Number - Fractions, decimals and percentages: identify and work with fractions in ratio problems

Overview

A ratio describes relative amounts, not automatically fractions of a whole. If red counters to blue counters are in the ratio 3:5, red represents three equal parts and blue represents five equal parts. The whole therefore contains eight parts. Red is 3/8 of all the counters, but 3/5 of the number of blue counters. Knowing which comparison a question asks for is the central skill in this topic.

Pearson Edexcel assesses this under N11: identify and work with fractions in ratio problems, with close links to R3, R4, R5, R6 and R8. The ideas apply at both Foundation and Higher tier. Candidates may identify a fraction, share a total, reconstruct a missing total from one share, or use the difference between shares. More demanding questions combine these operations with percentages, units or another ratio. 1 2

The reliable approach is to label the quantities, identify what the given number represents, find the value of one part, and build the required amount. This links fractions to proportional reasoning, probability and algebra. Clear working makes the method visible to the examiner and helps preserve available method or process marks if an arithmetic slip occurs. 3

Key Concepts

1. Ratio parts are equal units

In red:blue = 3:5, a part is an equal-sized unit. It might represent one counter, five counters or twenty counters. The ratio does not specify which until an actual amount is given. Writing red = 3k and blue = 5k captures this: k is the value of one part.

For example, 15 red counters and 25 blue counters fit the ratio because both quantities are five times their ratio parts. So do 6 red and 10 blue. Adding the same number to both quantities usually changes the ratio; multiplying both by the same positive number preserves it.

Order matters. Red:blue = 3:5 becomes blue:red = 5:3. Write the names above your ratio before calculating, especially when the question asks for the second quantity first.

Cover and recall: If apples:pears = 4:7, which fruit has more pieces? What does one part mean? Reveal: pears; one equal-sized unit shared by both quantities.

2. Part to whole is not part to part

For red:blue = 3:5, add the parts when comparing either colour with the whole: 3 + 5 = 8. The fraction red is 3/8; the fraction blue is 5/8. These fractions add to 1 because the two colours account for everything.

When comparing red with blue alone, do not add the parts. The number of red counters is 3/5 of the number of blue counters. Conversely, blue is 5/3 of red. A part-to-part fraction can exceed 1 because one share can be larger than another. A fraction of a non-negative whole cannot exceed 1.

The diagram's upper bar contains eight equal sections. The lower bars line up the two shares, so their two-part difference is visible. Colours help, but the labels and equal sections carry the mathematical meaning.

Why does this work? Fractions compare a selected amount with a reference amount. In 'fraction of all counters', the reference is eight parts. In 'fraction of blue counters', it is five parts. The numerator remains three; the denominator changes because the comparison changes.

3. Sharing a known total

Suppose forty counters are divided red:blue = 3:5. Eight parts represent forty counters, so one part is 40 ÷ 8 = 5. Red is 3 × 5 = 15 and blue is 5 × 5 = 25.

The fraction method gives exactly the same answers: red = (3/8) × 40 = 15 and blue = (5/8) × 40 = 25. The methods are equivalent because dividing by eight finds one part, then multiplying by three selects three parts.

Write at least one complete calculation rather than an unexplained pair of numbers. In a multi-step problem, candidates should identify the relevant total first. If there are other categories outside the named ratio, subtract them before sharing the remaining amount.

Check: 15 + 25 = 40 and 15:25 simplifies to 3:5. A sum check alone is insufficient: 20 and 20 also add to forty, but give the wrong ratio. A check is useful, not automatically a separate awarded mark.

4. Working backwards from one share

If red:blue = 3:5 and there are twenty-five blue counters, the twenty-five represents five parts, not eight. One part is 25 ÷ 5 = 5. Red is 3 × 5 = 15, and the total is 8 × 5 = 40.

The common error is 25 ÷ 8, treating a known share as the whole. Avoid it by writing 'blue: 5 parts = 25' before dividing. You can also use red = (3/5) × blue, provided you recognise that 3/5 compares red with blue rather than with the total.

What if? If blue doubled while the ratio remained unchanged, red and the total would also double. If blue doubled but red stayed fixed, the ratio would change.

5. Working backwards from a difference

If red:blue = 3:5 and blue exceeds red by twelve counters, the difference represents 5 − 3 = 2 parts. One part is 12 ÷ 2 = 6. Red is 18, blue is 30, and the total is 48.

Do not share twelve in the ratio 3:5. Twelve is the gap between the shares, not the combined amount. Use the sentence 'gap in quantities matches gap in parts' to select the correct divisor.

With three shares, use only the two shares mentioned to interpret a difference. For A:B:C = 2:3:7 and C exceeds A by thirty, the gap is 7 − 2 = 5 parts. One part is 6, so B is 18 and the total is 72. The middle share does not contribute to the C-minus-A difference.

Cover and recall: For 4:9 and a difference of £35, find one part without looking back. Reveal: £35 ÷ (9 − 4) = £7.

6. Converting a fraction of a whole into a ratio

If 2/7 of a group are left-handed and everyone else is right-handed, seven parts describe the whole. Two belong to the first category and 7 − 2 = 5 to the second. Therefore left-handed:right-handed = 2:5, not 2:7.

In general, if a fraction p/q belongs to A and the rest to B, A:B = p:(q − p). This requires two categories that are mutually exclusive and together exhaust the whole. If some people are in neither category, or a third category exists, the remainder is not automatically B.

A decimal or percentage can be converted first. If 35% prefer tea and the rest prefer coffee, tea:coffee = 35:65 = 7:13. As a fraction, 35% = 7/20, leaving 13/20. Both routes give the same ratio.

7. Ratios whose entries are fractions

The ratio 1/2:3/4 is not yet in simplest whole-number form. Multiply both entries by four, the lowest common multiple of the denominators: (1/2) × 4:(3/4) × 4 = 2:3.

This multiplication preserves the relationship because every entry receives the same scale factor. It does not mean that 1/2 and 3/4 are individually the fractions of the final mixture. After simplification, the first share is 2/5 of the whole and the second is 3/5.

For 2/3:5/6, multiply both entries by six to obtain 4:5. If a total is ninety, there are nine parts, one part is ten, and the shares are forty and fifty. An unsimplified equivalent ratio can still give a valid method, but simplest whole numbers reduce arithmetic errors and satisfy instructions to simplify.

8. Fractions, decimals and percentages

Once the fraction of the whole is correct, convert normally. For a 3:5 ratio, the first share is 3/8 = 0.375 = 37.5% of the whole. The second is 5/8 = 0.625 = 62.5%. These percentages add to 100%.

The first share is also 3/5 = 60% of the second share. This is not a contradiction: the reference amount differs. Always finish the sentence 'percentage of ...' before dividing.

For three categories in the ratio 2:3:5, the denominator for fractions of the whole is ten. Their fractions are 2/10, 3/10 and 5/10. Simplify individual fractions where appropriate, but preserve the labelled ratio order.

Mathematical Relationships

All relationships below: Must memorise. Here this means know or reconstruct the method. These elementary ratio relationships are not listed on Pearson's published formula aid; the existence of a formula sheet does not remove the need to understand them. Pearson has confirmed formula-aid support for 2025, 2026 and 2027. 4 5

For A:B = a:b, let T be their combined total, k the value of one part, and D the positive difference. Assume positive ratio entries.

Fraction of whole: A/T = a/(a + b), and B/T = b/(a + b). Use these only when T includes exactly A and B.

Part-to-part comparison: A/B = a/b, so A = (a/b)B. Reversing the comparison gives B/A = b/a.

Known total: k = T/(a + b); A = ak and B = bk.

Known share: k = A/a or k = B/b. Divide by the parts belonging to the given share.

Known difference: k = D/|b − a|; T = (a + b)k. The vertical lines mean absolute value. This method requires unequal ratio entries. For 4:4 the difference must be zero, and a zero difference does not determine the total.

Known fraction of whole: A/T = p/q implies A:B = p:(q − p), where B is all the remainder and 0 < p < q.

Three or more shares: total parts = a + b + c + ...; each share = T × (its parts)/(total parts). A difference between two shares uses the difference between their two part counts.

Equivalent ratios: a:b = ca:cb for any positive c. For fraction entries, multiply every entry by a common denominator, then divide every integer entry by their highest common factor.

Conversions: decimal = numerator ÷ denominator; percentage = fraction × 100%; percentage as a fraction = percentage number/100. None of these requires a special ratio formula from the sheet.

Exam Technique and Marks

Each 1MA1 paper has 80 marks in 90 minutes. Paper 1 is non-calculator; Papers 2 and 3 permit a calculator. A practical target is about one minute per mark, leaving checking time, not a Pearson requirement for every question. Aim for roughly three minutes on a three-mark item and six minutes on a six-mark item. 1

Read the command word and underline whether the given amount is the total, a share or a difference. Label a bar or write equal-part statements. Show the decisive division, then the multiplication, and state the requested quantity with units. Keep exact fractions until the final requested conversion or rounding.

Pearson mark schemes use M for method, P for process, A for dependent accuracy and B for an accuracy mark that does not require method. C identifies communication marks when used. Exact allocations depend on the individual question. Correct intermediate fractions, divisions or processes may earn credit where the relevant scheme allows, but there is no universal 'one mark for adding the parts' rule. 3

For a one-mark 'Write down' fraction item, a correct simplified fraction is normally enough. For 'Work out', show the calculation. For 'Explain', use a mathematical reason: '3/5 compares red with blue; red is 3/8 of the whole because there are eight equal parts.' Do not substitute a general opinion for numerical evidence.

Official benchmark: Pearson's teaching guidance includes a one-mark item with black:green = 2:5 asking for the fraction black, and a three-mark item with Frank:Mary:Seth = 4:5:7 where Seth has eighteen more sweets than Frank and candidates find the total. Their answers are 2/7 and 96 respectively. The longer examples in this guide are original exam-style composites, not quotations from an official paper, and their allocations are illustrative. 2

Practical Applications and Connected Skills

For a drink with concentrate:water = 1:4, concentrate is 1/5 of the final drink. A 750 ml mixture needs 150 ml concentrate and 600 ml water. The ratio 1:4 describes ingredients, not concentrate to finished drink. If 200 ml of concentrate is available, water needed is 800 ml and the total drink is 1 litre.

Use consistent units before forming a ratio: 0.6 litres:150 ml becomes 600 ml:150 ml = 4:1. Useful conversions are 1 litre = 1000 ml, 1 kg = 1000 g, 1 m = 100 cm and £1 = 100p. Ratios of like quantities have no units after simplification, but calculated amounts must retain the relevant units.

Probability is a natural link. If equally likely counters are in the ratio 3:5, the probability of drawing a red counter is 3/8, because probability compares favourable outcomes with all outcomes. This assumes random selection with every counter equally likely.

For a graph extension, let x be the number of blue counters and y the number of red counters. The relationship is y = (3/5)x, a straight line through the origin. Points include (0,0), (5,3) and (10,6). Put blue on the horizontal axis and red on the vertical axis; read the gradient as red divided by blue, not red divided by total. Physical counter amounts must be whole numbers consistent with the ratio. This connection is not automatically Higher-only. 2

There is no required practical for this mathematics topic. All core content here is suitable for both tiers; 'challenging' labels identify difficulty, not Higher-only specification content.

Podcast and Active Revision

The ten-minute episode runs from introduction to core ideas, exam technique, a retrieval quiz and a final recap. Listen with a pencil. When a question appears, pause and attempt it before listening to the answer. Then solve one known-total, one known-share and one difference problem without the guide. Revisit the recall questions tomorrow and later in the week; checking an answer after retrieval is more useful than copying it while visible.

References