Challenge · Percentages

13 marks

These problems use percentages of an amount, percentage change, and reverse percentages. Decide each time whether to multiply by a percentage, divide for a percentage change, or divide by a multiplier to reverse. A worked example shows how to start; hints are at the foot.

Worked example: how to start
A coat costs £80. In a sale it is reduced by $15\%$. Work out the sale price.
A $15\%$ decrease means keeping $85\%$, so multiply by $0.85$.
$£80\times0.85=£68$.
1A jacket costs £40. In a sale it is reduced by $15\%$. Work out the sale price.[2 marks]
Answer:
2A price increased from £25 to £30. Work out the percentage increase.[2 marks]
Answer:
3A phone costs £48 in a sale after a $20\%$ discount. Work out its original price.[3 marks]
Answer:
4A shop buys a bike for £80 and sells it for £100. Work out the percentage profit.[3 marks]
Answer:
5A £60 item can be reduced by $25\%$, or by a flat £16. Which is the bigger reduction, and by how much?[3 marks]
Answer:
★Extension: A salary of £24000 is increased by $5\%$, then increased by $5\%$ again the following year. Work out the salary after the two years.
Stuck? Hints (don't peek unless you need to)1. A $15\%$ decrease is a multiplier of $0.85$.2. Percentage change $=\dfrac{\text{change}}{\text{original}}\times100$.3. The £48 is $80\%$ of the original – divide by the multiplier.4. Profit $=$ selling $-$ cost; percentage profit is over the COST price.5. Work out $25\%$ of £60, then compare with £16.

Solutions & mark scheme · Percentages

Total: 13 marks

Award the marks shown for each correct step; many of these have more than one valid route, so give method marks for any correct working.

1A jacket costs £40. In a sale it is reduced by $15\%$. Work out the sale price.[2]
Model solution
Multiplier $=0.85$.
$£40\times0.85=£34$.
Answer: $£34$
Marks
✔1Multiplier $0.85$ (or finds $15\%=£6$)
✔1$=£34$
2A price increased from £25 to £30. Work out the percentage increase.[2]
Model solution
Change $=30-25=5$.
$\dfrac{5}{25}\times100=20\%$.
Answer: $20\%$
Marks
✔1Change $=5$ over original $25$
✔1$=20\%$
3A phone costs £48 in a sale after a $20\%$ discount. Work out its original price.[3]
Model solution
£48 is $80\%$ of the original, so multiplier $=0.8$.
Original $=48\div0.8=£60$.
Answer: $£60$
Marks
✔1Recognises £48 $=80\%$
✔1$48\div0.8$
✔1$=£60$
4A shop buys a bike for £80 and sells it for £100. Work out the percentage profit.[3]
Model solution
Profit $=100-80=£20$.
$\dfrac{20}{80}\times100=25\%$.
Answer: $25\%$
Marks
✔1Profit $=£20$
✔1Over cost $£80$
✔1$=25\%$
5A £60 item can be reduced by $25\%$, or by a flat £16. Which is the bigger reduction, and by how much?[3]
Model solution
$25\%$ of £60 $=£15$.
Compare £15 with £16.
The flat £16 is bigger, by $16-15=£1$.
Answer: $\text{the £16 reduction, by £1}$
Marks
✔1$25\%$ of £60 $=£15$
✔1Compares with £16
✔1£16, by £1
★Extension
After year 1: $24000\times1.05=£25200$.
After year 2: $25200\times1.05=£26460$.
(Or $24000\times1.05^2=£26460$.)
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