Year 6 Ratio and Proportion: check-up and SATs-style questions

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.