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A free printable book

Maths for classes 6 to 8

4 lessons, each with worked examples, a try-at-home activity and a practice sheet. The practice sheets are together after the lessons, and the answer keys are at the very back.

  • Maths
  • Classes 6-8
  • 4 lessons
  • 41 practice questions
  • Free to print

Contents

Free learning for kids, Gurukul Technicals Institute. Printed from engineergurukul.com/resources/kids/books/maths-6-8/

Contents

What is in this book

  1. 1Factors, multiples and prime numbersFind factors and multiples, spot prime and composite numbers, use divisibility tests, and work out HCF and LCM.5 min
  2. 2Decimals and percentagesUnderstand tenths and hundredths, change between fractions, decimals and percentages, and work out discounts.3 min
  3. 3Ratio and proportionCompare quantities with ratios, simplify them, check proportion and use the unitary method to solve everyday problems.4 min
  4. 4Algebra: first stepsUse letters for unknown numbers, write simple expressions and solve equations by keeping both sides balanced.4 min

Every lesson here can also be read on its own page, with links to related lessons. All printable books.

Lesson 1 of 4

Factors, multiples and prime numbers

Find factors and multiples, spot prime and composite numbers, use divisibility tests, and work out HCF and LCM.

Open this lesson on its own page · Practice sheet · Answer key · Contents

Let us begin

The big idea

Numbers have families. Some numbers divide others exactly, some are built from smaller numbers, and some cannot be broken down at all. Knowing these helps with fractions, ratios and puzzles.

A factor of a number divides it exactly, with nothing left over. The factors of 12 are 1, 2, 3, 4, 6 and 12. A multiple of a number is what you get by multiplying it by 1, 2, 3 and so on. The multiples of 4 are 4, 8, 12, 16 and so on.

A prime number has exactly two factors: 1 and itself. 2, 3, 5, 7, 11 and 13 are prime. A number with more than two factors is composite. 1 is neither prime nor composite. 2 is the only even prime.

Divisibility tests save time. A number is divisible by 2 if it ends in 0, 2, 4, 6 or 8. By 5 if it ends in 0 or 5. By 10 if it ends in 0. By 3 if the sum of its digits is divisible by 3. By 9 if the sum of its digits is divisible by 9.

Every composite number can be written as a product of primes in only one way, apart from the order. This is called prime factorisation. The HCF (highest common factor) of two numbers is the largest number that divides both. The LCM (lowest common multiple) is the smallest number that both divide into.

Prime numbers from 1 to 30A grid of the numbers 1 to 30 in three rows of ten. The prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29 are shaded and written in bold. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Shaded boxes are prime numbers. 1 is neither prime nor composite. 2 is the only even prime number. Every other number here has 3 or more factors.
The shaded numbers have exactly two factors, 1 and the number itself.

Worked examples

See how it works

  1. Factors of 36

    Find pairs that multiply to 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6. So the factors are 1, 2, 3, 4, 6, 9, 12, 18 and 36. That is 9 factors.

  2. Prime factorisation of 60

    60 = 2 × 30. 30 = 2 × 15. 15 = 3 × 5. So 60 = 2 × 2 × 3 × 5.

  3. HCF and LCM of 12 and 18

    Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. The largest common factor is 6, so HCF = 6. Multiples of 12: 12, 24, 36. Multiples of 18: 18, 36. The first common multiple is 36, so LCM = 36. Check: 6 × 36 = 216 = 12 × 18.

  4. Divisibility: is 738 divisible by 9?

    Add the digits: 7 + 3 + 8 = 18. 18 is divisible by 9 (and by 3). So 738 is divisible by 9 and by 3. It ends in 8, which is even, so it is also divisible by 2. Check: 738 ÷ 9 = 82.

Try it at home

Sieve of primes

You will need

  • A sheet of paper
  • A pencil
  • A ruler (optional)

What to do

  1. Write the numbers 1 to 50 in rows of 10. Cross out 1.
  2. Circle 2. Then cross out every second number after 2 (4, 6, 8 and so on).
  3. Circle the next number not crossed out, which is 3. Cross out every third number after it.
  4. Keep going with 5 and then 7. Cross out their multiples.
  5. The circled numbers and the ones left uncrossed are the primes. You should find 15 primes up to 50.

Ask a grown-up: No special help is needed. An adult can check that the answer is 15 primes.

Fun fact

2 is the only even prime number, because every other even number can be divided by 2.

Lesson 2 of 4

Decimals and percentages

Understand tenths and hundredths, change between fractions, decimals and percentages, and work out discounts.

Open this lesson on its own page · Practice sheet · Answer key · Contents

Let us begin

The big idea

Prices, marks and discounts all use decimals and percentages. They are just other ways of writing fractions, and you can move between them easily.

A decimal uses a point to show parts smaller than one. In 2.35 the 2 is ones, the 3 is tenths and the 5 is hundredths. So 2.35 is 2 + 3/10 + 5/100. Each place to the right is ten times smaller.

To compare decimals, compare the whole parts first, then tenths, then hundredths. 0.6 is the same as 0.60, which is bigger than 0.58. To add or subtract decimals, line up the decimal points.

Per cent means per hundred. 35% means 35 out of 100, which is 35/100 = 0.35. To change a fraction to a percentage, make the denominator 100 or multiply by 100. For example 3/4 = 75/100 = 75%.

To find a percentage of an amount, multiply by the percentage and divide by 100. 10% of an amount is that amount divided by 10. Discounts and marks use this.

35 out of 100 as a gridA square made of 100 small boxes in 10 rows of 10. The first 35 boxes are shaded. Beside it, the same amount is written as 35 out of 100, 35 per cent, 0.35 and 7 over 20. 35 out of 100 35% 0.35 35/100 = 7/20 Same amount, three ways.
35 shaded boxes out of 100 can be written as 35%, 0.35 or 35/100, which simplifies to 7/20.

Worked examples

See how it works

  1. Fraction, decimal and percentage: 3/4

    3 ÷ 4 = 0.75. And 0.75 = 75/100 = 75%. So 3/4 = 0.75 = 75%.

  2. Percentage of an amount

    Find 20% of Rs 450. 450 × 20 ÷ 100 = 9000 ÷ 100 = 90. So 20% of Rs 450 is Rs 90.

  3. Comparing decimals

    Which is bigger, 0.6 or 0.58? Write 0.6 as 0.60. Compare 60 hundredths with 58 hundredths. 0.60 is bigger. So 0.6 > 0.58.

  4. A discount

    A shirt costs Rs 600. There is a 15% discount. Discount = 600 × 15 ÷ 100 = 90. Price to pay = 600 − 90 = Rs 510.

Try it at home

Discount detective

You will need

  • An old shop bill, a newspaper advertisement or a list of prices you write yourself
  • A pencil and paper

What to do

  1. Choose three prices, for example Rs 80, Rs 250 and Rs 400.
  2. Find 10% of each by dividing by 10.
  3. Find 50% of each by halving. Then find 25% by halving again.
  4. Work out the price after a 10% discount for each item.
  5. Check one answer by working it out a different way.

Ask a grown-up: An adult can show a real shop bill. No money needs to be handled.

Fun fact

The word per cent comes from the Latin words per centum, which mean by the hundred.

Lesson 3 of 4

Ratio and proportion

Compare quantities with ratios, simplify them, check proportion and use the unitary method to solve everyday problems.

Open this lesson on its own page · Practice sheet · Answer key · Contents

Let us begin

The big idea

When you mix sharbat, scale a recipe or share sweets, you are using ratios. A ratio compares two quantities, and proportion says two ratios are equal.

A ratio compares two quantities of the same kind. If a sharbat has 2 glasses of syrup for every 3 glasses of water, the ratio of syrup to water is 2 : 3. The order matters.

Like a fraction, a ratio can be simplified by dividing both numbers by the same number. 12 : 18 becomes 2 : 3 when both are divided by 6. We can also multiply both numbers by the same number to get equivalent ratios.

Four numbers are in proportion when the two ratios are equal. 3 : 4 and 9 : 12 are in proportion because both simplify to 3 : 4. A quick check: the product of the outer numbers should equal the product of the inner numbers, 3 × 12 = 4 × 9.

The unitary method finds the value of one unit first and then the value of many. To share an amount in a ratio, add the parts to find the total number of parts, find one part, and then multiply.

Two jugs with the same ratioTwo rows of blocks. The first row has 2 gold blocks for syrup and 3 blue blocks for water, a ratio of 2 to 3. The second row has 4 gold blocks and 6 blue blocks, a ratio of 4 to 6, which is the same as 2 to 3. syrup water Small jug 2 : 3 Big jug Big jug: 4 : 6, which simplifies to 2 : 3 Both jugs taste the same.
Doubling both parts keeps the ratio the same: 2 : 3 = 4 : 6.

Worked examples

See how it works

  1. Simplify 12 : 18

    The biggest number that divides both 12 and 18 is 6. 12 ÷ 6 = 2 and 18 ÷ 6 = 3. So 12 : 18 = 2 : 3.

  2. Are 3 : 4 and 9 : 12 in proportion?

    3 × 3 = 9 and 4 × 3 = 12, so 3 : 4 = 9 : 12. Check with cross products: 3 × 12 = 36 and 4 × 9 = 36. They are equal, so yes.

  3. Unitary method

    5 notebooks cost Rs 120. One notebook costs 120 ÷ 5 = Rs 24. So 8 notebooks cost 8 × 24 = Rs 192.

  4. Sharing in a ratio

    Share Rs 500 in the ratio 2 : 3. Total parts = 2 + 3 = 5. One part = 500 ÷ 5 = 100. Shares are 2 × 100 = Rs 200 and 3 × 100 = Rs 300. Check: 200 + 300 = 500.

Try it at home

Sharbat ratio

You will need

  • Any fruit syrup or squash
  • Water
  • Two glasses or cups
  • A spoon
  • A small cup to use as a measure

What to do

  1. In glass 1 put 1 cup of syrup and 3 cups of water. Write the ratio 1 : 3.
  2. In glass 2 put 2 cups of syrup and 6 cups of water. Write the ratio 2 : 6.
  3. Simplify 2 : 6. It becomes 1 : 3, the same as glass 1.
  4. Taste both. They should taste the same because the ratio is the same.
  5. Predict how much syrup you need for 12 cups of water, keeping the ratio 1 : 3.

Ask a grown-up: An adult should help with pouring and supervise tasting, and check that nobody has an allergy to the syrup.

Lesson 4 of 4

Algebra: first steps

Use letters for unknown numbers, write simple expressions and solve equations by keeping both sides balanced.

Open this lesson on its own page · Practice sheet · Answer key · Contents

Let us begin

The big idea

Algebra uses letters to stand for numbers we do not know yet. It is like a puzzle: find the missing number so that both sides are equal.

A letter such as x or n that stands for a number is called a variable. If Ravi has x marbles and gets 5 more, he has x + 5 marbles. 3x means 3 times x. When we write 3x we leave out the × sign.

An expression such as 2m + 5 becomes a number when we put a value in place of the letter. If m = 6, then 2m + 5 = 2 × 6 + 5 = 17.

An equation says two things are equal, like x + 3 = 8. Think of a balance. To keep it balanced, whatever we do to one side we must do to the other side.

To solve x + 3 = 8, take away 3 from both sides. Then x = 8 − 3 = 5. To solve 3x = 21, divide both sides by 3. Then x = 7. Always check by putting your answer back into the equation.

A balance showing x + 3 = 8A balance beam level in the middle. On the left sit a box marked x and three small squares. On the right sit eight small squares. Both sides weigh the same, so x plus 3 equals 8, and x is 5.Both sides weigh the same x x + 3 = 8 Take 3 from each side: x = 8 − 3 = 5
Take 3 from each side to keep the balance: x = 8 − 3 = 5.

Worked examples

See how it works

  1. Writing an expression

    Ravi has x marbles and wins 5 more: x + 5. If x = 12, he has 12 + 5 = 17 marbles.

  2. Solve x + 7 = 15

    Subtract 7 from both sides: x = 15 − 7 = 8. Check: 8 + 7 = 15. Correct.

  3. Solve 3x = 21

    Divide both sides by 3: x = 21 ÷ 3 = 7. Check: 3 × 7 = 21. Correct.

  4. A number puzzle

    I think of a number, double it and add 4. The answer is 18. Call the number x. Then 2x + 4 = 18. Subtract 4: 2x = 14. Divide by 2: x = 7. Check: 2 × 7 + 4 = 18.

Try it at home

Secret number cups

You will need

  • Two paper cups or small bowls
  • About 20 pebbles or beans
  • Paper and a pencil

What to do

  1. Without telling your friend, put some pebbles under cup A, for example 5.
  2. Put 3 loose pebbles next to cup A. On the other side, put a pile with all the pebbles together, so 8.
  3. Ask your friend to find the secret number. They can take 3 pebbles away from each side to balance.
  4. Write the equation: x + 3 = 8, so x = 5. Lift the cup to check.
  5. Swap roles with a new secret number.

Ask a grown-up: Keep small pebbles away from younger children. An adult can play the secret-keeper.

Fun fact

The word algebra comes from al-jabr, a word in the title of a book written by the scholar al-Khwarizmi over 1,000 years ago.

Part two

Practice sheets

One sheet for each of the 4 lessons. Write your name and the date at the top, answer in the spaces, then check with the answer keys at the back.

Practice sheet

Factors, multiples and prime numbers

Practice sheet: Factors, multiples and prime numbers (Classes 6-8, Maths)

  1. List all the factors of 20.
  2. Write the first five multiples of 7.
  3. Is 29 a prime number?
  4. Is 51 a prime number? Give a reason.
  5. What is the smallest prime number?
  6. Write 84 as a product of prime factors.
  7. Find the HCF of 24 and 36.
  8. Find the LCM of 4 and 6.
  9. Is 3456 divisible by 9? Use the digit test.
  10. Two bells ring every 6 minutes and every 8 minutes. They ring together at 8:00 a.m. When will they next ring together?

Practice sheet

Decimals and percentages

Practice sheet: Decimals and percentages (Classes 6-8, Maths)

  1. Write 7/10 as a decimal.
  2. Write 0.25 as a fraction in its simplest form.
  3. Which is greater, 0.4 or 0.39?
  4. Find 2.5 + 1.75.
  5. Find 5.0 − 2.35.
  6. Write 1/2 as a percentage.
  7. What is 40% of 250?
  8. A student scores 45 out of 50. What is the percentage?
  9. A book costs Rs 250. There is a 10% discount. What do you pay?
  10. Write 0.08 as a percentage.
  11. Put in order, smallest first: 0.5, 0.05, 0.55.

Practice sheet

Ratio and proportion

Practice sheet: Ratio and proportion (Classes 6-8, Maths)

  1. Write the ratio of 4 red marbles to 6 blue marbles in simplest form.
  2. Simplify 15 : 25.
  3. Fill in the blank: 2 : 5 = 6 : ___.
  4. Are 4 : 5 and 12 : 15 in proportion?
  5. 3 kg of rice costs Rs 150. What do 5 kg cost?
  6. A class has 18 boys and 12 girls. Find the ratio of boys to girls in simplest form.
  7. Share 60 sweets between Asha and Ravi in the ratio 1 : 2.
  8. A recipe uses 2 cups of rice and 5 cups of water. How much water is needed for 6 cups of rice?
  9. A car goes 120 km in 2 hours at a steady speed. How far does it go in 5 hours?
  10. Red and white paint are mixed in the ratio 3 : 7 to make 40 ml. How much red paint is there?

Practice sheet

Algebra: first steps

Practice sheet: Algebra: first steps (Classes 6-8, Maths)

  1. Write "a number plus 9" using n for the number.
  2. Find the value of 3a when a = 4.
  3. Find the value of 2m + 5 when m = 6.
  4. Solve x + 6 = 14.
  5. Solve y − 4 = 9.
  6. Solve 5p = 35.
  7. Solve x ÷ 3 = 6.
  8. A square has side s. Write an expression for its perimeter and find it when s = 5.
  9. Solve 2x + 3 = 17.
  10. 3 boxes of mangoes and 2 extra mangoes make 26 mangoes. Each box has m mangoes. Write an equation and solve it.

Part three

Answer keys

The answers to every practice sheet, in the same order as the lessons. Parents and teachers may like to keep these pages separately.

Answer key

Answers: Factors, multiples and prime numbers

  1. 1, 2, 4, 5, 10, 20.
  2. 7, 14, 21, 28, 35.
  3. Yes. Its only factors are 1 and 29.
  4. No. 51 = 3 × 17, so it has more than two factors.
  5. 2
  6. 84 = 2 × 2 × 3 × 7.
  7. 12 (the common factors of 24 and 36 are 1, 2, 3, 4, 6 and 12).
  8. 12 (multiples of 4: 4, 8, 12; 12 is also a multiple of 6).
  9. Yes. 3 + 4 + 5 + 6 = 18, and 18 is divisible by 9. Indeed 3456 ÷ 9 = 384.
  10. At 8:24 a.m., because the LCM of 6 and 8 is 24 minutes.

Answer key

Answers: Decimals and percentages

  1. 0.7
  2. 1/4 (25/100 divided top and bottom by 25).
  3. 0.4 (0.40 is more than 0.39).
  4. 4.25
  5. 2.65 (5.00 − 2.35).
  6. 50%
  7. 100 (250 × 40 ÷ 100).
  8. 90% (45/50 = 90/100).
  9. Rs 225 (10% of 250 is 25, and 250 − 25 = 225).
  10. 8%
  11. 0.05, 0.5, 0.55.

Answer key

Answers: Ratio and proportion

  1. 2 : 3 (divide both by 2).
  2. 3 : 5 (divide both by 5).
  3. 15 (2 × 3 = 6, so 5 × 3 = 15).
  4. Yes. Multiply 4 : 5 by 3 to get 12 : 15.
  5. Rs 250 (one kg costs Rs 50, so 5 kg cost 5 × 50).
  6. 3 : 2 (divide both by 6).
  7. Asha gets 20 sweets and Ravi gets 40 (3 parts, 1 part = 20).
  8. 15 cups (6 cups is 3 times 2 cups, so 3 × 5 = 15).
  9. 300 km (speed 60 km per hour, 60 × 5 = 300).
  10. 12 ml (10 parts, 1 part = 4 ml, red = 3 × 4 = 12 ml).

Answer key

Answers: Algebra: first steps

  1. n + 9
  2. 12 (3 × 4).
  3. 17 (2 × 6 + 5).
  4. x = 8 (14 − 6).
  5. y = 13 (9 + 4).
  6. p = 7 (35 ÷ 5).
  7. x = 18 (6 × 3).
  8. 4s. When s = 5 the perimeter is 20.
  9. x = 7 (2x = 14, so x = 7).
  10. 3m + 2 = 26, so 3m = 24 and m = 8.

The end

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