Year 6 Ratio and Proportion: worked examples and questions with answers

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Divide both numbers by their highest common factor (HCF).

1. Write the ratio in the order the question gives. The first number stays first.
   5:7
2. Find the highest common factor (HCF) of the two numbers. It is the biggest number that divides into both.
   HCF of 5 and 7 is 1
3. Only 1 divides into both, so the ratio is already in its simplest form. Dividing by 1 changes nothing.
   5:7 is already simplest
4. Write the answer.
   5:7

Check. 5 × 1 = 5 and 7 × 1 = 7. ✓

Watch out. Divide both numbers by the same number. Keep the order the question gives.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Divide both numbers by their highest common factor (HCF).

1. Write the ratio in the order the question gives. The order of the numbers must not change.
   8:12
2. Find the highest common factor (HCF) of the two numbers. It is the biggest number that divides into both.
   HCF of 8 and 12 is 4
3. Divide both numbers by the HCF. This keeps the same ratio but uses smaller numbers.
   8 ÷ 4 = 2, and 12 ÷ 4 = 3
4. Write the answer.
   2:3

Check. 2 × 4 = 8 and 3 × 4 = 12. ✓

Watch out. Divide both numbers by the same number. Keep the order the question gives.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Divide both numbers by their highest common factor (HCF).

1. Write the ratio in the order the question gives. The order of the numbers must not change.
   20:12
2. Find the highest common factor (HCF) of the two numbers. It is the biggest number that divides into both.
   HCF of 20 and 12 is 4
3. Divide both numbers by the HCF. This keeps the same ratio but uses smaller numbers.
   20 ÷ 4 = 5, and 12 ÷ 4 = 3
4. Write the answer.
   5:3

Check. 5 × 4 = 20 and 3 × 4 = 12. ✓

Watch out. Divide both numbers by the same number. Keep the order the question gives.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Divide both numbers by their highest common factor (HCF).

1. Write the ratio in the order the question gives. The first number stays first.
   3:5
2. Find the highest common factor (HCF) of the two numbers. It is the biggest number that divides into both.
   HCF of 3 and 5 is 1
3. Only 1 divides into both, so the ratio is already in its simplest form. Dividing by 1 changes nothing.
   3:5 is already simplest
4. Write the answer.
   3:5

Check. 3 × 1 = 3 and 5 × 1 = 5. ✓

Watch out. Divide both numbers by the same number. Keep the order the question gives.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Divide both numbers by their highest common factor (HCF).

1. Write the ratio in the order the question gives. The order of the numbers must not change.
   2:6
2. Find the highest common factor (HCF) of the two numbers. It is the biggest number that divides into both.
   HCF of 2 and 6 is 2
3. Divide both numbers by the HCF. This keeps the same ratio but uses smaller numbers.
   2 ÷ 2 = 1, and 6 ÷ 2 = 3
4. Write the answer.
   1:3

Check. 1 × 2 = 2 and 3 × 2 = 6. ✓

Watch out. Divide both numbers by the same number. Keep the order the question gives.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Divide both numbers by their highest common factor (HCF).

1. Write the ratio in the order the question gives. The order of the numbers must not change.
   180:240
2. Find the highest common factor (HCF) of the two numbers. It is the biggest number that divides into both.
   HCF of 180 and 240 is 60
3. Divide both numbers by the HCF. This keeps the same ratio but uses smaller numbers.
   180 ÷ 60 = 3, and 240 ÷ 60 = 4
4. Write the answer.
   3:4

Check. 3 × 60 = 180 and 4 × 60 = 240. ✓

Watch out. Divide both numbers by the same number. Keep the order the question gives.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Divide both numbers by their highest common factor (HCF).

1. Write the ratio in the order the question gives. The order of the numbers must not change.
   18:27
2. Find the highest common factor (HCF) of the two numbers. It is the biggest number that divides into both.
   HCF of 18 and 27 is 9
3. Divide both numbers by the HCF. This keeps the same ratio but uses smaller numbers.
   18 ÷ 9 = 2, and 27 ÷ 9 = 3
4. Write the answer.
   2:3

Check. 2 × 9 = 18 and 3 × 9 = 27. ✓

Watch out. Divide both numbers by the same number. Keep the order the question gives.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Divide both numbers by their highest common factor (HCF).

1. Write the ratio in the order the question gives. The order of the numbers must not change.
   25:10
2. Find the highest common factor (HCF) of the two numbers. It is the biggest number that divides into both.
   HCF of 25 and 10 is 5
3. Divide both numbers by the HCF. This keeps the same ratio but uses smaller numbers.
   25 ÷ 5 = 5, and 10 ÷ 5 = 2
4. Write the answer.
   5:2

Check. 5 × 5 = 25 and 2 × 5 = 10. ✓

Watch out. Divide both numbers by the same number. Keep the order the question gives.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Keep the order of the question: apples first, then oranges. Then simplify.

1. Find the order in the question. The ratio is apples to oranges, so apples come first.
   4 apples : 10 oranges
2. Write the ratio in that order. Sam has the numbers the wrong way round.
   4 : 10
3. Simplify by dividing both numbers by 2. The HCF of 4 and 10 is 2.
   4 ÷ 2 = 2, and 10 ÷ 2 = 5
4. Write the answer.
   No. 4:10 simplifies to 2:5.

Check. The ratio of oranges to apples would be 5:2, but the question asks for apples to oranges. ✓

Watch out. The order must match the question. Sam swapped the numbers.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Simplify both ratios. If the simplest forms are the same, the ratios are equivalent.

1. Simplify 6:9. Divide both numbers by their HCF, 3.
   6:9 = 2:3
2. Simplify 3:2. Divide both numbers by their HCF, 1.
   3:2 = 3:2
3. Compare the two simplest forms. The same simplest form means the ratios are equivalent.
   2:3 and 3:2 are different
4. Write the answer.
   No

Check. 6 × 2 = 12 and 9 × 3 = 27, so the cross-products are not equal. ✓

Watch out. Compare the simplest forms, not the first numbers.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Simplify each pair of numbers, then compare with 2:3.

1. Simplify 10:15. The HCF of 10 and 15 is 5.
   10:15 = 2:3
2. Simplify 12:18. The HCF of 12 and 18 is 6.
   12:18 = 2:3
3. Compare with the club ratio, 2:3. Both simplify to the same ratio as the club.
   2:3 and 2:3 are the same
4. Write the answer.
   Yes, both. 10:15 and 12:18 both simplify to 2:3.

Check. 10 ÷ 5 = 2 and 15 ÷ 5 = 3; 12 ÷ 6 = 2 and 18 ÷ 6 = 3. ✓

Watch out. Both amounts can be right, because each is the same ratio with a different total.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 3:8, and the first number of the ratio that matches 12. Keep the numbers in the order of the ratio.
   3:8, and 12 matches 3
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   12 ÷ 3 = 4
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   8 × 4 = 32
4. Write the answer as a ratio, in the same order as the question. The ratio must keep its order.
   12:32

Check. 3 × 4 = 12, and the ratio 12:32 simplifies to 3:8. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Simplify both ratios. If the simplest forms are the same, the ratios are equivalent.

1. Simplify 4:10. Divide both numbers by their HCF, 2.
   4:10 = 2:5
2. Simplify 6:15. Divide both numbers by their HCF, 3.
   6:15 = 2:5
3. Compare the two simplest forms. The same simplest form means the ratios are equivalent.
   2:5 and 2:5 are the same
4. Write the answer.
   Yes

Check. 4 × 15 = 60 and 10 × 6 = 60, so the cross-products are equal. ✓

Watch out. Compare the simplest forms, not the first numbers.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 3:2, and the first number of the ratio that matches 12. Keep the numbers in the order of the ratio.
   3:2, and 12 matches 3
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   12 ÷ 3 = 4
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   2 × 4 = 8
4. Write the answer.
   8

Check. 3 × 4 = 12, and the ratio 12:8 simplifies to 3:2. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 4:7, and the second number of the ratio that matches 28. Keep the numbers in the order of the ratio.
   4:7, and 28 matches 7
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   28 ÷ 7 = 4
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   4 × 4 = 16
4. Write the answer.
   16

Check. 7 × 4 = 28, and the ratio 16:28 simplifies to 4:7. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 5:2, and the second number of the ratio that matches 6. Keep the numbers in the order of the ratio.
   5:2, and 6 matches 2
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   6 ÷ 2 = 3
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   5 × 3 = 15
4. Write the answer.
   15

Check. 2 × 3 = 6, and the ratio 15:6 simplifies to 5:2. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 3:4, and the second number of the ratio that matches 20. Keep the numbers in the order of the ratio.
   3:4, and 20 matches 4
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   20 ÷ 4 = 5
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   3 × 5 = 15
4. Write the answer as a ratio, in the same order as the question. The ratio must keep its order.
   15:20

Check. 4 × 5 = 20, and the ratio 15:20 simplifies to 3:4. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Check each statement by multiplying 3 and 5 by the same number.

1. Check 9:15. Multiply 3 and 5 by 3.
   3 × 3 = 9 and 5 × 3 = 15, so 9:15 is equivalent
2. Check 9:14. Find the number that 5 is multiplied by to give 14. 14 is not a multiple of 5.
   5 × ? = 14 has no whole-number answer
3. Write the answer.
   No. Only 9:15 is equivalent to 3:5.

Check. 3 × 3 = 9 and 5 × 3 = 15, so 9:15 matches. ✓

Watch out. Both numbers must be multiplied by the same number. 14 is not 5 times anything.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Multiply 6 and 8 by 1, 2, 3, and so on, until a number reaches 30.

1. Try multiplying by 1. Both numbers are below 30.
   6:8
2. Try multiplying by 2. Both numbers are below 30.
   12:16
3. Try multiplying by 3. Both numbers are below 30.
   18:24
4. Try multiplying by 4. The second number, 32, is 30 or more, so this one is not allowed.
   6 × 4 = 24 and 8 × 4 = 32, and 32 is not below 30
5. Write the answer. There are three ratios.
   3

Check. The three ratios are 6:8, 12:16 and 18:24. ✓

Watch out. Check both numbers. The second number 8 reaches 32 when multiplied by 4, which is too big.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 3:5, and the second number of the ratio that matches 40. Keep the numbers in the order of the ratio.
   3:5, and 40 matches 5
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   40 ÷ 5 = 8
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   3 × 8 = 24
4. Write the answer.
   24

Check. 5 × 8 = 40, and the ratio 24:40 simplifies to 3:5. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale from the red part, then multiply the green part by it.

1. Find the scale for red: 15 ÷ 3. The number of red sweets matches the 3 in the ratio.
   15 ÷ 3 = 5
2. Multiply the green part by 5. Green is 4 parts, so multiply it by the same scale.
   4 × 5 = 20
3. Write the answer.
   Yes. 20 green sweets.

Check. The ratio 15:20 divides by 5 to give 3:4. ✓

Watch out. The ratio must hold for every part, so the green sweets must be 20.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 2:5, and the second number of the ratio that matches 30. Keep the numbers in the order of the ratio.
   2:5, and 30 matches 5
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   30 ÷ 5 = 6
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   2 × 6 = 12
4. Write the answer.
   12

Check. 5 × 6 = 30, and the ratio 12:30 simplifies to 2:5. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 3:4, and the second number of the ratio that matches 36. Keep the numbers in the order of the ratio.
   3:4, and 36 matches 4
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   36 ÷ 4 = 9
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   3 × 9 = 27
4. Write the answer.
   27

Check. 4 × 9 = 36, and the ratio 27:36 simplifies to 3:4. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Divide both numbers by their highest common factor (HCF).

1. Write the ratio in the order the question gives. The order of the numbers must not change.
   16:24
2. Find the highest common factor (HCF) of the two numbers. It is the biggest number that divides into both.
   HCF of 16 and 24 is 8
3. Divide both numbers by the HCF. This keeps the same ratio but uses smaller numbers.
   16 ÷ 8 = 2, and 24 ÷ 8 = 3
4. Write the answer.
   2:3

Check. 2 × 8 = 16 and 3 × 8 = 24. ✓

Watch out. Divide both numbers by the same number. Keep the order the question gives.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 5:2, and the first number of the ratio that matches 35. Keep the numbers in the order of the ratio.
   5:2, and 35 matches 5
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   35 ÷ 5 = 7
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   2 × 7 = 14
4. Write the answer.
   14

Check. 5 × 7 = 35, and the ratio 35:14 simplifies to 5:2. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 4:3, and the first number of the ratio that matches 24. Keep the numbers in the order of the ratio.
   4:3, and 24 matches 4
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   24 ÷ 4 = 6
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   3 × 6 = 18
4. Write the answer.
   18

Check. 4 × 6 = 24, and the ratio 24:18 simplifies to 4:3. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 2:5, and the second number of the ratio that matches 40. Keep the numbers in the order of the ratio.
   2:5, and 40 matches 5
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   40 ÷ 5 = 8
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   2 × 8 = 16
4. Write the answer.
   16

Check. 5 × 8 = 40, and the ratio 16:40 simplifies to 2:5. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 2:7, and the second number of the ratio that matches 14. Keep the numbers in the order of the ratio.
   2:7, and 14 matches 7
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   14 ÷ 7 = 2
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   2 × 2 = 4
4. Write the answer.
   4

Check. 7 × 2 = 14, and the ratio 4:14 simplifies to 2:7. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 3:7, and the second number of the ratio that matches 42. Keep the numbers in the order of the ratio.
   3:7, and 42 matches 7
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   42 ÷ 7 = 6
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   3 × 6 = 18
4. Write the answer.
   18

Check. 7 × 6 = 42, and the ratio 18:42 simplifies to 3:7. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Simplify both ratios. If the simplest forms are the same, the ratios are equivalent.

1. Simplify 12:18. Divide both numbers by their HCF, 6.
   12:18 = 2:3
2. Simplify 6:8. Divide both numbers by their HCF, 2.
   6:8 = 3:4
3. Compare the two simplest forms. The same simplest form means the ratios are equivalent.
   2:3 and 3:4 are different
4. Write the answer.
   No

Check. 12 × 8 = 96 and 18 × 6 = 108, so the cross-products are not equal. ✓

Watch out. Compare the simplest forms, not the first numbers.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 5:6, and the second number of the ratio that matches 48. Keep the numbers in the order of the ratio.
   5:6, and 48 matches 6
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   48 ÷ 6 = 8
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   5 × 8 = 40
4. Write the answer.
   40

Check. 6 × 8 = 48, and the ratio 40:48 simplifies to 5:6. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale from the second number, then find the first number. Check the other option.

1. Find the scale: 27 ÷ 9. The 27 matches the 9 in the ratio 4:9.
   27 ÷ 9 = 3
2. Multiply the first number by 3. The first number is 4 parts.
   4 × 3 = 12
3. Check the other option: could 27 be the first number? Then the second number would be 27 × 9 ÷ 4, which is not whole.
   27 × 9 ÷ 4 = 60.75
4. Write the answer. The other number must be 12.
   No. The other number would be 12, not 13.

Check. 4:9 is the same as 12:27, because 12 ÷ 4 = 3 and 27 ÷ 9 = 3. ✓

Watch out. Check which number the given value matches before you find the other number.

Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.

Method. Multiply the old length by the scale factor.

1. Write the scale factor. It tells us what to multiply by.
   scale factor 3
2. Multiply the old length by the scale factor. A scale factor below 1 makes the shape smaller.
   5 × 3 = 15
3. Write the answer.
   15

Check. 15 ÷ 3 = 5. ✓

Watch out. Multiply the length by the scale factor. Do not add it.

Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.

Method. The scale factor is new length divided by old length.

1. Write the old length and the new length that match. Match the sides that are the same.
   6 cm → 24 cm
2. Divide the new length by the old length. This gives the scale factor.
   24 ÷ 6 = 4
3. Write the answer.
   4

Check. 6 × 4 = 24. ✓

Watch out. Divide the new length by the old length, not the other way round.

Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.

Method. The scale factor is new length divided by old length.

1. Write the old length and the new length that match. Match the sides that are the same.
   9 cm → 3 cm
2. Divide the new length by the old length. This gives the scale factor.
   3 ÷ 9 = 1/3
3. Write the answer.
   1/3

Check. 9 × 1/3 = 3. ✓

Watch out. Divide the new length by the old length, not the other way round.

Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.

Method. Find the scale factor from one matching pair, then scale the other length.

1. Find the scale factor: divide the new length by the old length. The scale factor multiplies every length.
   45 ÷ 15 = 3
2. Multiply the other old length by the scale factor. Every length in a similar shape is scaled in the same way.
   10 × 3 = 30
3. Write the answer.
   30

Check. 30 ÷ 3 = 10, so the scale factor is right. ✓

Watch out. Use the scale factor on every length, not only on the side you were given.

Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.

Method. The scale factor is new length divided by old length.

1. Write the old length and the new length that match. Match the sides that are the same.
   6 cm → 450 cm
2. Divide the new length by the old length. This gives the scale factor.
   450 ÷ 6 = 75
3. Write the answer.
   75

Check. 6 × 75 = 450. ✓

Watch out. Divide the new length by the old length, not the other way round.

Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.

Method. Find the scale factor from one matching pair, then scale the other length.

1. Find the scale factor: divide the new length by the old length. The scale factor multiplies every length.
   12 ÷ 4 = 3
2. Multiply the other old length by the scale factor. Every length in a similar shape is scaled in the same way.
   9 × 3 = 27
3. Write the answer.
   27

Check. 27 ÷ 3 = 9, so the scale factor is right. ✓

Watch out. Use the scale factor on every length, not only on the side you were given.

Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.

Method. Find the scale factor for the width and for the height. If they match, the shapes are similar.

1. Find the scale factor for the width: the new width divided by the old width. Matching sides give the scale factor.
   6 ÷ 2 = 3
2. Find the scale factor for the height: the new height divided by the old height. The shapes are similar only if this is the same.
   15 ÷ 5 = 3
3. Compare the two scale factors. They are both 3, so the sides are in the same ratio.
   3 = 3
4. Write the answer.
   Yes. Both lengths are multiplied by 3.

Check. 2 × 3 = 6 and 5 × 3 = 15. ✓

Watch out. Check both sides, not just one.

Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.

Method. Find the scale factor for the width and for the height. If they are different, the shapes are not similar.

1. Find the scale factor for the width: the new width divided by the old width. Matching sides give the scale factor.
   4 ÷ 3 = 4/3
2. Find the scale factor for the height: the new height divided by the old height. The shapes are similar only if this is the same.
   10 ÷ 8 = 1.25
3. Compare the two scale factors. They are different, so the sides do not grow in the same way.
   4/3 ≠ 1.25
4. Write the answer.
   No. The scale factors are different: 4/3 and 5/4.

Check. Using 3 and 4 gives 4/3, but 8 and 10 gives 5/4. ✓

Watch out. The scale factors must be equal for the shapes to be similar.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Share 40 into 8 equal parts, then give each share its number of parts.

1. Add the parts of the ratio to find the total number of parts. Each number in the ratio counts equal parts.
   3 + 5 = 8 parts
2. Divide the total by 8 to find the value of one part. Every part is the same size.
   40 ÷ 8 = 5
3. Multiply each number in the ratio by the value of one part. Each share is its number of parts.
   3 × 5 = 15; 5 × 5 = 25
4. Check: the shares add up to the total. If they do not, check the working.
   15 + 25 = 40
5. Write the share that the question asks for: Second. Second gets 25.
   25

Check. The ratio of the shares is 3:5, and they add up to 40. ✓

Watch out. Find the value of one part first. Do not share the total equally.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Share 90 into 9 equal parts, then give each share its number of parts.

1. Add the parts of the ratio to find the total number of parts. Each number in the ratio counts equal parts.
   4 + 5 = 9 parts
2. Divide the total by 9 to find the value of one part. Every part is the same size.
   90 ÷ 9 = 10
3. Multiply each number in the ratio by the value of one part. Each share is its number of parts.
   4 × 10 = 40; 5 × 10 = 50
4. Check: the shares add up to the total. If they do not, check the working.
   40 + 50 = 90
5. Write the share that the question asks for: Ben. Ben gets 50.
   50

Check. The ratio of the shares is 4:5, and they add up to 90. ✓

Watch out. Find the value of one part first. Do not share the total equally.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Share 36 into 9 equal parts, then give each share its number of parts.

1. Add the parts of the ratio to find the total number of parts. Each number in the ratio counts equal parts.
   5 + 4 = 9 parts
2. Divide the total by 9 to find the value of one part. Every part is the same size.
   36 ÷ 9 = 4
3. Multiply each number in the ratio by the value of one part. Each share is its number of parts.
   5 × 4 = 20; 4 × 4 = 16
4. Check: the shares add up to the total. If they do not, check the working.
   20 + 16 = 36
5. Write the share that the question asks for: Ava. Ava gets 20.
   20

Check. The ratio of the shares is 5:4, and they add up to 36. ✓

Watch out. Find the value of one part first. Do not share the total equally.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Share 15 into 5 equal parts, then give each share its number of parts.

1. Add the parts of the ratio to find the total number of parts. Each number in the ratio counts equal parts.
   2 + 3 = 5 parts
2. Divide the total by 5 to find the value of one part. Every part is the same size.
   15 ÷ 5 = 3
3. Multiply each number in the ratio by the value of one part. Each share is its number of parts.
   2 × 3 = 6; 3 × 3 = 9
4. Check: the shares add up to the total. If they do not, check the working.
   6 + 9 = 15
5. Write the share that the question asks for: white. white gets 9.
   9

Check. The ratio of the shares is 2:3, and they add up to 15. ✓

Watch out. Find the value of one part first. Do not share the total equally.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Find the value of one part from the known share, then find the amount asked for.

1. Find the value of one part: 14 ÷ 2. The known share is some number of parts.
   14 ÷ 2 = 7
2. Find Ben. Ben has 5 parts.
   7 × 5 = 35
3. Write the answer.
   35

Check. 2 parts is 14, so one part is 7. ✓

Watch out. Find the value of one part first, then multiply. Do not divide the total by 2.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Find the value of one part from the known share, then find the amount asked for.

1. Find the value of one part: 14 ÷ 2. The known share is some number of parts.
   14 ÷ 2 = 7
2. Find the total. The ratio has 7 parts altogether.
   7 × 7 = 49
3. Write the answer.
   49

Check. 2 parts is 14, so one part is 7. ✓

Watch out. Find the value of one part first, then multiply. Do not divide the total by 2.

Key words. A ratio is made of parts. Every part is the same size. The total is all the parts added together.

Method. Find the total parts, then the fraction that Sam has.

1. Add the parts of the ratio: 1 + 2. The total number of parts is 3.
   1 + 2 = 3 parts
2. Sam has 2 of the 3 parts. So Sam has 2 out of 3 equal parts of the money.
   2 parts out of 3
3. Write the answer as a fraction. The fraction is the parts Sam has over the total parts.
   2/3

Check. 1/3 + 2/3 = 3/3, which is the whole amount. ✓

Watch out. The fraction uses the parts of the ratio, not the money. Sam has the bigger share, so the fraction is bigger than 1/2.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Share 50 into 5 equal parts, then give each share its number of parts.

1. Add the parts of the ratio to find the total number of parts. Each number in the ratio counts equal parts.
   2 + 3 = 5 parts
2. Divide the total by 5 to find the value of one part. Every part is the same size.
   50 ÷ 5 = 10
3. Multiply each number in the ratio by the value of one part. Each share is its number of parts.
   2 × 10 = 20; 3 × 10 = 30
4. Check: the shares add up to the total. If they do not, check the working.
   20 + 30 = 50
5. Write the share that the question asks for: Big share. Big share gets 30.
   30

Check. The ratio of the shares is 2:3, and they add up to 50. ✓

Watch out. Find the value of one part first. Do not share the total equally.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Check the shares: they must be in the ratio 1:2 and add up to 36.

1. Find the total parts: 1 + 2 = 3. Each share is a number of these parts.
   1 + 2 = 3 parts
2. Find one part: 36 ÷ 3. One part is the total divided by the number of parts.
   36 ÷ 3 = 12
3. Find the two shares: 1 × 12 and 2 × 12. Each share is its number of parts.
   1 × 12 = 12, and 2 × 12 = 24
4. Write the answer. The shares add up to 36, so Sam is right.
   Yes. 12 and 24 are in the ratio 1:2, and 12 + 24 = 36.

Check. 12 + 24 = 36, and 24 ÷ 12 = 2. ✓

Watch out. Check that the shares add up to the total before you agree.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Add the parts, then check whether 50 divides exactly into that number of parts.

1. Add the parts: 2 + 3 + 4. The total number of parts is 9.
   2 + 3 + 4 = 9 parts
2. Divide 50 by 9. For the shares to be whole numbers, 9 must divide exactly into 50.
   50 ÷ 9 = 5 remainder 5
3. Write the answer. There is a remainder, so the shares are not whole numbers.
   No. 50 is not a multiple of 9.

Check. 9 × 5 = 45, and 50 - 45 = 5 is left over. ✓

Watch out. The total must be a multiple of the number of parts.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Find the difference in parts, then the value of one part, then the total.

1. Find the difference in parts: 7 − 3 = 4. Ben has 4 more parts than Adam.
   7 − 3 = 4 parts
2. Find one part: 36 ÷ 4. The difference of £36 is 4 parts.
   36 ÷ 4 = 9
3. Find the total number of parts: 3 + 7 = 10. The total is made of 10 parts.
   3 + 7 = 10 parts
4. Find the total money: 10 × 9. Each part is £9.
   10 × 9 = 90
5. Write the answer.
   90

Check. Adam has 3 × 9 = 27 and Ben has 7 × 9 = 63. 63 - 27 = 36, and 27 + 63 = 90. ✓

Watch out. The difference is in parts, not in pounds. Find one part first.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Share 48 into 12 equal parts, then give each share its number of parts.

1. Add the parts of the ratio to find the total number of parts. Each number in the ratio counts equal parts.
   5 + 7 = 12 parts
2. Divide the total by 12 to find the value of one part. Every part is the same size.
   48 ÷ 12 = 4
3. Multiply each number in the ratio by the value of one part. Each share is its number of parts.
   5 × 4 = 20; 7 × 4 = 28
4. Check: the shares add up to the total. If they do not, check the working.
   20 + 28 = 48
5. Write the share that the question asks for: Max. Max gets 28.
   28

Check. The ratio of the shares is 5:7, and they add up to 48. ✓

Watch out. Find the value of one part first. Do not share the total equally.

Key words. A ratio is made of parts. Every part is the same size. The total is all the parts added together.

Method. Find the number of groups, then the amount in one group, then multiply.

1. Find the number of groups: divide 20 by the group size, 4. The groups must fill exactly.
   20 ÷ 4 = 5 groups
2. Find how many red pens are in one group. The ratio parts add up to 4, so each group has 4 parts.
   4 ÷ 4 = 1 per part, and 1 × 3 = 3
3. Multiply by the number of groups, 5. Every group has the same amount.
   3 × 5 = 15
4. Write the answer.
   15

Check. 5 × 4 = 20. ✓

Watch out. Check that the groups fill exactly. A group must be the whole ratio, such as 2 + 3 = 5 parts.

Key words. A ratio is made of parts. Every part is the same size. The total is all the parts added together.

Method. Find how many are in one group, then multiply by the number of groups.

1. Write how many yellow tiles are in one group. The ratio part is the number in one group.
   5 in one group
2. Multiply by the number of groups, 7. Every group has the same amount.
   5 × 7 = 35
3. Write the answer.
   35

Check. 7 groups of 5 gives 35. ✓

Watch out. Multiply by the number of groups. Do not add the ratio parts.

Key words. A ratio is made of parts. Every part is the same size. The total is all the parts added together.

Method. Find the number of groups, then the amount in one group, then multiply.

1. Find the number of groups: divide 56 by the group size, 8. The groups must fill exactly.
   56 ÷ 8 = 7 groups
2. Find how many chocolate cakes are in one group. The ratio parts add up to 8, so each group has 8 parts.
   8 ÷ 8 = 1 per part, and 1 × 3 = 3
3. Multiply by the number of groups, 7. Every group has the same amount.
   3 × 7 = 21
4. Write the answer.
   21

Check. 7 × 8 = 56. ✓

Watch out. Check that the groups fill exactly. A group must be the whole ratio, such as 2 + 3 = 5 parts.

Key words. A ratio is made of parts. Every part is the same size. The total is all the parts added together.

Method. Find the number of groups, then the amount in one group, then multiply.

1. Find the number of groups: divide 60 by the group size, 6. The groups must fill exactly.
   60 ÷ 6 = 10 groups
2. Find how many fruity sweets are in one group. The ratio parts add up to 3, so each group has 3 parts.
   6 ÷ 3 = 2 per part, and 2 × 2 = 4
3. Multiply by the number of groups, 10. Every group has the same amount.
   4 × 10 = 40
4. Write the answer.
   40

Check. 10 × 6 = 60. ✓

Watch out. Check that the groups fill exactly. A group must be the whole ratio, such as 2 + 3 = 5 parts.

Key words. A ratio is made of parts. Every part is the same size. The total is all the parts added together.

Method. Find the number of groups, then the amount in one group, then multiply.

1. Find the number of groups: divide 20 by the group size, 5. The groups must fill exactly.
   20 ÷ 5 = 4 groups
2. Find how many blue counters are in one group. The ratio parts add up to 5, so each group has 5 parts.
   5 ÷ 5 = 1 per part, and 1 × 3 = 3
3. Multiply by the number of groups, 4. Every group has the same amount.
   3 × 4 = 12
4. Write the answer.
   12

Check. 4 × 5 = 20. ✓

Watch out. Check that the groups fill exactly. A group must be the whole ratio, such as 2 + 3 = 5 parts.

Key words. A ratio is made of parts. Every part is the same size. The total is all the parts added together.

Method. Find the number of groups, then the amount in one group, then multiply.

1. Find the number of groups: divide 80 by the group size, 8. The groups must fill exactly.
   80 ÷ 8 = 10 groups
2. Find how many girls are in one group. The ratio parts add up to 4, so each group has 4 parts.
   8 ÷ 4 = 2 per part, and 2 × 1 = 2
3. Multiply by the number of groups, 10. Every group has the same amount.
   2 × 10 = 20
4. Write the answer.
   20

Check. 10 × 8 = 80. ✓

Watch out. Check that the groups fill exactly. A group must be the whole ratio, such as 2 + 3 = 5 parts.

Key words. A ratio is made of parts. Every part is the same size. The total is all the parts added together.

Method. Find the number of groups, then the amount in one group, then multiply.

1. Find the number of groups: divide 90 by the group size, 9. The groups must fill exactly.
   90 ÷ 9 = 10 groups
2. Find how many red pencils are in one group. The ratio parts add up to 9, so each group has 9 parts.
   9 ÷ 9 = 1 per part, and 1 × 4 = 4
3. Multiply by the number of groups, 10. Every group has the same amount.
   4 × 10 = 40
4. Write the answer.
   40

Check. 10 × 9 = 90. ✓

Watch out. Check that the groups fill exactly. A group must be the whole ratio, such as 2 + 3 = 5 parts.

Key words. A ratio is made of parts. Every part is the same size. The total is all the parts added together.

Method. Find the number of groups, then the amount in one group, then multiply.

1. Find the number of groups: divide 24 by the group size, 6. The groups must fill exactly.
   24 ÷ 6 = 4 groups
2. Find how many boys are in one group. The ratio parts add up to 3, so each group has 3 parts.
   6 ÷ 3 = 2 per part, and 2 × 1 = 2
3. Multiply by the number of groups, 4. Every group has the same amount.
   2 × 4 = 8
4. Write the answer.
   8

Check. 4 × 6 = 24. ✓

Watch out. Check that the groups fill exactly. A group must be the whole ratio, such as 2 + 3 = 5 parts.

Key words. A ratio is made of parts. Every part is the same size. The total is all the parts added together.

Method. Find the red sweets in one pack, then in 3 packs.

1. Find the red sweets in one pack. The ratio 1:2 shares 6 sweets into 3 parts, so one part is 2 sweets.
   6 ÷ 3 = 2, and 1 × 2 = 2 red
2. Multiply by the number of packs, 3. Every pack has the same number of red sweets.
   2 × 3 = 6 red sweets
3. Compare with 4. Three packs have 6 red sweets, which is not 4.
   6 ≠ 4
4. Write the answer.
   No. 3 packs have 6 red sweets, not 4.

Check. Each pack has 2 red sweets, so 3 packs have 3 × 2 = 6 red sweets. ✓

Watch out. Find the number in one pack first. Then multiply by the number of packs.

Key words. A ratio is made of parts. Every part is the same size. The total is all the parts added together.

Method. Find the number of groups, then the amount in one group, then multiply.

1. Find the number of groups: divide 40 by the group size, 8. The groups must fill exactly.
   40 ÷ 8 = 5 groups
2. Find how many mint sweets are in one group. The ratio parts add up to 8, so each group has 8 parts.
   8 ÷ 8 = 1 per part, and 1 × 3 = 3
3. Multiply by the number of groups, 5. Every group has the same amount.
   3 × 5 = 15
4. Write the answer.
   15

Check. 5 × 8 = 40. ✓

Watch out. Check that the groups fill exactly. A group must be the whole ratio, such as 2 + 3 = 5 parts.

Key words. A ratio is made of parts. Every part is the same size. The total is all the parts added together.

Method. Find the number of groups, then the amount in one group, then multiply.

1. Find the number of groups: divide 60 by the group size, 10. The groups must fill exactly.
   60 ÷ 10 = 6 groups
2. Find how many boys are in one group. The ratio parts add up to 5, so each group has 5 parts.
   10 ÷ 5 = 2 per part, and 2 × 2 = 4
3. Multiply by the number of groups, 6. Every group has the same amount.
   4 × 6 = 24
4. Write the answer.
   24

Check. 6 × 10 = 60. ✓

Watch out. Check that the groups fill exactly. A group must be the whole ratio, such as 2 + 3 = 5 parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the total boys and girls, then simplify the ratio.

1. Find the total number of boys: 10 teams with 4 boys each. Each team has 4 boys.
   10 × 4 = 40 boys
2. Find the total number of girls: 10 teams with 6 girls each. Each team has 6 girls.
   10 × 6 = 60 girls
3. Write the ratio of boys to girls. The order is boys first.
   40:60
4. Simplify by dividing both numbers by their HCF, which is 20. This gives the simplest form.
   40 ÷ 20 = 2, and 60 ÷ 20 = 3
5. Write the answer.
   2:3

Check. 2 × 20 = 40 and 3 × 20 = 60. ✓

Watch out. Simplify the total ratio, not the ratio in one team. Both give 2:3 here.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Divide both numbers by their highest common factor (HCF).

1. Write the ratio in the order the question gives. The order of the numbers must not change.
   15:9
2. Find the highest common factor (HCF) of the two numbers. It is the biggest number that divides into both.
   HCF of 15 and 9 is 3
3. Divide both numbers by the HCF. This keeps the same ratio but uses smaller numbers.
   15 ÷ 3 = 5, and 9 ÷ 3 = 3
4. Write the answer.
   5:3

Check. 5 × 3 = 15 and 3 × 3 = 9. ✓

Watch out. Divide both numbers by the same number. Keep the order the question gives.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 4:6, and the first number of the ratio that matches 10. Keep the numbers in the order of the ratio.
   4:6, and 10 matches 4
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   10 ÷ 4 = 5/2
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   6 × 5/2 = 15
4. Write the answer as a ratio, in the same order as the question. The ratio must keep its order.
   10:15

Check. 4 × 5/2 = 10, and the ratio 10:15 simplifies to 4:6. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 3:7, and the second number of the ratio that matches 21. Keep the numbers in the order of the ratio.
   3:7, and 21 matches 7
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   21 ÷ 7 = 3
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   3 × 3 = 9
4. Write the answer.
   9

Check. 7 × 3 = 21, and the ratio 9:21 simplifies to 3:7. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Share 35 into 7 equal parts, then give each share its number of parts.

1. Add the parts of the ratio to find the total number of parts. Each number in the ratio counts equal parts.
   2 + 5 = 7 parts
2. Divide the total by 7 to find the value of one part. Every part is the same size.
   35 ÷ 7 = 5
3. Multiply each number in the ratio by the value of one part. Each share is its number of parts.
   2 × 5 = 10; 5 × 5 = 25
4. Check: the shares add up to the total. If they do not, check the working.
   10 + 25 = 35
5. Write the share that the question asks for: Second person. Second person gets 25.
   25

Check. The ratio of the shares is 2:5, and they add up to 35. ✓

Watch out. Find the value of one part first. Do not share the total equally.

Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.

Method. The scale factor is new length divided by old length.

1. Write the old length and the new length that match. Match the sides that are the same.
   2 cm → 10 cm
2. Divide the new length by the old length. This gives the scale factor.
   10 ÷ 2 = 5
3. Write the answer.
   5

Check. 2 × 5 = 10. ✓

Watch out. Divide the new length by the old length, not the other way round.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 7:4, and the second number of the ratio that matches 20. Keep the numbers in the order of the ratio.
   7:4, and 20 matches 4
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   20 ÷ 4 = 5
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   7 × 5 = 35
4. Write the answer.
   35

Check. 4 × 5 = 20, and the ratio 35:20 simplifies to 7:4. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio is made of parts. Every part is the same size. The total is all the parts added together.

Method. Find how many are in one group, then multiply by the number of groups.

1. Write how many sour sweets are in one group. The ratio part is the number in one group.
   2 in one group
2. Multiply by the number of groups, 8. Every group has the same amount.
   2 × 8 = 16
3. Write the answer.
   16

Check. 8 groups of 2 gives 16. ✓

Watch out. Multiply by the number of groups. Do not add the ratio parts.

Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.

Method. Multiply the old length by the scale factor.

1. Write the scale factor. It tells us what to multiply by.
   scale factor 1/3
2. Multiply the old length by the scale factor. A scale factor below 1 makes the shape smaller.
   18 × 1/3 = 6
3. Write the answer.
   6

Check. 6 ÷ 1/3 = 18. ✓

Watch out. Multiply the length by the scale factor. Do not add it.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Simplify both ratios. If the simplest forms are the same, the ratios are equivalent.

1. Simplify 3:5. Divide both numbers by their HCF, 1.
   3:5 = 3:5
2. Simplify 12:20. Divide both numbers by their HCF, 4.
   12:20 = 3:5
3. Compare the two simplest forms. The same simplest form means the ratios are equivalent.
   3:5 and 3:5 are the same
4. Write the answer.
   Yes

Check. 3 × 20 = 60 and 5 × 12 = 60, so the cross-products are equal. ✓

Watch out. Compare the simplest forms, not the first numbers.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Share 81 into 9 equal parts, then give each share its number of parts.

1. Add the parts of the ratio to find the total number of parts. Each number in the ratio counts equal parts.
   4 + 5 = 9 parts
2. Divide the total by 9 to find the value of one part. Every part is the same size.
   81 ÷ 9 = 9
3. Multiply each number in the ratio by the value of one part. Each share is its number of parts.
   4 × 9 = 36; 5 × 9 = 45
4. Check: the shares add up to the total. If they do not, check the working.
   36 + 45 = 81
5. Write the share that the question asks for: Smaller share. Smaller share gets 36.
   36

Check. The ratio of the shares is 4:5, and they add up to 81. ✓

Watch out. Find the value of one part first. Do not share the total equally.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Divide both numbers by their highest common factor (HCF).

1. Write the ratio in the order the question gives. The order of the numbers must not change.
   12:20
2. Find the highest common factor (HCF) of the two numbers. It is the biggest number that divides into both.
   HCF of 12 and 20 is 4
3. Divide both numbers by the HCF. This keeps the same ratio but uses smaller numbers.
   12 ÷ 4 = 3, and 20 ÷ 4 = 5
4. Write the answer.
   3:5

Check. 3 × 4 = 12 and 5 × 4 = 20. ✓

Watch out. Divide both numbers by the same number. Keep the order the question gives.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 5:3, and the first number of the ratio that matches 15. Keep the numbers in the order of the ratio.
   5:3, and 15 matches 5
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   15 ÷ 5 = 3
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   3 × 3 = 9
4. Write the answer.
   9

Check. 5 × 3 = 15, and the ratio 15:9 simplifies to 5:3. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Divide both numbers by their highest common factor (HCF).

1. Write the ratio in the order the question gives. The order of the numbers must not change.
   6:15
2. Find the highest common factor (HCF) of the two numbers. It is the biggest number that divides into both.
   HCF of 6 and 15 is 3
3. Divide both numbers by the HCF. This keeps the same ratio but uses smaller numbers.
   6 ÷ 3 = 2, and 15 ÷ 3 = 5
4. Write the answer.
   2:5

Check. 2 × 3 = 6 and 5 × 3 = 15. ✓

Watch out. Divide both numbers by the same number. Keep the order the question gives.

Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.

Method. Find the scale first, then multiply the other number of the ratio by the same scale.

1. Write the ratio 4:6, and the first number of the ratio that matches 10. Keep the numbers in the order of the ratio.
   4:6, and 10 matches 4
2. Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.
   10 ÷ 4 = 5/2
3. Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.
   6 × 5/2 = 15
4. Write the answer.
   15

Check. 4 × 5/2 = 10, and the ratio 10:15 simplifies to 4:6. ✓

Watch out. Find the scale first. Do not add the same number to both parts.

Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.

Method. Find the scale factor from one matching pair, then scale the other length.

1. Find the scale factor: divide the new length by the old length. The scale factor multiplies every length.
   12 ÷ 4 = 3
2. Multiply the other old length by the scale factor. Every length in a similar shape is scaled in the same way.
   7 × 3 = 21
3. Write the answer.
   21

Check. 21 ÷ 3 = 7, so the scale factor is right. ✓

Watch out. Use the scale factor on every length, not only on the side you were given.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Share 56 into 7 equal parts, then give each share its number of parts.

1. Add the parts of the ratio to find the total number of parts. Each number in the ratio counts equal parts.
   3 + 4 = 7 parts
2. Divide the total by 7 to find the value of one part. Every part is the same size.
   56 ÷ 7 = 8
3. Multiply each number in the ratio by the value of one part. Each share is its number of parts.
   3 × 8 = 24; 4 × 8 = 32
4. Check: the shares add up to the total. If they do not, check the working.
   24 + 32 = 56
5. Write the share that the question asks for: Ali. Ali gets 24.
   24

Check. The ratio of the shares is 3:4, and they add up to 56. ✓

Watch out. Find the value of one part first. Do not share the total equally.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Share 36 into 9 equal parts, then give each share its number of parts.

1. Add the parts of the ratio to find the total number of parts. Each number in the ratio counts equal parts.
   4 + 5 = 9 parts
2. Divide the total by 9 to find the value of one part. Every part is the same size.
   36 ÷ 9 = 4
3. Multiply each number in the ratio by the value of one part. Each share is its number of parts.
   4 × 4 = 16; 5 × 4 = 20
4. Check: the shares add up to the total. If they do not, check the working.
   16 + 20 = 36
5. Write the share that the question asks for: Girls. Girls gets 20.
   20

Check. The ratio of the shares is 4:5, and they add up to 36. ✓

Watch out. Find the value of one part first. Do not share the total equally.

Key words. The scale factor is the number that every length is multiplied by. It is bigger than 1 for an enlargement and smaller than 1 for a reduction.

Method. Find the scale factor from one matching pair, then scale the other length.

1. Find the scale factor: divide the new length by the old length. The scale factor multiplies every length.
   9 ÷ 6 = 1.5
2. Multiply the other old length by the scale factor. Every length in a similar shape is scaled in the same way.
   3 × 1.5 = 4.5
3. Write the answer.
   4.5

Check. 4.5 ÷ 1.5 = 3, so the scale factor is right. ✓

Watch out. Use the scale factor on every length, not only on the side you were given.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Share 96 into 8 equal parts, then give each share its number of parts.

1. Add the parts of the ratio to find the total number of parts. Each number in the ratio counts equal parts.
   2 + 3 + 3 = 8 parts
2. Divide the total by 8 to find the value of one part. Every part is the same size.
   96 ÷ 8 = 12
3. Multiply each number in the ratio by the value of one part. Each share is its number of parts.
   2 × 12 = 24; 3 × 12 = 36; 3 × 12 = 36
4. Check: the shares add up to the total. If they do not, check the working.
   24 + 36 + 36 = 96
5. Write the share that the question asks for: Joe. Joe gets 36.
   36

Check. The ratio of the shares is 2:3:3, and they add up to 96. ✓

Watch out. Find the value of one part first. Do not share the total equally.

Key words. To share in a ratio means to split a total into shares that have the same ratio as the numbers given.

Method. Share the money first, then work out what Ali gives to Ben.

1. Find the total parts: 3 + 5 = 8 parts. Find one part: 120 ÷ 8 = 15. The total is shared into 8 equal parts.
   120 ÷ 8 = 15
2. Find Ali and Ben’s shares: Ali 3 × 15 = 45, Ben 5 × 15 = 75. Each share is its number of parts.
   3 × 15 = 45, and 5 × 15 = 75
3. Ali gives 1/3 of £45 to Ben: 45 ÷ 3 = 15. One third of Ali’s money is 15 pounds.
   45 ÷ 3 = 15
4. Add the money to Ben’s share: 75 + 15. Ben now has his own money plus the gift.
   75 + 15 = 90
5. Write the answer.
   90

Check. Ali has 45 - 15 = 30 left, and Ben has 90. Together: 30 + 90 = 120. ✓

Watch out. Find each person’s share before you change the amounts.