Skip to content
Cambridge IGCSE0607

Percentages

International Mathematics 0607 Chapter Notes

What this chapter covers

Percentages
ShareWhatsAppPost
Percentages notes

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Percentages notes as text: skim, search, and jump between subtopics.

~8 min read

In this chapter

  1. 1A percentage of a quantity
  2. 2One quantity as a percentage of another
  3. 3Percentage increase, decrease, profit and loss
  4. 4Simple interest
  5. 5Compound interest and repeated change
  6. 6Reverse percentages
  7. 7Reverse percentages with repeated change

1. A percentage of a quantity

A percentage is a fraction out of 100. So 16% means 16/100.

To find a percentage of a quantity, divide by 100 and multiply by the percentage. Or build it from 10% and 1%. For 15% of 240, take 10% = 24 and 5% = 12, so 15% = 36.

To compare a percentage of an amount with a fraction of the same amount, work out both amounts. Then subtract.

x% of Q = x/100 × Q

Examiner insight

“A number of candidates could not find 7% of 45. A few worked out 0.7 × 45 instead of 0.07 × 45.”

Worked example 13 marks

A is 16% of 360 dollars. B is 2/9 of 360 dollars.

Work out which is greater, A or B, and by how much. Show how you decide.

  1. 1

    A = 16/100 × 360 = 57.60 [1]

  2. 2

    B = 2/9 × 360 = 80 [1]

  3. 3

    B is greater, by 80 - 57.60 = 22.40dollars [1]

2. One quantity as a percentage of another

Write the part over the whole, then multiply by 100. The whole is the quantity you compare with. It is the number after the words “as a percentage of”.

A fraction becomes a percentage the same way. 13/20 = 65/100 = 65%.

percentage = part/whole × 100

Examiner insight

“Many candidates knew to divide 93 by 150 and multiply by 100.” “Some candidates found 93% of 150.”

Worked example 11 mark

(b) Write 93 dollars as a percentage of 150 dollars.

  1. 1

    93/150 × 100 = 62% [1]

3. Percentage increase, decrease, profit and loss

Find the change first, then divide it by the original value and multiply by 100. The original is the value before the change.

A multiplier does the change in one step. Increase by 12%: multiply by 1.12. Decrease by 12%: multiply by 0.88. A percentage can be over 100%. 250% of a price is 2.5 times it.

Profit is selling price minus cost price. Loss is the other way round. Percentage profit is the profit over the cost price. A discount is the amount taken off. The sale price is what is left.

percentage change = change/original × 100

new value = original × (1 ± x/100)

Examiner insight

“Some, however, incorrectly divided the reduction by the sale price or even 100.”

Worked example 14 marks

Mina buys a car for 13 500 dollars. On the same day, Mina buys a painting for 9000 dollars.

(a) At the end of 1 year the value of the car has decreased by 18%. Work out the value of the car at the end of 1 year.

(b) The value of the painting at the end of 1 year is the same as the value of the car at the end of 1 year. Work out the percentage increase in the value of the painting.

  1. 1

    (a) multiplier 1 - 0.18 = 0.82, so 13 500 × 0.82 [1]

  2. 2

    value of the car = 11 070dollars [1]

  3. 3

    (b) increase = 11 070 - 9000 = 2070 [1]

  4. 4

    2070/9000 × 100 = 23% [1]

4. Simple interest

Simple interest is the same amount every year. It is a percentage of the original sum invested.

Interest for n years is the sum times the rate over 100, times n. The value of the investment is the sum plus the interest. Check which one the question asks for. To find the rate, find the interest first: final value minus the sum invested.

I = (P × r × n)/100

value = P + I

Examiner insight

“A significant number did not go on to find the value of investment.” “Another common error was to multiply 400 by 2, but to omit the division by 100, reaching a value of $800 for the interest.”

Worked example 13 marks

Alain invests 200 dollars at a rate of r% simple interest per year. At the end of 2 years the value of Alain’s investment is 212 dollars.

Work out the value of r.

  1. 1

    interest = 212 - 200 = 12dollars [1]

  2. 2

    (200 × r × 2)/100 = 12, so 4r = 12 [1]

  3. 3

    r = 3 [1]

5. Compound interest and repeated change

With compound interest the interest is added each year. The next year's interest is on the new, larger total. A repeated percentage change multiplies by the same multiplier each time.

Write the rate as a multiplier first: 3% is 1 + 3/100 = 1.03. Do not put the 100 over the whole bracket.

On a graphic display calculator, enter 600 × 1.03^(X) as a function and read the table. Find the year just before the target and the year just after it.

value = P(1 + r/100)ⁿ

compound interest = value - P

Examiner insight

“A few tried to use the formula but put the 1 + 1.2 all over 100.” “When using trials candidates should try to show their calculations in an orderly way and to find the number of years just before the required target and the number of years just after the target.”

Worked example 16 marks

(b) Ami invests 600 dollars at a rate of 3% per year compound interest.

(i) Calculate the value of Ami’s investment at the end of 5 years.

(ii) Find the number of complete years it takes for the value of Ami’s investment of 600 dollars to be first more than 850 dollars.

  1. 1

    (i) 600 × 1.03⁵ [1]

  2. 2

    = 695.56dollars [1]

  3. 3

    (ii) value after n years is 600 × 1.03ⁿ [1]

  4. 4

    table: n = 11 gives 830.54, which is less than 850 [1]

  5. 5

    n = 12 gives 855.46, which is more than 850 [1]

  6. 6

    12 years [1]

6. Reverse percentages

In a reverse percentage question you are given the value after the change. Find the original.

Write the new value as a percentage of the original. A 35% reduction leaves 65%. A 15% increase makes 115%. Then divide the given value by that multiplier.

Do not find 35% of the sale price and add it on. The 35% was taken off the original price, not the sale price.

original = (new value)/multiplier

Examiner insight

“The response 20 400 × 1.15 was seen at least as often as the correct answer.” “A significant number of candidates found 35% of 31.07 and added it on to 31.07.”

Worked example 12 marks

In a sale, all prices are reduced by 35%. The sale price of a dress is 31.07 dollars.

Calculate the original price of the dress.

  1. 1

    sale price is 100% - 35% = 65% of the original, so original = 31.07 ÷ 0.65 [1]

  2. 2

    original price = 47.80dollars [1]

7. Reverse percentages with repeated change

When there are two changes, multiply the two multipliers first. Then divide the final value by the product.

Do not add the percentages. An increase of 8% then 5% is not 13%. Divide by 1.08 and by 1.05. Never multiply the final value by 0.92 and 0.95.

In a profit question the cost price is the original. A profit of 20% of the cost makes the selling price 120% of the cost.

Examiner insight

“Many candidates reduced the given value by 10% and then by 6%, multiplying by 0.9 and then 0.94 instead of dividing by 1.10 and then by 1.06.” “Some of the weaker candidates added the two percentages and treated the question as an increase of 16%.”

Worked example 13 marks

Raul bought a house on 1 January 2024. On 1 January 2025 the value of the house increased by 8%. On 1 January 2026 the value of the house increased by 5% of its value on 1 January 2025.

The value of the house on 1 January 2026 was 90 720 dollars.

(a) Calculate the amount Raul paid for the house on 1 January 2024.

  1. 1

    multiplier for both years = 1.08 × 1.05 = 1.134 [1]

  2. 2

    original = 90 720 ÷ 1.134 [1]

  3. 3

    = 80 000dollars [1]

Recap

  • x% of Q = x ÷ 100 × Q.
  • Percentage = part ÷ whole × 100. The whole is what you compare with.
  • Percentage change = change ÷ original × 100.
  • Profit and loss are percentages of the cost price.
  • Simple interest: I = Prn/100. Value = sum + interest.
  • Compound interest: multiply by the same multiplier each year.
  • Reverse percentage: divide by the multiplier. Never add percentages.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. (a) Find 35% of 240 dollars. (b) Write 18 as a percentage of 40.4 marks
  2. Do not use a calculator. A jacket costs 80 dollars. In a sale the price is reduced by 15%. Find the sale price of the jacket.2 marks
  3. Do not use a calculator. A shop buys a bike for 200 dollars and sells it for 250 dollars. Work out the percentage profit.2 marks
  4. Do not use a calculator. A coin cost 60 dollars. It is now worth 250% of its cost. (a) Find the value of the coin now. (b) Find how much more than 60 dollars this is.3 marks
  5. Do not use a calculator. Mia invests 500 dollars at r% simple interest per year. After 3 years the value of her investment is 545 dollars. Work out r.3 marks
  6. Sana invests 2500 dollars at 4% compound interest per year for 3 years. Calculate the interest she earns.3 marks
  7. Do not use a calculator. In a sale a jacket is reduced by 20%. The sale price is 36 dollars. Find the original price.2 marks
  8. A machine loses 15% of its value in its first year. In its second year it loses 10% of its value at the start of that year. After 2 years it is worth 13 005 dollars. Calculate its value when new.3 marks
  9. A shop marks a tent at 40% above its cost price. In a sale the shop gives a discount of d% of the marked price. It still makes a profit of 19% of the cost price. (a) Find the value of d. (b) The discount is 63 dollars. Calculate the cost price of the tent.5 marks
  10. Ravi and Sofia each invest the same sum of money on 1 January 2020. Ravi’s account pays 6% per year simple interest. Sofia’s account pays 5% per year compound interest. On 1 January 2023 the value of Sofia’s investment is 9261 dollars. (a) Calculate the sum of money each of them invested. (b) Calculate how much more Ravi’s investment is worth than Sofia’s on 1 January 2023. (c) Find the year in which the value of Sofia’s investment on 1 January is first more than the value of Ravi’s.7 marks

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Types of numberSetsPowers and rootsFractions, decimals and percentagesOrderingThe four operations