-
Non-calculator 2 marks
The price of a sofa increases by 50%.
This 50% increase adds £211 to the price of the sofa.
Work out the price of the sofa before the increase.
Show answer
\(\text{£}422\)The £211 is 50% of the price before the increase.
\(100 \div 50 = 2\), so the price before the increase (100%) is 2 times as much as the £211.
\(£211 \times 2 = £422\)
The price before the increase was £422.
- M1the \(£211\) is 50%, so multiplies by \(2\) to get 100%
- A1price before the increase \(£422\)
-
Non-calculator 2 marks
In a sale, the normal price of a wardrobe is reduced by 62.5%.
The sale price is £243.
Work out the normal price.
You must not use a calculator.
Show answer
\(\text{£}648\)The price was reduced by 62.5%, so the sale price is \(37.5\%\) of the normal price.
\(37.5\%\) is 3 blocks of \(12.5\%\), so 3 blocks are worth £243.
One block of \(12.5\%\) is \(243 \div 3 = 81\), so it is £81.
\(100\%\) is 8 blocks of \(12.5\%\):
\(81 \times 8 = 648\)
The normal price is £648.
- M1sale price divided by the blocks left, one block of \(12.5\%\) is £81
- A1multiplied by \(8\) blocks, normal price £648
-
Either paper 3 marks
In a sale, the price of every item in a shop is reduced by 25%.
The sale price of a garden bench is £75.
Work out the normal price of the garden bench.
Show answer
\(\text{£}100\)It is tempting to work out 25% of £75 and add it back on. That gives \(75 \times \frac{ 25 }{100} = 18.75\) and a normal price of £93.75, and it is wrong.
The reason is that 25% of the sale price is not the same amount of money as 25% of the normal price, and it was the normal price the reduction was taken from. Taking a percentage of the wrong starting number can never undo it.
Use a multiplier instead. Taking 25% off leaves 75% of the normal price, so
\(\text{normal price} \times 0.75 = 75\)
Multiplying was done to get here, so divide to get back:
\(75 \div 0.75 = 100\)
The normal price is £100. Check it: \(100 \times 0.75 = 75\).
- M1multiplier \(0.75\) for \(75\)% of the normal price
- M1dividing \(75\) by \(0.75\)
- A1normal price £\(100\)
-
Either paper 3 marks
A shop sells a bookshelf for £294.
This price includes VAT at 20%.
(a)Work out the price of the bookshelf before VAT was added.
(b)Work out the amount of VAT included in the £294.
Show answer
(a) \(\text{£}245\)
(b) \(\text{£}49\)Adding VAT at 20% multiplies the price before VAT by
\(1 + \frac{20}{100} = 1.2\)
so \(\text{price before VAT} \times 1.2 = 294\).
Divide to undo the multiplication:
\(294 \div 1.2 = 245\)
The price before VAT is £245.
Note that 20% of £294 is \(294 \times \frac{20}{100} = 58.80\), which is not the VAT — that is 20% of the price with VAT already in it.
\(£294 - £245 = £49\)
Check: 20% of £245 is \(245 \times \frac{20}{100} = 49\).
- (a) M1dividing \(294\) by \(1.2\)
- (a) A1price before VAT £\(245\)
-
Either paper 4 marks
During last year the number of members of Westgate Chess Club increased by 25%.
The increase was 130 members.
(a)Work out how many members Westgate Chess Club had at the start of last year.
(b)Work out how many members Westgate Chess Club has now.
Show answer
(a) \(520\ \text{members}\)
(b) \(650\ \text{members}\)The 130 extra members are 25% of the number at the start of the year, not 25% of the number now.
\(\frac{ 25 }{100} \times \text{members at the start} = 130\)
So the number at the start is
\(130 \times \frac{100}{ 25 } = 520\)
Westgate Chess Club had 520 members at the start of the year.
\(520 + 130 = 650\)
- (a) M1increase 130 is 25% of the original number
- (a) M1\(130 \times \frac{100}{ 25 }\)
- (a) A1520 members
-
Either paper 3 marks
In a sale, the price of every item in a shop is reduced by 25%.
Zainab sees a laptop with a sale price of £225. She works out the normal price like this.
25% of £225 = £56.25
Normal price = £225 + £56.25 = £281.25
Zainab is wrong.
(a)Explain the mistake Zainab has made.
(b)Work out the correct normal price of the laptop.
Show answer
(a) Zainab has found 25% of the sale price, but the reduction was 25% of the normal price, which is a larger amount of money.
(b) \(\text{£}300\)The 25% was taken off the normal price, so 25% of the normal price is the size of the reduction. Zainab has instead found 25% of the sale price, which is a smaller amount of money, so adding it back on does not return to where the reduction started.
You can see it fails by checking the answer: 25% off £281.25 is not £225.
Reducing by 25% leaves 75% of the normal price, so it multiplies the normal price by
\(1 - \frac{ 25 }{100} = 0.75\)
\(\text{normal price} \times 0.75 = 225\)
Divide to undo it:
\(225 \div 0.75 = 300\)
The normal price is £300.
- (b) M1divides the sale price \(£225\) by the multiplier \(0.75\) (75% of the normal price)
- (b) A1normal price \(£300\)
-
Either paper 3 marks
A shop reduces the price of everything it sells by 10%.
Erin chooses a sound bar. She pays the reduced price, and uses a £25 gift voucher towards it, so she hands over £425.
Work out the price of the sound bar before the shop reduced it.
Show answer
\(\text{£}500\)Undo the steps in the opposite order to the one they were done in. The voucher came off last, so put it back on first.
\(£425 + £25 = £450\)
That £450 is the reduced price. Taking 10% off leaves 90% of the price before the reduction, so it multiplies that price by
\(1 - \frac{ 10 }{100} = 0.9\)
\(\text{price before the reduction} \times 0.9 = 450\)
That multiplier is the last thing between £450 and the answer, so divide by it.
\(450 \div 0.9 = 500\)
The price before the shop reduced it was £500.
The order matters. Dividing the £425 by 0.9 straight away gives the wrong answer, because £425 is not the reduced price — the voucher has already come off it. The voucher has to go back on before the reduction is undone.
- M1voucher put back on \(450\)
- M1\(450 \div 0.9\)
- A1price \(500\)
-
Either paper 3 marks
The number of books borrowed from a village library this year was 160% of the number last year.
This year 3104 books were borrowed.
(a)Work out how many books were borrowed last year.
(b)Write down the percentage increase from last year to this year.
Show answer
(a) \(1940\)
(b) \(60\%\)"160% of last year's number" means last year's number has been multiplied by
\(\frac{ 160 }{100} = 1.6\)
so \(\text{last year} \times 1.6 = 3104\).
Going forward a year multiplies by \(1.6\), so going back a year divides by it.
\(3104 \div 1.6 = 1940\)
Last year 1940 books were borrowed.
Check it: \(1940 \times 1.6 = 3104\).
Last year's number counts as 100%, and this year's is 160% of it, so the increase is
\(160 - 100 = 60\)
a 60% increase. There is nothing to divide here: the increase is already written into the 160%.
- (a) M1divides this year's figure \(3104\) by the multiplier \(1.6\)
- (a) A1last year's figure \(1940\)
-
Either paper 3 marks
A shop has taken 25% off the price of all its garden chairs.
Leila buys 8 identical garden chairs and pays £186.00 altogether.
Work out what one garden chair cost before the shop took the 25% off.
Show answer
\(\text{£}31\)Taking 25% off leaves 75%, so every price has been multiplied by
\(1 - \frac{ 25 }{100} = 0.75\)
That multiplier applies to the total as well as to one garden chair, so undo it on the total first.
\(£186.00 \div 0.75 = £248\)
That £248 is what 8 garden chairs would have cost before the reduction, so divide by 8.
\(£248 \div 8 = £31\)
One garden chair cost £31.
Dividing by 8 first works just as well. The sale price of one is \(£186.00 \div 8 = £23.25\), and undoing the reduction on that gives \(£23.25 \div 0.75 = £31\).
- M1total divided by 0.75 \(248\)
- M1divided by 8
- A1price \(31\)
-
Either paper 3 marks
A shop has a sale. Every price in the shop is reduced by 30%.
A log burner costs £492.10 in the sale.
Before the sale started, Yusuf had £692 set aside for it.
(a)Work out the price of the log burner before the sale.
(b)Could Yusuf have bought the log burner at the price it was before the sale?
Give a reason for your answer.
Show answer
(a) \(\text{£}703\)
(b) No — £692 is £11 less than the £703 it cost before the saleThe £492.10 is the price after the reduction. Reducing by 30% leaves 70% of the old price, so it multiplies the old price by
\(1 - \frac{ 30 }{100} = 0.7\)
Dividing undoes that.
\(£492.10 \div 0.7 = £703\)
Compare the £692 with the price before the sale, not with the price in the sale.
£692 is £11 less than £703, so Yusuf could not have bought it.
- (a) M1dividing \(492.10\) by \(0.7\)
- (a) A1price before the sale £\(703\)