Printable · GCSE Foundation · ages 14-16
Ratio notation and simplest form worksheet — GCSE Foundation
Fifteen questions on "ratio notation and simplest form" — DfE statement R4. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
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Ratio notation and simplest form worksheet — GCSE Foundation
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- 1.Write the ratio 27 : 45 in its simplest form.
- 2.A gardener mixes 300 ml of plant feed concentrate with 1.2 litres of water to make a spray. Write the ratio of concentrate to water in its simplest form.
- 3.Simplify the ratio 45 : 30 : 75 to its simplest form.
- 4.Which of these ratios is equivalent to 5 : 4?
- 5.Simplify the ratio 12 : 18 : 30 to its simplest form.
- 6.Write the ratio 2 : 1/4 as a ratio of whole numbers in its simplest form.
- 7.Write 350 ml : 1.4 l as a ratio in its simplest form.
- 8.A class has 28 pupils. 16 of the pupils are girls and the rest are boys. Write the ratio of girls to boys in its simplest form.
- 9.Which of these ratios is equivalent to 6 : 10 : 14?
- 10.A recipe uses flour, sugar and butter in the ratio 8 : 3 : 5. Write the ratio of flour to the rest of the mixture (sugar and butter combined) in its simplest form.
- 11.A mortar mix is made from sand and cement in the ratio 5 : 3. Write the ratio of sand to the total mix in its simplest form.
- 12.Which of these ratios is already written in its simplest form?
- 13.Write the ratio 4x : 6x in its simplest form, where x is a positive number.
- 14.A jar contains 24 green sweets and 16 orange sweets. Write the ratio of green sweets to orange sweets in its simplest form.
- 15.A fruit bowl contains 9 apples and 21 oranges. Write down the ratio of apples to oranges in its simplest form.
Answer key
- (b) 3 : 5 — The highest common factor of 27 and 45 is 9. Divide both parts by 9: 27 ÷ 9 = 3 and 45 ÷ 9 = 5, giving 3 : 5. Giving 9 : 15 divides by 3, a common factor but not the highest one, so it is not yet in simplest form. Giving 5 : 3 swaps the order of the two parts. Giving 27 : 45 has not been simplified at all.
- (c) 1 : 4 — Convert to the same unit: 1.2 litres = 1200 ml, since 1 litre = 1000 ml. This gives the ratio 300 : 1200. Divide both parts by their highest common factor, 300, to get 1 : 4. Giving 3 : 12 divides by 100 only, which is a common factor but not the highest one. Giving 1 : 1.2 has not converted 1.2 litres into millilitres, so the two parts are not in the same unit. Giving 4 : 1 swaps the order, comparing water to concentrate instead of concentrate to water.
- (d) 3 : 2 : 5 — The highest common factor of 45, 30 and 75 is 15. Divide each part by 15: 45 ÷ 15 = 3, 30 ÷ 15 = 2 and 75 ÷ 15 = 5, giving 3 : 2 : 5. Giving 9 : 6 : 15 divides by 5, a common factor but not the highest one. Giving 15 : 10 : 25 divides by 3 only, even further from simplest form. Giving 2 : 3 : 5 has the first two parts swapped.
- (a) 15 : 12 — Multiply both parts of the ratio 5 : 4 by the same number, 3, to get an equivalent ratio: 5 × 3 = 15 and 4 × 3 = 12, giving 15 : 12. Giving 15 : 16 multiplies the two parts by different scale factors (×3 and ×4), which changes the ratio. Giving 9 : 8 adds 4 to each part instead of multiplying, which also changes the ratio. Giving 4 : 5 swaps the order of the two parts.
- (d) 2:3:5 — The highest common factor of 12, 18 and 30 is 6. Divide every part by 6: 12 ÷ 6 = 2, 18 ÷ 6 = 3, 30 ÷ 6 = 5, giving 2 : 3 : 5. Dividing by 2 instead of 6 gives 6 : 9 : 15, which still shares a common factor of 3, so it is not fully simplified. Dividing by 3 instead of 6 gives 4 : 6 : 10, which still shares a common factor of 2, so it is not fully simplified either. Swapping the first two parts gives 3 : 2 : 5, the parts in the wrong order.
- (a) 8:1 — Multiply both parts of the ratio by 4 to clear the fraction: 2 × 4 = 8 and 1/4 × 4 = 1, giving 8 : 1. Getting 1 : 8 has the two parts the wrong way round. Getting 2 : 4 comes from writing down the denominator of the fraction (4) as the second part instead of multiplying through by it. Getting 8 : 4 comes from multiplying only the first part of the ratio by 4 and leaving the second part as the fraction's denominator.
- (d) 1:4 — Convert 1.4 l to millilitres: 1.4 l = 1400 ml. The ratio is 350 : 1400. Divide both parts by 350: 350 ÷ 350 = 1 and 1400 ÷ 350 = 4, giving 1 : 4. Misreading 1.4 l as 14 (moving the decimal point) gives 350 : 14, which simplifies to 25 : 1 — a very different, implausible ratio. Dividing by 175 instead of 350 gives 2 : 8, which still shares a common factor of 2, so it is not fully simplified. Swapping the order gives 4 : 1, litres to millilitres the wrong way round.
- (d) 4:3 — Boys = 28 − 16 = 12. Girls to boys is 16 : 12. The highest common factor of 16 and 12 is 4. Divide both parts by 4: 16 ÷ 4 = 4 and 12 ÷ 4 = 3, so the ratio is 4 : 3. Dividing by 2 instead of 4 gives 8 : 6, which still shares a common factor of 2, so it is not fully simplified. Swapping the order gives 3 : 4, boys to girls instead of girls to boys. Writing girls to the whole class, 16 : 28, simplifies to 4 : 7 — that compares girls to everyone, not girls to boys.
- (c) 9:15:21 — 6 : 10 : 14 simplifies to 3 : 5 : 7 (divide every part by 2). Multiplying every part of 3 : 5 : 7 by 3 gives 9 : 15 : 21, so 9 : 15 : 21 is equivalent to 6 : 10 : 14. Adding 2 to every part of 6 : 10 : 14 gives 8 : 12 : 16, which is not equivalent — ratios are equivalent when every part is multiplied by the same number, not when the same number is added to every part. Doubling only the first two parts, 6 × 2 = 12 and 10 × 2 = 20, but leaving the third part unchanged at 14, gives 12 : 20 : 14 — a scaling applied to two parts and not the third. Cancelling the first two parts correctly, 6 ÷ 2 = 3 and 10 ÷ 2 = 5, then treating the three numbers as a sequence and making the third part the sum of the first two, 3 + 5 = 8, gives 3 : 5 : 8 — the third part was never divided by 2 at all.
- (d) 1 : 1 — Sugar and butter together make 3 + 5 = 8 parts of the mixture. Comparing flour to this, 8 : 8, simplifies to 1 : 1. Giving 1 : 2 compares flour with the whole mixture (8 + 3 + 5 = 16 parts, giving 8 : 16 = 1 : 2) instead of with the rest of the mixture. Giving 3 : 5 is the ratio of sugar to butter, not of flour to the rest of the mixture. Giving 8 : 3 compares flour only with sugar, leaving butter out altogether.
- (b) 5:8 — The mix has 5 parts sand and 3 parts cement, so 5 + 3 = 8 parts in total. Sand to total is 5 : 8, and since the highest common factor of 5 and 8 is 1, this is already in its simplest form. Giving 5 : 3 answers sand to cement, not sand to the total mix. Giving 3 : 8 is cement to total, the wrong part of the mix. Giving 8 : 5 has the total and the sand swapped round.
- (b) 4:9 — 4 : 9 has no common factor other than 1, so it is already in its simplest form. 6 : 8 can be divided by 2 to give 3 : 4, so it is not simplest. 10 : 15 can be divided by 5 to give 2 : 3, so it is not simplest. 7 : 14 can be divided by 7 to give 1 : 2, so it is not simplest.
- (c) 2:3 — Divide both parts of the ratio by their highest common factor, 2x: 4x ÷ 2x = 2 and 6x ÷ 2x = 3, giving 2 : 3. Writing 4 : 6 has not been simplified at all. Writing 2x : 3x has cancelled the common factor of 2 but left the x in, so it is not written as a ratio of whole numbers. Writing 3 : 2 has the two parts the wrong way round.
- (a) 3:2 — There are 24 green sweets and 16 orange sweets. The highest common factor of 24 and 16 is 8. Divide both numbers by 8: 24 ÷ 8 = 3 and 16 ÷ 8 = 2, so the ratio is 3 : 2. Dividing by 4 instead of 8 gives 6 : 4, which still has a common factor of 2, so it is not fully simplified. Writing green sweets to the total number of sweets, 24 : 40, simplifies to 3 : 5 — that compares green to everything, not green to orange, so it answers a different question. Swapping the order gives 2 : 3, green and orange the wrong way round.
- (d) 3 : 7 — There are 9 apples and 21 oranges, giving the ratio 9 : 21. The highest common factor of 9 and 21 is 3: 9 ÷ 3 = 3 and 21 ÷ 3 = 7, so the simplest form is 3 : 7. Giving 9 : 21 has not been simplified. Giving 7 : 3 swaps the order, comparing oranges to apples instead of apples to oranges. Giving 9 : 12 compares the number of apples with the difference between the two amounts (21 − 9 = 12), not with the number of oranges.
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