Simplifying and Using Ratios

Ratios are one of the most frequently examined topics in the Edexcel GCSE Mathematics (1MA1) specification. They appear across all three papers — Paper 1 (non-calculator), Paper 2 (calculator) and Paper 3 (calculator). This lesson covers how to write, simplify and use ratios, including dividing quantities in a given ratio and combining ratios.

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What Is a Ratio?

A ratio compares two or more quantities and shows their relative sizes. Ratios are written using a colon, for example 3 : 5, and they have no units — they simply tell us how many times bigger or smaller one quantity is compared to another.

Concept	Explanation	Example
Ratio	A comparison of two or more quantities	Boys to girls = 3 : 5
Parts	Each number in the ratio represents a "part"	3 parts and 5 parts = 8 parts total
Order	The order of the numbers matters	3 : 5 is NOT the same as 5 : 3
Units	Both quantities must be in the same unit before writing the ratio	Convert cm and m to the same unit first

Writing a Ratio

When writing a ratio, you must ensure the quantities are in the same units.

Worked Example 1

Write the ratio 40 cm to 2 m in its simplest form.

1. Convert to the same units: 2 m = 200 cm
2. Write as a ratio: 40 : 200
3. Simplify by dividing both by the HCF (40): 1 : 5

Edexcel Exam Tip: Always check the units before writing a ratio. A very common mistake is to write 40 : 2 instead of converting metres to centimetres first. On Paper 1 (non-calculator) you must do all simplifying by hand, so look for common factors.

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Simplifying Ratios

Simplifying a ratio works exactly like simplifying a fraction — you divide every part by the highest common factor (HCF).

Method

1. Find the HCF of all the numbers in the ratio.
2. Divide each part by the HCF.
3. The result is the ratio in its simplest form.

Worked Example 2

Simplify 24 : 36.

• HCF of 24 and 36 = 12
• 24 / 12 = 2
• 36 / 12 = 3
• Simplified ratio = 2 : 3

Worked Example 3

Simplify 15 : 25 : 45.

• HCF of 15, 25 and 45 = 5
• 15 / 5 = 3, 25 / 5 = 5, 45 / 5 = 9
• Simplified ratio = 3 : 5 : 9

Simplifying Ratios Involving Fractions and Decimals

If a ratio contains fractions, multiply every part by the lowest common denominator (LCD). If it contains decimals, multiply by a power of 10 to eliminate the decimal places, then simplify.

Worked Example 4

Simplify 0.6 : 1.5.

• Multiply both by 10: 6 : 15
• Divide both by HCF (3): 2 : 5

Worked Example 5

Simplify 1/3 : 1/2.

• LCD of 3 and 2 is 6
• Multiply both by 6: 2 : 3
• Simplified ratio = 2 : 3

Worked Example 6

Simplify 2 : 3/4.

• Multiply both by 4: 8 : 3
• Simplified ratio = 8 : 3

Edexcel Exam Tip: On Paper 1 (non-calculator), ratio-with-fractions questions appear regularly. Multiply through by the LCD to clear all fractions in one step.

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Equivalent Ratios

Two ratios are equivalent if one can be obtained by multiplying (or dividing) every part of the other by the same number. This is identical to the concept of equivalent fractions.

Original Ratio	Multiply by	Equivalent Ratio
2 : 3	x 4	8 : 12
5 : 2	x 3	15 : 6
12 : 8	/ 4	3 : 2

Worked Example 7

The ratio of red beads to blue beads is 3 : 7. If there are 21 blue beads, how many red beads are there?

• 7 parts = 21, so 1 part = 3
• Red beads = 3 parts = 3 x 3 = 9 red beads

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Unit Ratios (1 : n and n : 1)

A unit ratio expresses one part of the ratio as 1. This is useful for comparing and converting scales.

Worked Example 8

Express 8 : 5 in the form 1 : n.

• Divide both by 8: 1 : 5/8 = 1 : 0.625

Worked Example 9

Express 8 : 5 in the form n : 1.

• Divide both by 5: 8/5 : 1 = 1.6 : 1

Edexcel Exam Tip: Map scale questions often require you to express a ratio in the form 1 : n. You may also be asked to find a real-life distance given a map scale — always check the units of your final answer.

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Dividing a Quantity in a Given Ratio

This is one of the most common ratio question types in the Edexcel exam.

Method

1. Add the parts of the ratio to find the total number of parts.
2. Divide the total amount by the total number of parts to find the value of one part.
3. Multiply each part of the ratio by the value of one part.

Worked Example 10

Divide 180 in the ratio 2 : 3 : 4.

1. Total parts = 2 + 3 + 4 = 9
2. One part = 180 / 9 = 20
3. The three shares are: 2 x 20 = 40, 3 x 20 = 60, 4 x 20 = 80

Check: 40 + 60 + 80 = 180

Worked Example 11

Alex and Beth share 240 pounds in the ratio 5 : 7. How much does Beth receive?

1. Total parts = 5 + 7 = 12
2. One part = 240 / 12 = 20
3. Beth's share = 7 x 20 = 140 pounds

Worked Example 12

Two numbers are in the ratio 3 : 8. The difference between them is 30. Find both numbers.

1. Difference in parts = 8 - 3 = 5
2. One part = 30 / 5 = 6
3. Smaller number = 3 x 6 = 18; Larger number = 8 x 6 = 48

Check: 48 - 18 = 30

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Combining Ratios

Sometimes you need to combine two separate ratios into a single ratio.

Worked Example 13

A : B = 2 : 3 and B : C = 4 : 5. Find A : B : C.

1. B appears in both ratios but has different values (3 and 4).
2. Find the LCM of 3 and 4 = 12.
3. Scale A : B so that B = 12: multiply by 4 to give A : B = 8 : 12.
4. Scale B : C so that B = 12: multiply by 3 to give B : C = 12 : 15.
5. Combined ratio: A : B : C = 8 : 12 : 15

Worked Example 14

X : Y = 5 : 3 and Y : Z = 6 : 7. Find X : Y : Z.

1. LCM of 3 and 6 = 6.
2. X : Y = 10 : 6 and Y : Z = 6 : 7.
3. Combined ratio: X : Y : Z = 10 : 6 : 7

Edexcel Exam Tip: Combining ratios questions are typically worth 3-4 marks and appear on both Foundation and Higher papers. Always make the common term equal before combining.

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Practice Problems

1. Simplify the ratio 36 : 48. Answer: 3 : 4
2. Write 0.75 : 1.25 in its simplest form. Answer: 3 : 5
3. Simplify 2/5 : 3/4. Answer: 8 : 15
4. Divide 450 in the ratio 2 : 3 : 4. Answer: 100, 150, 200
5. P : Q = 3 : 4 and Q : R = 6 : 5. Find P : Q : R. Answer: 9 : 12 : 10
6. The ratio of cats to dogs at a shelter is 5 : 8. There are 40 dogs. How many cats are there? Answer: 25
7. Two amounts of money are in the ratio 7 : 3. The larger amount is 28 pounds more than the smaller. What is the total? Answer: 70 pounds
8. Express the ratio 4 : 9 in the form 1 : n. Answer: 1 : 2.25

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Common Mistakes

Mistake	How to Avoid It
Forgetting to convert to the same units	Always check units before writing a ratio
Dividing by different numbers when simplifying	Always use the same HCF for all parts
Getting the order wrong in the ratio	Read the question carefully — the order matters
Using the difference instead of the total when dividing in a ratio	Re-read whether the question gives a total amount or a difference
Not simplifying fully	Check there is no further common factor

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Summary

• A ratio compares quantities using the same units.
• Simplify by dividing all parts by the HCF.
• Equivalent ratios are found by multiplying or dividing all parts by the same number.
• Unit ratios (1 : n or n : 1) are useful for scales and comparisons.
• Divide in a ratio by finding the value of one part first.
• Combine ratios by making the common term equal using the LCM.
• Always show your working and check your answer adds back to the original total.

Edexcel Exam Tip: Ratio questions can appear as part of bigger problems involving fractions, percentages or algebra. Be ready to use ratio skills alongside other topics.

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Extended Worked Examples

The following examples cover the full range of ratio techniques you may meet on Edexcel Paper 1, Paper 2 or Paper 3. Work through each one in your notebook without peeking at the solution, then compare.

Worked Example A — Sharing in a Given Ratio

Alex, Bea and Chen share $\pounds 840$£840 in the ratio $3 : 4 : 5$3:4:5. Calculate how much each person receives.

1. Find the total number of parts: $3 + 4 + 5 = 12$3+4+5=12 parts.
2. Find the value of one part: $\dfrac{840}{12} = \pounds 70$12840​=£70.
3. Multiply each share by the value of one part:
   • Alex: $3 \times 70 = \pounds 210$3×70=£210
   • Bea: $4 \times 70 = \pounds 280$4×70=£280
   • Chen: $5 \times 70 = \pounds 350$5×70=£350
4. Check: $210 + 280 + 350 = 840$210+280+350=840. Correct.

Head of Maths tip: The check step is not optional. If the three shares do not add back to the original total, a single arithmetic slip has occurred — and the whole question is salvageable by re-doing only one multiplication.

Worked Example B — "Given the Difference" Ratio Problem

Priya and Quinn share a prize in the ratio $7 : 4$7:4. Priya receives $\pounds 126$£126 more than Quinn. Work out the total prize.

1. The difference between Priya's and Quinn's parts is $7 - 4 = 3$7−4=3 parts.
2. Those 3 parts are worth $\pounds 126$£126, so one part $= \dfrac{126}{3} = \pounds 42$=3126​=£42.
3. Total parts in the ratio $= 7 + 4 = 11$=7+4=11.
4. Total prize $= 11 \times 42 = \pounds 462$=11×42=£462.

Common misread: Many students assume the $\pounds 126$£126 is the total. Read the question again — it is the difference between the two shares, not the sum.

Worked Example C — Scale Drawings and Maps

A map is drawn to a scale of $1 : 25,000$1:25,000. Two villages are $8.4$8.4 cm apart on the map. What is the real distance between them, in kilometres?

1. Real distance in cm $= 8.4 \times 25,000 = 210,000$=8.4×25,000=210,000 cm.
2. Convert cm to metres: $210,000 \div 100 = 2,100$210,000÷100=2,100 m.
3. Convert metres to kilometres: $2,100 \div 1,000 = 2.1$2,100÷1,000=2.1 km.

Answer: $2.1$2.1 km.

A helpful rule for $1 : 25,000$1:25,000 maps is that $1$1 cm on the map represents $250$250 m in reality. So $8.4 \text{ cm} \times 250 \text{ m} = 2,100$8.4 cm×250 m=2,100 m, giving the same $2.1$2.1 km.

Worked Example D — Combining Two Ratios

In a mixed netball club, the ratio of juniors to seniors is $5 : 3$5:3, and among the juniors the ratio of girls to boys is $4 : 1$4:1. If there are $64$64 juniors, how many senior members are there, and how many junior boys?

1. Juniors : Seniors $= 5 : 3$=5:3. The $5$5 parts represent $64$64 juniors, so one part $= \dfrac{64}{5} = 12.8$=564​=12.8.
2. Seniors $= 3 \times 12.8 = 38.4$=3×12.8=38.4. Since we cannot have a fractional member, we conclude the ratio cannot correspond to exactly $64$64 juniors. Alternatively, interpret "ratio $5 : 3$5:3" as meaning juniors are a multiple of $5$5 — then $64$64 juniors is inconsistent, and the question would need the ratio to be re-scaled.

Head of Maths note: This kind of worked example is valuable because it teaches pupils to recognise impossible ratios. Edexcel examiners sometimes include a "show that the ratio cannot be exact" style question. The reasoning — that shares in a ratio must be whole numbers of a single part — is the mark-winner.

Worked Example E — Ratio Problems with Unit Conversion

The masses of copper and zinc in a brass alloy are in the ratio $3 : 2$3:2. A rod of brass has a mass of $750$750 g. Calculate the mass of zinc in kilograms.

1. Total parts $= 3 + 2 = 5$=3+2=5.
2. One part $= \dfrac{750}{5} = 150$=5750​=150 g.
3. Zinc $= 2 \times 150 = 300$=2×150=300 g.
4. Convert to kg: $300 \text{ g} = 0.3$300 g=0.3 kg.

Answer: $0.3$0.3 kg.

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Answering at different grade levels

Exam-style question: Samira and Tomas share $\pounds 924$£924 in the ratio $5 : 6$5:6. How much more does Tomas receive than Samira? (3 marks)

Grade 3–4 response: "I added $5 + 6 = 11$5+6=11. I did $924 \div 11 = 84$924÷11=84. Samira got $5 \times 84 = \pounds 420$5×84=£420 and Tomas got $6 \times 84 = \pounds 504$6×84=£504. The difference is $504 - 420 = \pounds 84$504−420=£84."

Teacher feedback: Correct method, clearly communicated, full marks. Uses the precise term "ratio" but does not yet label the strategy as "one part = value".

Grade 5–6 response: "Using the ratio $5 : 6$5:6, there are $11$11 equal parts. One part has value $\pounds 924 \div 11 = \pounds 84$£924÷11=£84. The difference between the two shares is $6 - 5 = 1$6−5=1 part, which is $\pounds 84$£84."

Teacher feedback: This candidate has spotted the shortcut — the difference in parts is $1$1, so the money difference equals one part. Secure method marks and saves time.

Grade 7–9 response: "Let one part of the ratio have value $p$p. Then Samira receives $5p$5p and Tomas receives $6p$6p. The total satisfies $11p = 924$11p=924, giving $p = 84$p=84. The difference $(6p - 5p) = p = \pounds 84$(6p−5p)=p=£84. This uses the fact that the difference between consecutive ratio shares is always equal to the unit part, which generalises to any ratio of the form $n : (n+1)$n:(n+1)."

Teacher feedback: Algebraic framing, correct use of terminology, and a generalisation that demonstrates deep understanding. This is the response that will also help the candidate on harder combined-ratio questions later in the paper.

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Edexcel alignment: This content is aligned with Edexcel GCSE Mathematics (1MA1) specification — specifically Topic R [R1 Ratio notation, R2 Division of a quantity in a given ratio, R3 Relating ratios to fractions, R5 Scale drawings and maps]. Assessed on Papers 1, 2, 3.