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Cambridge IGCSE0580

Percentages

Mathematics 0580 Chapter Notes

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Percentages
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In this chapter

  1. 1A percentage of a quantity
  2. 2One quantity as a percentage of another
  3. 3Percentage increase and decrease
  4. 4Profit, loss and percentages over 100%
  5. 5Simple interest
  6. 6Compound interest
  7. 7Repeated change and finding the rate
  8. 8Reverse percentages

1. A percentage of a quantity

To find a percentage of a quantity, write the percentage as a fraction of 100 or as a decimal, then multiply. So 18% is 0.18.

Some questions ask for an amount, not a percentage, such as the remaining cost after a deposit.

percentage of a quantity = percentage/100 × quantity

Examiner insight

“The most common error was to simply give the percentage not occupied (16 per cent) rather than the number of seats.” “In their working, candidates were required to write 0.18 or 18/100 and not just 18% or alternately 0.82 or 82/100 and not just 82%.”

Worked example 13 marks

A television costs char"24 1260. Liesl pays a deposit of 35% of the cost of the television. She then pays monthly payments of char"24 45.50 for the remaining cost of the television. Find the number of monthly payments that Liesl pays.

  1. 1

    deposit = 35/100 × 1260 = 441 [1]

  2. 2

    remaining cost = 1260 - 441 = 819 [1]

  3. 3

    number of payments = 819 ÷ 45.50 = 18 [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 are comparing with, the one that follows "of" in the question.

A percentage can be more than 100% when the part is larger than the whole.

percentage = part/whole × 100

Examiner insight

“Common errors included inaccuracies when copying the numbers, dividing the numbers the wrong way round, finding the answer as a fraction rather than a percentage and giving the answer inaccurately as 4.5 or 4.54.” “Some candidates found the cost of running the club as a percentage of the sum of the running cost and the membership fees.”

Worked example 12 marks

Do not use a calculator. Write char"24 30 as a percentage of char"24 120.

  1. 1

    30/120 = 1/4 [1]

  2. 2

    1/4 × 100 = 25% [1]

3. Percentage increase and decrease

Find the change, divide by the original amount, then multiply by 100. The original is the amount before the change happened.

To increase or decrease an amount in one step, multiply by a multiplier. Increasing by 10% uses 1.10. Decreasing by 24% uses 0.76.

percentage change = change/original × 100

increase by p%: × (1 + p/100) decrease by p%: × (1 - p/100)

Examiner insight

“Common errors were dividing by the new amount (19 780), dividing correctly but not multiplying by 100 or using a trial and improvement method to attempt to find the percentage, often leading to answers of 7% or 8%.” “The most common errors included finding the percentage of $980 rather than the percentage increase (not deducting 100%), or not converting from a decimal to a percentage (not multiplying by 100).”

Worked example 14 marks

Do not use a calculator.(a) In a restaurant the cost of a pizza is char"24 25. The restaurant reduces the cost of the pizza to char"24 19. Work out the percentage decrease in the cost.

(b) In another restaurant, the cost of a pizza is char"24 15. This restaurant adds a service charge of 10% to each pizza. Find the cost of a pizza, including the service charge.

  1. 1

    (a) decrease = 25 - 19 = 6 [1]

  2. 2

    percentage decrease = 6/25 × 100 = 24% [1]

  3. 3

    (b) service charge = 10/100 × 15 = 1.50 [1]

  4. 4

    cost = 15 + 1.50 = char"24 16.50 [1]

4. Profit, loss and percentages over 100%

Profit is selling price minus cost price. A loss is cost price minus selling price. Give a profit or loss as a percentage of the cost price.

A percentage over 100% means the new amount is bigger than the original. A new amount that is 150% of the original is found by multiplying by 1.5.

percentage profit = (selling price - cost price)/(cost price) × 100

Examiner insight

“Those opting to calculate the actual profit tended to make more errors, usually basing the profit on the selling price and not the cost price.”

Worked example 12 marks

A man buys a painting for char"24 1200. He sells the painting for char"24 1360. Calculate the percentage profit the man makes on the painting.

  1. 1

    profit = 1360 - 1200 = 160 [1]

  2. 2

    percentage profit = 160/1200 × 100 = 13.3% [1]

5. Simple interest

With simple interest the same amount is added every year. The interest is worked out on the original sum only.

The interest is the extra money earned. The total value is the original sum plus the interest. Read the question to see which one it asks for.

I = PRT/100

total value = P + I

Examiner insight

“Those candidates who did not score full marks often found the correct interest, $345 but did not add it to the initial investment.” “The most common error was to attempt to use compound interest.”

Worked example 12 marks

Dave invests char"24 8500 at a rate of 4.5% per year simple interest. Calculate the total interest earned at the end of 3 years.

  1. 1

    interest for 1 year = 4.5/100 × 8500 = 382.50 [1]

  2. 2

    interest for 3 years = 3 × 382.50 = char"24 1147.50 [1]

6. Compound interest

With compound interest each year's interest is added to the sum, so the next year's interest is worked out on the new, larger amount.

Use a multiplier and a power. The multiplier for r% a year is 1 + r/100. For 3.5% it is 1.035, not 1.35. Do not round the multiplier or its power part-way through.

V = P(1 + r/100)ⁿ

interest = V - P

Examiner insight

“Common errors included using simple interest, writing the multiplier as 1.35 instead of 1.035, or premature rounding of the 1.0355 value, which led to an inaccurate answer.” “Another common error was that candidates did not understand that they needed to subtract the original value of $4000 to ‘calculate the total interest’ as asked in the question.”

Worked example 13 marks

Joseph invests char"24 36 000 at a rate of 6.2% per year compound interest. Calculate the total interest earned at the end of 4 years.

  1. 1

    multiplier = 1 + 6.2/100 = 1.062 [1]

  2. 2

    value = 36 000 × 1.062⁴ = 45 793.15… [1]

  3. 3

    interest = 45 793.15 - 36 000 = char"24 9793 [1]

7. Repeated change and finding the rate

Repeated percentage change works like compound interest. A decrease of r% each year uses the multiplier 1 - r/100 raised to the number of years.

To find an unknown rate, divide the final value by the original, take the nth root, then subtract 1 and multiply by 100. Keep the full calculator value until the last step.

V = P(1 - r/100)ⁿ

r = ( √[n]V/P - 1 ) × 100

Examiner insight

“It was noted that the majority of candidates working with compound decrease did so by use of the multiplier approach, with only a minority attempting to do the longer stepwise process of working year by year.” “A common incorrect approach was to subtract 1500 from 1656.73 and then divided by 8 to give 19.6 per cent, whilst others incorrectly divided 1656.73 by 8 and then by 1500 to get 0.138 before converting this to a percentage 13.8 per cent.”

Worked example 13 marks

Behnaz invests char"24 4500 at a rate of r% per year compound interest. At the end of 5 years, the value of the investment is char"24 5420. Calculate the value of r.

  1. 1

    4500(1 + r/100)⁵ = 5420 [1]

  2. 2

    1 + r/100 = √[5]5420/4500 = 1.0379… [1]

  3. 3

    r = 3.79 [1]

8. Reverse percentages

In a reverse percentage question you are given the amount after the change and must find the original. The given amount is not 100%. It is 100% plus or minus the change.

Divide the given amount by the multiplier. Do not find the percentage of the given amount and add or subtract it.

original = (new amount)/multiplier

Examiner insight

“Calculating 24 per cent of the sale price and either adding it to or subtracting it from the sale price were the two most common errors.” “The most common error involved increasing 120 by 4 per cent leading to 124.8, which was often rounded to 125, and this earned no marks.”

Worked example 14 marks

Yasmin collects coins. 10% of her coins are gold. 25% of the remaining coins are silver. The other 54 coins are copper.(a) Calculate the number of silver coins.

(b) Calculate the total number of coins.

  1. 1

    (a) copper is 100 - 25 = 75% of the remaining coins, so remaining coins = 54 ÷ 0.75 = 72 [1]

  2. 2

    silver coins = 72 - 54 = 18 [1]

  3. 3

    (b) the 72 remaining coins are 100 - 10 = 90% of all her coins [1]

  4. 4

    total = 72 ÷ 0.9 = 80 [1]

Recap

  • Percentage of a quantity: percentage ÷ 100 × quantity.
  • Part as a percentage of the whole: part ÷ whole × 100.
  • Percentage change = change ÷ original × 100. Profit and loss are a percentage of the cost price.
  • Simple interest: I = PRT/100, the same each year. Add it to the original for the total value.
  • Compound interest: V = P(1 + r/100)ⁿ. Subtract P for the interest.
  • Extended: a repeated decrease uses 1 - r/100. Find a rate with an nth root.
  • Extended: for a reverse percentage, divide the new amount by the multiplier.

End-of-chapter exercise

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

  1. Do not use a calculator. Find 15% of 240.2 marks
  2. Do not use a calculator. Write 18 as a percentage of 45.2 marks
  3. Do not use a calculator. A shop reduces the price of a jacket from char"24 60 to char"24 45. Work out the percentage decrease.3 marks
  4. Do not use a calculator. A laptop costs char"24 640. In a sale the price is reduced by 12.5%. A customer pays a deposit of 20% of the sale price. Find the deposit.3 marks
  5. Amira invests char"24 2400 at a rate of 3.5% per year simple interest. Find the total value of her investment at the end of 6 years.3 marks
  6. Do not use a calculator. A trader buys a bag for char"24 40 and sells it for char"24 52. Work out her percentage profit.2 marks
  7. A club has 40 members. Next year the number of members is 250% of this year's number. Find the number of members next year.2 marks
  8. A bank pays 3% per year compound interest. Ravi invests char"24 5000. Calculate the total interest he earns in 4 years. Give your answer to the nearest dollar.3 marks
  9. Do not use a calculator. A shop adds 60% to the cost price of a coat to get its marked price. In a sale the marked price is reduced by 25%, and the coat is sold for char"24 96. (a) Find the cost price of the coat. (b) Find the shop's percentage profit on the coat.5 marks
  10. A car costs char"24 25 000 new. Its value falls by 18% in the first year. In each of the next 4 years its value falls by r% of its value at the start of that year. At the end of the 5 years the car is worth char"24 12 000. Calculate the value of r. Give your answer correct to 3 significant figures.4 marks

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