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IGCSE Mathematics 0580 · Chapter 2 notes · ~21 min read

Ratio, percentages & money

Ratio, proportion, speed, percentages, interest and exchange rates: the everyday maths that fills IGCSE Paper 1 and Paper 3, taught with bar models, multipliers and full worked examples.

RatioProportionRates & speedPercentagesReverse percentageInterestGrowth & decay
RSby Riyaz Sir
mentorriyaz.in
No tag = Core (everyone)EXTENDED Extended onlyhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • Ratio compares parts. Find the value of one part first (a bar model helps), then multiply. Proportion is the same idea: find the value of 1, then scale.
  • Every percentage question is a multiplier: increase 20% → × 1.2, decrease 15% → × 0.85. To go backwards (reverse percentage) you divide by the multiplier.
  • Compound interest and growth and decay use multipliern. For speed use the triangle D, S, T, and keep the units matching.

By the end you can

  • Simplify ratios, divide a quantity in a given ratio and solve ratio problems.
  • Use the unitary method and scale factors for direct proportion, and find best buys.
  • Calculate with speed, density and other rates, and convert between units of speed.
  • Find percentages of amounts, percentage change, profit and loss, and use percentage multipliers.
  • Solve reverse percentage problems.
  • Calculate simple and compound interest.
  • Work with time (12-hour and 24-hour clocks), timetables and currency conversion.
  • Use direct and inverse proportion with x, x², x³, √x, 1/x and 1/x². EXT
  • Solve exponential growth and decay problems. EXT

1Ratio

Simplifying a ratio

Simplify: divide both parts by the HCF12 : 18÷ 62 : 3simplest form:no common factor leftDifferent units: convert first, then simplify50 cm : 2 msame units50 : 200÷ 501 : 4Write as 1 : n: divide both parts by the first2 : 5÷ 21 : 2.5
Divide both parts by the same number. For different units, convert first.

Ratio — a comparison of quantities in the same units, written a : b. Divide both parts by the same number and you get an equivalent ratio.

  • Simplest form: divide by the HCF (see Number toolkit). 12 : 18 = 2 : 3.
  • Fractions or decimals: multiply both sides to clear them. 23 : 12 = 4 : 3 (multiply by 6). 1.2 : 0.8 = 12 : 8 = 3 : 2.
  • 1 : n form: divide both by the first number. 2 : 5 = 1 : 2.5.
  • A ratio has no units once simplified: 50 cm : 2 m = 50 : 200 = 1 : 4.
Common mistakeSimplifying 50 cm : 2 m as 25 : 1 by ignoring the units. Convert to the same unit first.

Sharing in a ratio

£60 in total£12£12£12£12£122 parts = £243 parts = £361. total parts: 2 + 3 = 52. one part: £60 ÷ 5 = £123. shares: 2 × £12 = £24 and 3 × £12 = £36
Add the parts, find one part, then multiply. The shares must add back to the total.
  • Method: (1) add the parts; (2) divide the total by this to find one part; (3) multiply each part by that number; (4) check that the shares add to the total.
  • A ratio of 2 : 3 also means the shares are 25 and 35 of the total.
Worked example

Share £480 in the ratio 3 : 5 : 4.

  1. total parts = 3 + 5 + 4 = 12
  2. one part = 480 ÷ 12 = £40
  3. shares: 3 × 40 = £120, 5 × 40 = £200, 4 × 40 = £160
  4. check: 120 + 200 + 160 = 480 ✓
Worked example

The ratio of boys to girls is 4 : 5. There are 36 boys. How many girls?

  1. 4 parts = 36 boys
  2. one part = 36 ÷ 4 = 9
  3. girls = 5 parts = 5 × 9 = 45
Worked example

A and B share money in the ratio 3 : 7. B gets £48 more than A. How much is shared altogether?

  1. difference in parts = 7 − 3 = 4 parts
  2. 4 parts = £48, so one part = £12
  3. total parts = 10, so total = 10 × 12 = £120
Exam tipWhen you are given one share (not the total), use that share to find one part. When you are given the difference, use the difference in parts.

2Proportion

Unitary method and best buys

Unitary method: go to 1 first5 pens£3.50÷ 51 pen£0.70× 88 pens£5.60Scale factor: when it divides nicely4 people300 g flour× 1.56 people450 g flourscale factor6 ÷ 4 = 1.5
Go to one first (divide), then to the amount you want (multiply). If the numbers divide nicely, use a scale factor.

Direct proportion — when one quantity doubles, the other doubles too. More pens cost more money.

  • Unitary method: find the value of one, then multiply. Scale factor: multiply both quantities by the same number.
  • Best buy: work out the price per unit (per gram, per litre) for each, then compare. The lower price per unit is the better value.
Worked example

5 pens cost £3.50. Find the cost of 8 pens.

  1. one pen = 3.50 ÷ 5 = £0.70
  2. 8 pens = 8 × 0.70 = £5.60
Worked example

Which is the better buy: 500 g for £2.40 or 750 g for £3.45?

  1. price per gram: 240p ÷ 500 = 0.48p, 345p ÷ 750 = 0.46p
  2. the 750 g pack costs less per gram
  3. the 750 g pack is the better buy

Direct and inverse proportionEXTENDED

xy123452468100xy12345369120Direct proportion: y = kxstraight line through (0, 0)Inverse proportion: y = k ÷ xa curve: y falls as x risesdouble x → double ydouble x → half y
Direct proportion is a straight line through the origin. Inverse proportion is a curve that falls as x rises.

Inverse proportion — EXT when one quantity doubles, the other halves. More workers means fewer days.

  • EXT Direct: y = kx. The graph is a straight line through (0, 0). Inverse: y = k ÷ x. The graph is a curve that never touches the axes.
  • EXT Method: (1) write the equation with k; (2) use the given pair to find k; (3) rewrite the equation with that k; (4) substitute the new value.
Worked example

EXT y is directly proportional to x, and y = 12 when x = 4. Find y when x = 7.

  1. write y = kx
  2. substitute: 12 = k × 4, so k = 3
  3. equation: y = 3x
  4. when x = 7, y = 3 × 7 = 21
Worked example

EXT y is inversely proportional to x, and y = 6 when x = 4. Find x when y = 3.

  1. write y = kx
  2. 6 = k4, so k = 24
  3. equation: y = 24x
  4. when y = 3: 3 = 24x, so x = 24 ÷ 3 = 8
Worked example

EXT 6 workers take 10 days to build a wall. How long would 4 workers take?

  1. more workers means fewer days: inverse proportion
  2. total work = 6 × 10 = 60 worker-days
  3. days = 60 ÷ 4 = 15 days

Other forms of proportionEXTENDED

xyy ∝ xy = kxx × 2 → y × 2xyy ∝ x²y = kx²x × 2 → y × 4xyy ∝ 1/xy = k ÷ xx × 2 → y ÷ 2xyy ∝ √xy = k√xx × 4 → y × 2
Match the words to the equation. The graph shape tells you which form it is.
StatementEquationIf x doubles
y is proportional to xy = kxy doubles
y is proportional to x²y = kx²y × 4
y is proportional to x³y = kx³y × 8
y is proportional to √xy = k√xy × √2
y is inversely proportional to xy = k/xy halves
y is inversely proportional to x²y = k/x²y ÷ 4
  • EXT "Proportional to" always means a constant k. The same four steps work for every form: find k first.
Worked example

EXT y is proportional to x², and y = 50 when x = 5. Find y when x = 8.

  1. y = kx²
  2. 50 = k × 25, so k = 2
  3. y = 2x²
  4. when x = 8: y = 2 × 64 = 128
Worked example

EXT y is inversely proportional to the square of x. y = 4 when x = 3. Find y when x = 6.

  1. y = kx²
  2. 4 = k9, so k = 36
  3. y = 36x²
  4. when x = 6: y = 3636 = 1
Memory trickDirect → multiply, Inverse → divide. If the question says "square of", put the square on x, not on y.

3Rates and speed

Speed, distance and time

DSTcover the one you wantcover DD = S × Tcover SS = D ÷ Tcover TT = D ÷ Saverage speed =total distance ÷ total timeunits must match: km, h, km/h or m, s, m/s
Cover the quantity you want to find. What is left shows the calculation.

Rate — a quantity per unit of another, for example km per hour, pence per gram, litres per minute.

Speed = DistanceTime
so Distance = Speed × Time and Time = Distance ÷ Speed
  • Units must match: km, h and km/h or m, s and m/s. Convert the time first.
  • Minutes to hours: divide by 60. 1 h 30 min = 1.5 h (not 1.30). 2 h 15 min = 2.25 h.
  • Average speed = total distance ÷ total time for the whole journey. It is not the average of the speeds.
  • km/h to m/s: multiply by 1000 and divide by 3600. So 72 km/h = 72 × 1000 ÷ 3600 = 20 m/s. (Divide by 3.6.)
Worked example

A car travels 150 km in 2 hours 30 minutes. Find its average speed.

  1. change the time: 2 h 30 min = 2.5 h
  2. speed = 1502.5 = 60 km/h
Worked example

A car drives 60 km at 40 km/h, then 60 km at 60 km/h. Find the average speed for the whole journey.

  1. time for part 1 = 60 ÷ 40 = 1.5 h
  2. time for part 2 = 60 ÷ 60 = 1 h
  3. total distance = 120 km, total time = 2.5 h
  4. average speed = 120 ÷ 2.5 = 48 km/h (not 50)
Common mistakeAveraging the two speeds: (40 + 60) ÷ 2 = 50. The car spends more time at the slower speed, so the average is below 50.

Other rates

  • Density = massvolume, measured in g/cm³ or kg/m³. Flow rate = volumetime (for example litres per minute).
  • Pro rata means "in proportion": £7.50 for 3 hours is £2.50 per hour, so 5 hours costs £12.50.
Worked example

A metal block has mass 540 g and volume 200 cm³. Find its density.

  1. density = mass ÷ volume
  2. = 540 ÷ 200
  3. = 2.7 g/cm³
Worked example

A tank holds 600 litres. A tap fills it at 25 litres per minute. How long does it take?

  1. time = amount ÷ rate
  2. = 600 ÷ 25
  3. = 24 minutes
Memory trickDraw the triangle for any rate: the product goes on top. Density has mass on top, and speed has distance on top. See also distance–time graphs.

4Percentages

Percentage multipliers

increase 20%× 1.2100% + 20% = 120%decrease 15%× 0.85100% − 15% = 85%increase 3.5%× 1.035100% + 3.5% = 103.5%decrease 8%× 0.92100% − 8% = 92%multiplier = new percentage ÷ 100
Work out the new percentage first, then divide by 100. The multiplier does an increase or a decrease in one step.

Percentage — a fraction out of 100. 35% means 35100 = 0.35.

  • Percentage of an amount: change to a decimal and multiply. 15% of 80 = 0.15 × 80 = 12.
  • Express as a percentage: write a fraction, then multiply by 100. 18 out of 72 = 1872 × 100 = 25%.
  • Increase / decrease: use a multiplier. Increase by 12% → × 1.12. Decrease by 35% → × 0.65.
Worked example

Increase £250 by 12%.

  1. new percentage = 100% + 12% = 112%
  2. multiplier = 1.12
  3. 250 × 1.12 = £280
Worked example

Decrease 640 by 35%.

  1. new percentage = 100% − 35% = 65%
  2. multiplier = 0.65
  3. 640 × 0.65 = 416

Percentage change and profit

A price rises from £80 to £92before£80original = 100%after£8012← change = £12percentage change = change ÷ original × 10012 ÷ 80 × 100 = 15% increaseDivide by the ORIGINAL (the "before" amount), not the new one.
Always divide the change by the original value, which is the starting amount.
% change = changeoriginal × 100
change = new − old. Positive = increase, negative = decrease
  • Profit or loss: % profit = profitcost price × 100. The cost price is the starting value.
  • A discount or VAT is just a percentage change on the price.
Worked example

A phone costs £450. In a sale it costs £369. Find the percentage decrease.

  1. change = 450 − 369 = £81
  2. 81450 × 100 = 18%
  3. answer: 18% decrease
Worked example

A trader buys a chair for £40 and sells it for £52. Find the percentage profit.

  1. profit = 52 − 40 = £12
  2. 1240 × 100 = 30%
  3. divide by the cost price, not the selling price

Reverse percentages

After a 20% rise the price is £96. Find the original.20%20%20%20%20%100% = ?20%20%20%20%20%+20%120% = £96beforeafter1. 120% = £962. ÷ 6 → 20% = £16 (or ÷ 120 → 1% = £0.80)3. × 5 → 100% = £80, the original priceNot 96 − 20% of 96: the 20% was added to the ORIGINAL.
The percentage was taken from the original, so divide by the multiplier to get back to 100%.
  • Method: (1) write the given amount as a percentage of the original; (2) find 1% (divide); (3) multiply by 100. Or divide by the multiplier.
Worked example

A jacket costs £51 after a 15% discount. Find the original price.

  1. sale price = 100% − 15% = 85% of the original
  2. 85% = £51, so 1% = 51 ÷ 85 = £0.60
  3. 100% = 0.60 × 100 = £60
  4. (check: 60 × 0.85 = 51 ✓)
Worked example

A price including 20% VAT is £90. Find the price before VAT.

  1. price with VAT = 120% of the original
  2. original = 90 ÷ 1.2
  3. = £75
Common mistakeWorking out 15% of the sale price and adding it back: 51 + 7.65 = 58.65. The 15% was taken from the original, not from £51, so divide by 0.85 to get £60.
Exam tipCheck by forward calculation: multiply your answer by the multiplier. You should get the amount in the question.

5Interest, growth and decay

Simple and compound interest

Interest earned on £1000 at 5% a year, over 20 years£5101520500100015000yearssimple: the same £50 each yearcompound: × 1.05 each yearcompound£1653.30simple£1000
Simple interest adds the same amount each year. Compound interest earns interest on the interest too, so it curves upwards.

Simple interest — interest on the original amount only, the same every year.

Compound interest — interest on the original amount plus the interest already added.

Simple interest I = P × R × T100
P = principal, R = rate % per year, T = years
Compound amount A = P(1 + r100)n
n = number of years. The interest earned = A − P
Worked example

Find the simple interest on £800 at 4.5% a year for 3 years.

  1. I = 800 × 4.5 × 3100
  2. = £108
  3. total amount = 800 + 108 = £908
Worked example

Find the compound amount when £2000 is invested at 3% a year for 4 years.

  1. multiplier = 1 + 3100 = 1.03
  2. A = 2000 × 1.034
  3. = £2251.02
  4. interest earned = 2251.02 − 2000 = £251.02
Worked example

£5000 is invested at 2% compound interest for 3 years. How much interest is earned?

  1. A = 5000 × 1.023
  2. = £5306.04
  3. interest = 5306.04 − 5000 = £306.04
Exam tip"Interest" and "total amount" are different. Read the last line of the question: do they want A or A − P?

Exponential growth and decayEXTENDED

n£2468101000200030000n£24681050010000Growth: +10% a year× 1.1 each year, A = 1000 × 1.1nDecay: −20% a year× 0.8 each year, A = 1000 × 0.8n£2594£107£1000 grows to £2594 in 10 yearsnever reaches 0, just gets smaller
Repeated multiplication gives a curve. A multiplier above 1 grows, and a multiplier between 0 and 1 decays.
Final amount = initial × multipliern
multiplier = 1 + rate for growth, and 1 − rate for decay
  • EXT Growth: +r% a year → multiplier 1 + r100 (population, investments). Decay: −r% a year → multiplier 1 − r100 (depreciation, radioactive decay).
  • EXT This is the same maths as compound interest: only the context changes. Decay curves never reach zero.
Worked example

EXT A town has 12 000 people. The population grows by 2.5% a year. Find the population after 5 years.

  1. multiplier = 1.025
  2. 12 000 × 1.0255
  3. = 13 577 (nearest whole number)
Worked example

EXT A car costing £15 000 loses 18% of its value each year. Find its value after 3 years.

  1. multiplier = 1 − 0.18 = 0.82
  2. 15 000 × 0.823
  3. = £8270.52
Worked example

EXT A culture of bacteria doubles every 3 hours. There are 500 at the start. How many are there after 12 hours?

  1. 12 ÷ 3 = 4 doublings
  2. multiplier for each doubling = 2
  3. 500 × 24 = 500 × 16 = 8000

6Time and money

Time

24-hour clock: afternoon times are the 12-hour time + 12ampm00:0006:0012:0018:0024:00Time taken from 08:40 to 11:15: count on08:4009:0011:0011:15+ 20 min+ 2 h+ 15 min20 min + 2 h + 15 min = 2 h 35 min
In the 24-hour clock, afternoon times are the 12-hour time plus 12. To find a time taken, count on to the next hour.
  • Hours and minutes: there are 60 minutes in an hour, not 100. 3.4 hours = 3 h + 0.4 × 60 min = 3 h 24 min.
  • 12-hour to 24-hour: pm times add 12 (9:05 pm = 21:05). 24-hour to 12-hour: subtract 12 after 12:59 (17:45 = 5:45 pm).
  • Time taken: count on to a whole hour, then whole hours, then the rest.
Worked example

A film starts at 19:50 and lasts 2 hours 35 minutes. When does it finish?

  1. 19:50 + 10 min = 20:00
  2. 20:00 + 2 h = 22:00
  3. 22:00 + 25 min = 22:25
  4. (10 + 25 = 35 minutes ✓)
Worked example

Change 0.35 hours into minutes.

  1. minutes = 0.35 × 60
  2. = 21 minutes

Currency conversion

Exchange rate: £1 = $1.25£240× 1.25pounds → dollars: multiply$300$300÷ 1.25dollars → pounds: divide£240Sense check: here £1 buys MORE than 1 dollar, so £ → $ gives a bigger number
Multiply to change into the currency with the bigger number, divide to come back. Always do a sense check.
  • Write the rate first (£1 = $1.25). Decide multiply or divide, then check that the answer is sensible.
Worked example

A holiday costs €630. The rate is £1 = €1.40. How much is this in pounds?

  1. euros → pounds: divide by 1.40
  2. 630 ÷ 1.40
  3. = £450
  4. (check: 450 × 1.40 = 630 ✓)
Worked example

A £600 laptop is bought on credit: 20% deposit and 12 payments of £42. How much extra is paid?

  1. deposit = 20% of 600 = £120
  2. payments = 12 × 42 = £504
  3. total paid = 120 + 504 = £624
  4. extra = 624 − 600 = £24

7Test yourself

Test yourself — cover the answers first

Quick check

  1. Divide 90 in the ratio 2 : 3 : 1.
  2. Simplify 45 : 60.
  3. Write 2 : 5 in the form 1 : n.
  4. Increase 60 by 15%.
  5. Decrease 80 by 25%.
  6. What percentage is 12 of 48?
  7. A price rises from £80 to £92. Find the percentage increase.
  8. A price after a 10% rise is £66. Find the original price.
  9. Find the simple interest on £500 at 4% a year for 2 years.
  10. A car travels 45 km in 30 minutes. Find its speed in km/h.
  11. Change 36 km/h into m/s.
  12. Write 2.4 hours in hours and minutes.
  13. Write 17:45 in 12-hour time.
  14. Change £80 to dollars when £1 = $1.30.
  15. EXT y is directly proportional to x. y = 10 when x = 4. Find y when x = 10.
  16. EXT y is inversely proportional to x. y = 2 when x = 9. Find y when x = 6.
  17. EXT y is proportional to x². y = 18 when x = 3. Find y when x = 5.
  18. EXT What multiplier means a decrease of 4.5%?
  19. EXT £2000 grows by 5% a year. Find the value after 3 years.
  20. EXT A population of 5000 falls by 10% each year. Find it after 2 years.

Exam-style

  1. Ali, Beth and Cara share £360 in the ratio 2 : 3 : 7. Work out Cara's share. [3]
  2. A coat normally costs £85. In a sale it is reduced by 24%. Find the sale price. [2]
  3. The price of a TV including 20% VAT is £510. Find the price before VAT. [3]
  4. A car travels 90 km in 1 hour 15 minutes. Find its average speed in km/h. [2]
  5. A train leaves at 09:48 and the journey takes 3 hours 27 minutes. At what time does it arrive? [2]
  6. Sam invests $4000 at 2.5% compound interest each year for 3 years. Calculate the amount at the end, to the nearest cent. [3]
  7. EXT y is inversely proportional to x. y = 8 when x = 3. Find y when x = 12. [3]
  8. EXT A machine costs £25 000 and loses 12% of its value each year. Find its value after 4 years, to the nearest pound. [3]
  9. EXT A price falls from £2.50 to £2.20. Find the percentage decrease. [2]
Show answers
  1. 30, 45, 15
  2. 3 : 4
  3. 1 : 2.5
  4. 69
  5. 60
  6. 25%
  7. 15%
  8. £60
  9. £40
  10. 90 km/h
  11. 10 m/s
  12. 2 h 24 min
  13. 5:45 pm
  14. $104
  15. 25
  16. 3
  17. 50
  18. 0.955
  19. £2315.25
  20. 4050
  21. total parts = 2 + 3 + 7 = 12
    one part = 360 ÷ 12 = 30
    Cara = 7 × 30 = £210
  22. multiplier = 1 − 0.24 = 0.76
    85 × 0.76 = £64.60
  23. price with VAT = 120%
    510 ÷ 1.2
    = £425
  24. 1 h 15 min = 1.25 h
    90 ÷ 1.25 = 72 km/h
  25. 09:48 + 12 min = 10:00
    10:00 + 3 h = 13:00, then + 15 min
    13:15
  26. multiplier = 1.025
    4000 × 1.0253
    = $4307.56
  27. y = k ÷ x, so 8 = k ÷ 3, k = 24
    y = 24 ÷ x
    x = 12 gives y = 2
  28. multiplier = 0.88
    25 000 × 0.884
    = £14 992
  29. change = 0.30
    0.302.50 × 100
    = 12%
Practise this chapter
Ratio & proportionRatesPercentagesTimeMoneyExponential growth & decayProportion
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