Pearson Edexcel International GCSE Mathematics A 4MA1: Percentages, ratio and proportion – Revision Notes

Full explanations and longer worked examples are in the Percentages, ratio and proportion study guide. These notes are for the final weeks.

They cover sections 1.6 (Percentages) and 1.7 (Ratio and proportion) of the Pearson Edexcel International GCSE Mathematics A (4MA1) specification, Issue 2 (November 2017), for the January and June series. Section 1.6 has two statements marked Higher tier only: repeated percentage change and solving compound interest problems. Section 1.7 is the same for both tiers. A calculator may be used on every 4MA1 paper, and no percentage or interest formula is printed on either formulae sheet.

For the rest of topic 1, use the topic 1 revision notes. The course hub is Edexcel IGCSE Mathematics and the printable checklist lists every statement.

Definitions

• Percentage (1.6A): number of parts per 100. 7% = 7/100 = 0.07.
• Multiplier (1.6D): the decimal you multiply by to apply a percentage. It is the percentage of the original you end up with, divided by 100.
• Compound interest (1.6G): interest added each year, so the next year’s interest is on the new total.
• Depreciation (1.6G): a fall in value by a fixed percentage each year.
• Ratio (1.7A): a comparison of quantities in the same units, written a : b.
• Direct proportion (1.7D): y = kx. When one quantity is multiplied by a number, the other is multiplied by the same number.
• Map scale 1 : n (1.7E): 1 unit on the map stands for n of the same units in real life.

Formulas and methods

You want	Use
Percentage to decimal	÷ 100
a as a percentage of b	(a ÷ b) × 100, same units first
x% of A	(x ÷ 100) × A
Increase by r%	× (1 + r/100)
Decrease by r%	× (1 − r/100)
Percentage change	(change ÷ original) × 100
Original before a change	final ÷ multiplier
Compound growth, n years	P × (1 + r/100)ⁿ
Depreciation, n years	P × (1 − r/100)ⁿ
Repeated change (Higher tier only)	multiply all the multipliers
One part of a ratio	known amount ÷ number of parts it represents
Direct proportion	k = y ÷ x, then y = kx
Map to real	map length × n, then convert units
Real to map	real length (in map units) ÷ n

Useful conversions: 1 m = 100 cm, 1 km = 100 000 cm, 1 kg = 1000 g, 1 litre = 1000 ml.

Method in steps

Percentage increase or decrease

1. Write the new percentage of the original (e.g. 100 - 18 = 82%).
2. Convert to a multiplier (0.82).
3. Multiply the original by the multiplier.

Small reminder: 18% of 450 = 0.18 × 450 = 81.

Reverse percentage

1. Decide the multiplier for the change that already happened.
2. Divide the final value by it.
3. Check by multiplying back.

Small reminder: a price is 172.80 after an 8% rise, so the original is 172.80 ÷ 1.08 = 160.

Compound interest or depreciation

1. Write the yearly multiplier.
2. Raise it to the power of the number of years.
3. Multiply by the starting amount.
4. If asked for interest or loss, subtract.

Solving compound interest problems (Higher tier only)

Original:   divide the final value by multiplier^n.
Rate:       final / start = m^n, take the nth root, rate = (m - 1) x 100.
Years:      try n = 1, 2, 3, ... and write the values either side of the target.

Dividing in a ratio

1. Add the parts (or subtract them, if the difference is given).
2. Find one part.
3. Multiply each ratio number by one part.
4. Check the shares add to the total.

Small reminder: share 96 in the ratio 5 : 3. Total 8 parts, one part 12, shares 60 and 36.

Direct proportion table

1. Use a complete pair to find k = y / x.
2. Write y = kx.
3. Substitute for each gap: multiply for y, divide for x.

Ratio essentials

• Simplify by dividing all parts by the HCF. Clear decimals first by multiplying all parts by 10 or 100.
• 1 : n divide every part by the first part. n : 1 divide by the second part.
• If a : b = 2 : 7, then a is 2/9 of the total and a is 2/7 of b.
• Three-part ratios work the same way. To combine a : b and b : c, scale both so the b values match.

Equivalents worth knowing (1.6C)

Percentage	Fraction	Decimal
5%	1/20	0.05
20%	1/5	0.2
25%	1/4	0.25
28%	7/25	0.28
37.5%	3/8	0.375
75%	3/4	0.75
33⅓%	1/3	0.333…
66⅔%	2/3	0.666…

Knowing these lets you check a calculator answer quickly. For example, 25% of a quantity is the same as dividing it by 4.

Maps and scale diagrams (1.7E)

• Scale 1 : n has no units. Both sides are in the same unit, usually cm.
• Real distance = map distance × n. Map distance = real distance ÷ n.
• Convert to cm before dividing by n, and convert back to m or km after multiplying.
• Sense check: the real distance must always be much bigger than the map distance.
• A plan or scale drawing works the same way. A scale of 1 : 50 means 1 cm on the drawing is 50 cm in real life.

Must-know distinctions

This	Is not the same as
Change ÷ original	Change ÷ new value
Final ÷ multiplier (reverse)	Final × (1 − r/100)
Compound: P × mⁿ	Simple: P + n × (r% of P)
Value after n years	Interest or loss (the change)
Total number of parts	Number of people sharing
Difference between shares	Total of the shares
Repeated change: multiply multipliers	Adding the percentages

Higher tier reminder: a 6% rise then a 6% fall gives 1.06 × 0.94 = 0.9964, an overall decrease of 0.36%, not zero.

Quick self-test

1. Write 2.5% as a decimal and as a fraction in its simplest form.
2. Express 36 as a percentage of 240.
3. Decrease 85 by 12%.
4. A price rises from £125 to £140. Find the percentage increase.
5. After a 40% reduction, a coat costs £27. Find the original price.
6. £3200 is invested at 2.5% per year compound interest. Find its value after 2 years.
7. Write 12 : 30 in the form 1 : n.
8. Share 1.8 kg of flour in the ratio 4 : 5. Give your answers in grams.
9. y varies directly as x, and y = 21 when x = 6. Find y when x = 10.
10. On a map with scale 1 : 20 000, a path is 4.5 cm long. Find its real length in km.
11. (Higher tier only) A value rises by 40% and then falls by 25%. Find the overall percentage change.
12. (Higher tier only) A machine’s value falls from £50 000 to £40 500 in 2 years at a fixed rate of depreciation. Find the rate.

Answers

1. 0.025 and 25/1000 = 1/40.
2. 36/240 × 100 = 15%.
3. 85 × 0.88 = 74.8.
4. Change 15; 15/125 × 100 = 12%.
5. Multiplier 0.6; 27 ÷ 0.6 = £45.
6. 3200 × 1.025² = £3362.
7. 30 ÷ 12 = 2.5, so 1 : 2.5.
8. 1800 g ÷ 9 = 200 g per part; 800 g and 1000 g.
9. k = 21/6 = 3.5; y = 3.5 × 10 = 35.
10. 4.5 × 20 000 = 90 000 cm = 0.9 km.
11. 1.4 × 0.75 = 1.05, so an overall increase of 5%.
12. m² = 40 500/50 000 = 0.81, m = 0.9, so 10% per year.

Where marks are usually lost

• Dividing the change by the new value in a percentage change question.
• In a reverse percentage, subtracting r% of the sale price instead of dividing by the multiplier.
• Using the multiplier 0.4 for a 40% reduction, when it should be 0.6.
• Giving the final amount when the question asks for the interest earned, or the other way round.
• Rounding the value after each year of compound interest, so the final answer drifts.
• Not converting units before writing a ratio or a percentage (cm with m, g with kg).
• Dividing a quantity by the number of people rather than the total number of ratio parts.
• Writing 1 : n with the parts the wrong way round, or simplifying 12 : 30 to 2 : 5 when 1 : n was asked for.
• Leaving a map distance in cm when the question asks for km.
• (Higher tier only) Adding successive percentage changes instead of multiplying the multipliers.

When you are ready, work through the practice questions, which have full mark schemes, and the topic 1 practice questions for wider number work. The free 10-minute diagnostics help you find other weak spots.

Official syllabus

Pearson Edexcel International GCSE in Mathematics (Specification A) (4MA1), Specification, Issue 2, November 2017, Pearson Education Limited (first assessment June 2018). Topic 1, Numbers and the number system, sections 1.6 Percentages and 1.7 Ratio and proportion (Foundation Tier content and Higher Tier content).