°, D = 128°.
c) Vertical pairs: ( A, C) and ( B, D).
d) 64° and 64°.
Page 89
16. a) y = 48.
b) A = 72°, B = 108°, C = 72°, D = 108°.
c) Acute: A, C; obtuse: B, D.
Page 90
17. a) B = 114°, C = 68°, D = 114°.
b) 136°; c) 224°; d) 44°.
Page 91
18. a) k = 28.
b) A = 76°, B = 46°, C = 88°, D = 66°,
E = 86°.
c) Largest: C. Acute: all five angles.
d) 184°.
Page 92
19. a) AOB = 58°.
b) 88°; c) 103°; d) 88° — acute; 103° — obtuse
Page 93
20. a) AOB = 38°, BOC = 110°; b) 148°.
c) 72°; d) 34°; Acute.
21. a) x = 18; P = 50°, Q = 70°.
b) 88°; c) x = 36; d) Yes.
Page 94
22. a) y = 19.
b) AOB = 69°, BOC = 34°, COD = 89°.
c) COD.
Page 95
23. a) Corresponding.
b) Alternate interior.
c) Corresponding.
d) Alternate exterior.
Page 96
24. a) Corresponding: A and E, B and H, C and G, D and F.
b) Alternate interior: B and F, C and E.
c) Alternate exterior: A and G, D and H.
d) Same-side interior: B and E, C an F.
Page 97
25. a) x = 15°.
b) A = 68°, E = 68°.
c) B = 112°.
d) F = 112°, H = 112°.
Page 98
26. a) x = 30°.
b) C = 100°, E = 100°.
c) D = 80°, H = 80°.
Page 99
27. a) x = 18°.
b) B = 120°, E = 60°.
c) A = 60°, G = 60°.
Page 100
28. a) B = 142°, C = 38°, D = 142°.
b) E = 38°, F = 142°.
c) J = 38° and L = 38°.
Page 101
29. a) x = 16°.
b) ABC = 72°, BCD = 108°.
c) Acute: ABC; Obtuse: BCD.
d) 72°.
Page 102
30. a) No; b) Yes.
Page 103
31. a) No, C = 110°.
b) No, C = 96°.
c) Yes, C = 152°.
7. TRIANGLES AND THEIR PROPERTIES
Page 104
1. a) by sides: equilateral; by angles: acute.
b) by sides: scalene; by angles: right.
c) by sides: isosceles; by angles: acute.
d) by sides: isosceles; by angles: obtuse.
e) by sides: isosceles; by angles: obtuse.
f) by sides: scalene; by angles: right.
g) by sides: scalene; by angles: acute.
h) by sides: scalene; by angles: acute.
Page 105
2. a) by sides: scalene; by angles: right.
b) by sides: isosceles; by angles: acute.
c) by sides: scalene; by angles: obtuse.
d) by sides: equilateral; by angles: acute.
e) by sides: isosceles; by angles: right.
3. a) median.
b) centroid.
c) altitude.
d) orthocenter.
e) centroid.
f) orthocenter (at the right-angled vertex).
Page 106
4. a) AG = 14 cm, GD = 7 cm.
b) GD = 6 cm, AD = 18 cm.
c) BG = 20 cm, GE = 10 cm.
Page 107
5. a) H is inside; acute.
b) H at a vertex; right.
c) H is outside; obtuse.
Page 108
6. a) AD = 15 cm, BE = 21 cm.
b) AG = 18 cm, GD = 9 cm, BE = 27 cm.
Page 107
7. (a) B = C = 72°.
(b) A = 76°, C = 52°.
(c) A = B = 38°, C = 104°.
Page 108
8. a) C = 71°.
b) B = 60°.
c) A = 45°.
d) 36°, 60°, 84°.
e) 72°, 72°, 36°.
f) C = 185° – 3x.
Page 109
9. a) exterior at C = 108°; C = 72°.
b) B = 68°, C = 62°.
c) 3x + 28 = 136 Þ x = 36; then A = 72°, B = 64°, C = 44°.
10. Picture 1.
Page 110
11. a) B = C = 66°.
b) A = 62°, B = C = 59°.
c) k = 24; A = 55°, B = C = 63°.
Page 111
12. a) F. Correct: an exterior angle equals the sum of the two non-adjacent interior angles.
b) F. Correct: the two opposite interior angles sum to 120°.
c) F. Correct: each exterior angle is greater than either of the two non-adjacent interior angles.
d) T. e) T. f) T.
Page 112
13. a) ACB = 155° – x.
(b) BCD = 52°.
Page 113
14. a) x = 44; B = 58°.
b) y = 26; C = 58°.
Page 114
15. a) C = 66°, B = 66°.
b) x = 34; A = 72°, B = C = 54°.
Page 115
16. a) exterior at B = 114°; exterior at C = 114°.
b) A = 42°, B = 42°, C = 96°.
17. a) Yes; b) Yes; c) No; d) Yes.
18. a) c {6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}.
b) c {7, 8, 9, …, 23}.
c) c {8, 9, 10, …, 28}.
19. a) 10 cm; b) 15 cm.
Page 116
20. a) 12 cm; b) 24 cm.
21. a) right; b) acute; c) obtuse.
Page 117
22. a) 15 cm; b) 15 cm
23. 12 m.
Page 118
24. BC = 12 cm, AC = 15 cm.
25. A E, B F, C D; Criterion: SAS.
Page 119
26. A D, B E, C F; Criterion: ASA.
27. a) Yes; b) No.
28. A D, B E, C F; Criterion: AAS
Page 120
29. SSS.
30. Corresponding pairs as marked; AC=DF.
Page 121
31. AC = 7 cm.
32. DB = 9 cm.
33. a) ABC ~ ACH ~ CBH.
b) BC = 12 cm.
c) AB = 25 cm.
Page 122
34. AE = 5 cm.
35. a) x = 15; b) x = 21.
36. a) 50 cm; b) 90 cm².
8. RATIO AND PROPORTION
Page 123
1. b) 6 9 = 6 : 9 = “6 to 9”
c) 15 10 = 15 : 10 = “15 to 10”
d) 24 36 = 24 : 36 = “24 to 36”
2. (a) 2 : 3 (b) 3 : 8 (c-i) 2 : 5 (c-ii) 1 : 2
Page 124
3. a) 2 : 3 b) 5 : 7 c) 4 : 5 d) 2 : 3
4. a) 6 : 5 b) 16 : 3 c) 5 : 3 d) 3 : 8
Page 125
5. a) Yes b) Yes c) Yes d) Yes
6. a) 2 : 3 b) 4 : 5 c) 3 : 5 d) 1 : 2
7. a) 2 : 3 : 5 b) 2 : 3 : 5 c) 2 : 3 : 4
Page 126
8. a) 5 : 3 b) 3 : 1 c) 3 : 5 d) 2 : 3
9. Flour: 450 g, Sugar: 300 g, Butter: 150 g
Page 127
10. a) < b) > c) = d) >
11. Basket A
12. 5 : 12 1, 7 : 15 2, 9 : 16 3, 11 : 18 4
13. a) 60, 120 b) 150, 200 c) 200, 360
Page 128
14. A: ₹120, B: ₹180, C: ₹420
15. 12 cm, 20 cm
16. 8 : 12 : 15
Page 129
17. 6 : 14 : 63 18. 40, 60, 100 19. $18,000
Page 130
20. 40°, 60°, 80°
21. a) Yes b) Yes c) Yes d) Yes
22. a) 15 b) 24 c) 15
23. 18 cm, 24 cm, 30 cm
Page 131
24. 36 km 25. 1 26. 3
Page 132
27. 1 28. 15 29. 16 cm 30. 9
Page 133
31. ₹98 32. ₹240 33. 600 34. 16 days
Page 134
35. 300 km 36. 21 km/h 37. 15 m/s
Page 135
38. 18 km/h 39. 50 km/h 40. 9.6 km/h
Page 136
41. 30 km/h 42. 2.5 hours 43. 42 km/h
Page 137
44. 4 hours 45. 60 km/h 46. 9:54 am
Page 138
47. 13 L 48. 8 days 49. 3 hours
Page 139
50. 13 L 51. 8 days
Page 140
52. 5.25 L 53. 10 days
9. PERCENTAGE, PROFIT AND LOSS, SIMPLE INTEREST
Page 141
1. a) 1 5 ; b) 20%; (c) 70.
Page 142
2. a) 4 : 5; (b) 40%; (c) 45.
Page 143
3. a) Science; (b) 20%; (c) 120.
4. a) 75%; (b) 28%.
Page 144
5. a) 9 20 ; b) 5 4 .
6. a) 7%; (b) 150%
7. a) 0.08; (b) 1.5.
8. a) 25%; 400%; b) 50%; 200%.
Page 145
9. a) 40; b) 40
10. 38% < 40% < 45%
Page 146
11. a) 3 : 2; b) 60%
12. a) 9 : 11; b) 55%
13. a) 12; b) 60%.
14. a) 7 : 3; b) 140.
Page 147
15. a) 90; b) 18.
16. a) 25%; b) 60%.
17. a) 60; b) 60.
Page 148
18. a) 25%; (b) 25%.
19. a) 15%; (b) 85%.
Page 149
20. a) 240; b) 150; (c) 35%.
21. a) ₹40; b) 20%.
22. a) 5000; b) 10%.
Page 150
23. a) 25%; b) 20%.
24. a) 72 is 10% less than 80; b) 96 is 20% more than 80.
Page 151
25. a) +8%; b) ₹10,800.
26. a) n30; b) 20%.
Page 152
27. a) ₹50; (b) 20%.
28. ₹80
29. ₹1,500.
Page 153
30. ₹10,200
31. a) ₹150; b) ₹1,350.
32. a) ₹960; b) 20%.
Page 154
33. ₹1,200
34. a) 28%; b) ₹1,460
Page 155
35. a) ₹160; b) 10% and 10%.
36. a) ₹120; (b) 20%.
Page 156
37. ₹800
38. 9%
39. 2 years
10. RATIONAL NUMBERS
Page 157
1. a) − 2 3 ; b) 1 2 ; c) 35 88 ; d) − 3 2 .
Page 158 2.
3. a) 20 24 ; b) − 21 24 ; c) 22 24 .
4. a) 12 20 , 21 35 ; b) − 8 14 , − 12 21 .
5. a) − 3 4 ; b) − 3 4 ; c) 3 4 ; d) − 4 5 .
Page 159
6. a) − 2 7 ; b) 10 13 .
7. a) 11 4 ; b) − 23 6 ; c) 23 6 ; d) − 25 8 .
8. a) 11 6 ; b) − 13 4 ; c) 0.
9. a) 13 14 ; b) 9 10 .
Page 160
10. a) − 5 7 ; b) 11 12 .
11. 4 3 < − 7 6 < − 9 8 < − 5 6 .
12. 11 12 < − 7 9 < − 3 8 < 2 3 < 5 6 .
13. a) −15; b) −21; c) 14.
Page 161
14. a) Yes; b) Yes; c) Yes.
15. 4 7 .
16. (3,8), (6,16), (9,24), (12,34), (15,40).
Page 162
17. 5 6 < − 3 4 ; 7 12 < 5 8 ; − 9 10 = − 27 30 .
18. Yes.
19. m = −13 (so x = − 13 24 ).
20. r1 = 1 4 , r2 = 5 24 , r3 = 7 24
Page 163
21. 18 40 , − 17 40 .
22. 5 36 , 6 36 , 7 36 . 23. 36 60 .
24. a) Yes; b) No; c) Yes.
Page 164
25. a) 11 12 ; b) − 3 10 ; c) 19 54 ; d) − 11 8 .
26. 53 60 . 27. 71 72 .
28. x = − 1 4 .
Page 165
29. x = − 1 20 .
30. a) − 5 6 ; b) − 2 9 .
31. a) − 3 4 ; b) 1 6
32. a) 31 36 ; b) 7 40 ; c) − 2 7 ; d) − 2 3 ; e) 5 36 ; f) 11 12 ; g) 2 5 ; h) 25 42 ; i) 17 30 ; j) 7 10 ; k) 2 3 ; l) 3 11 .
Page 166
33. 11 20 m2.
34. 2.
Page 167
35. 23 30 .
36. 11 18 m.
37. − 3 ° 8 C.
Page 168
38. 21 25 .
39. Yes.
40. x = − 3 4 .
11. CONSTRUCTION OF TRIANGLES
Page 169
2. Each half is 20°.
3. a) Yes; b) Yes; c) Yes.
Page 170
4. a) 3; b) side.
5. a) Yes; b) Yes; c) No; d) No; e) No.
6. Scalene.
Page 171
7. ∠A = 60 ° , ∠B = 60 ° , ∠C = 60 ° .
8. AB = 4 cm, BC = 6 cm, AC = 8 cm.
9. SAS.
Page 172
10. Obtuse.
11. ∠A = 75 ° .
12. Yes
Page 173
13. AC (opposite the largest angle B = 80°).
14. 10.
15. SABC : SPQR = 1 : 9.
Page 174
16. a) Yes; b) Yes; c) Yes; d) No.
17. c).
18. Yes.
12. PERIMETER AND AREA
Page 175
1. Perimeter = 32 cm; Area = 64 cm2.
2. Side = 11 cm; Area = 121 cm2.
Page 176
3. Other side = 15 cm; Perimeter = 54 cm.
4. Perimeter = 40 cm.
Page 177
5. 39 grams.
Page 178
6. Area 180 cm2.
7. Area 200 cm2.
Page 179
8. Area 116 cm2.
9. Area 320 cm2; Perimeter 96 cm.
Page 180
10. Perimeter = 48 cm.
11. 180 cm2.
Page 181
12. Area(APQD) Area(ABCD) = AP AB = 3 5 AP = 3 5 × 24 = 14.4 cm.
13. 12 cm and 15 cm.
Page 182
14. Area(APQD) : Area(PQCB) = AP : PD = 1 : 1.
15. Height = 2 × 132 12 = 22 cm.
Page 183
16. Their areas are equal.
17. Areas are proportional to bases: 12 : 8 = 3 : 2.
Page 184
18. 24 cm2
19. 21 cm2.
Page 185
20. 12 cm.
21. 54 cm2.
Page 186
22. 32 cm2.
23. 38.5 cm2
Page 187
24. 56 cm2.
25. Man: 84 m; dog: 132 m; difference: 48 m.
Page 188
26. Adhya: 112 m; Aarav: 128 m; difference: 16 m.
13. ALGEBRAIC EXPRESSIONS
Page 189
1. a) 14, 17, 20; b) 21, 24, 27; c) 19, 22, 25; d) −8, −11, −14.
2. a) 43; b) 209; c) 59.
3. a) 55; b) 210; c) 1275.
Page 190
4. a) 3a; b) 4a; c) 5a; d) 6a.
5. 2b − l − 2.
6. Triangle: 9a; Square: 8a; Circle: A.
7. a) 7x, −3y, 5; b) −4a, 2ab, −b; c) 3p2q, −2q, 9.
Page 191
8. a) Numerical: 2.5; Algebraic: b.
b) Numerical: −7; Algebraic: 1.
c) Numerical: − 1 3 . ; Algebraic: b.
9. a) 9; b) none; c) 7.
10.
Page 192 11. a)
12. a) x + 2y; b) z 3 + 3 4 ; c) 3n − n 4 ; d) t − 5.
Page 193
13. a) 2l + 2b; b) a2; c) 3a.
14. a) m + 50; b) x + y; c) 2r 3 .
15. na.
16. a) x-terms: 3x, −2x, x; y-terms: 5y, y. b) ab-terms: ab, −3ab, 4ab; others: a2b, b2. b) c)
17. a) Like; b) Unlike; c) Like.
Page 194
18. a) 6x + y − 4; b) −2ab + a + b.
19. a) monomial; b) binomial; c) trinomial; d) quadrinomial.
20. Odd: p + q + r; Reason: trinomial.
21. a) Terms: 2m, −3n, 5; Type: trinomial.
b) Terms: −a, b; Type: binomial.
c) Terms: 7; Type: monomial.
22. a) 4x + 3y; b) 3a + b + 2.
Page 195
23. a) 2p + 2q − 5; b) 3m − 2n + 7.
24. a) 3x − 2y + 4; b) 4ab − 3a + b.
25. 2x + 4.
26. 3x + 50.
27. 2l + 2b + 4a.
Page 196
28. a) 3x − 4y; b) −a + 5b − 8.
29. a) 2p + 4q − 7; b) −2m − 3n + 5.
30. −x + 4y.
31. 3a − 5b + 8.
Page 197
32. 50.
33. 3r 4 − 3.
34. a) x − y + 3; b) a + b − 1.
35. a) −x + 7; b) 2y − 1.
Page 198
36. a) −9; b) 12; c) −23.
37. a) 11; b) 5; c) 14.
Page 199
38. a) 2l + 2b − 4a; b) (k + 2)2.
39. 2x + 2.
40. a) monomial; b) binomial; c) trinomial; d) quadrinomial.
41. a) Like; b) Unlike.
Page 200
42. a) 5x − 2y − 3; b) 4m − 3n − 1.
43. b + h + c.
14. EXPONENTS AND POWERS
Page 201
1. a) 105; b) 107; c) 10.
2. 100 hundreds
1000 tens
10 lakhs
44. a) 13; b) 8; c) 1. 10000 1000000 10000
3. a) 5, 00, 000; b) 3, 00, 00, 000; c) 2, 50, 00, 000.
4. 101 = 10; 102 = 100; 103 = 1000.
5. Yes.
Page 202
6. a) 2 × 2 × 2 × 2 × 2; b) 7 × 7 × 7; c) 10 × 10 × 10 × 10.
7. a) base 5, power 7; b) base −3, power 4; c) base 10, power 9.
8. 81; 25; 64.
9. a) 121; b) 144; c) 343.
10. a) >; b) >; c) <.
11. a; 1; 0.
Page 203
12. −9; 9; −32.
13. 4, 8, 16; 9, 27, 81; 16, 64, 256; 25, 125, 625.
14. a) False; b) True; c) True.
15. 8 × 2 × 2 = 32.
16. −16; 16; −32.
17. a) Positive; b) Negative; c) Positive.
Page 204
18. Yes.
19. a) >; b) >.
20. a) 23 × 33; b) 28 × 3; c) 22 × 3 × 53.
21. a) 22 × 33 × 5; b) 2 × 5 × 72.
Page 205
22. a) HCF = 22 × 32 = 36, LCM = 23 × 33 × 5 = 1080; b) HCF = 24 × 7 = 112, LCM = 25 × 3 × 72 = 4704.
23. 216 = 23 × 33; groups = 24.
24. Pairs for 12: (1, 12), (2, 6), (3, 4).
25. Pairs for 36: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6); middle pair is the square root.
26. 28 = 256; 56 = 15625.
Page 206
27. 152 = 225; 143 = 2744.
28. 1; 1; 1.
29. 24 = 16; 54 = 625.
30. 9 16 ; 16.
31. 212 = 4096; 56 = 15625.
32. 192.
Page 2207
33. x = 3; t = 3.
34. p = 1 .
35. n = 5.
36. a) ( 2 3 )×( 2 3 )×( 2 3 )×( 2 3 ); b) (− 3 5 )×(− 3 5 )×(− 3 5 ).
37. 16 81 ; − 27 125 .
38. a) ( q p )n ; b) 1 (−a)n ; c) 125 8 .
Page 208
39. a) 4.5 × 104; b) 8 × 106.
40. a) 360000; b) 72000000.
41. a) >; b) <.
42. 8, 300, 000 7 digits.
43. (2.5 × 103) × (4 × 102) = 10.0 × 105 = 1.0 × 106.
Page 209
44. 1, 250, 000, 000.
45. 243.
46. Score after n rounds = 5 × 2n.
47. 1 GB = 230 bytes; 1 TB = 240 bytes.
15. SYMMETRY
Page 210
1. a; c. 2. a.
3. b. 4. a; c; d.
Page 211 5. 6. a) 1; b) 2; c) 3.
Page 212
7. a) 2; b) 2. 8. c. 9. a) e)
Page 213 10.
Page 214
16. VISUALISING SOLID SHAPES
Page 215 1.
circle square ellipse rectangle oval trapezium
a, f, g, h b, c, d, e
Page 216 3.
4. A. 1.Triangle 2. Circle 3.Square 4.Rectangle B. 1.Cube 2. Pyramid 3.Sphere 4.Cone
Page 217
5. a) 4; b) 3; c) 1; d) 4; e) 2.
Page 218
6. a) 1; b) 2; c) 2; d) 1; e) 2.