Mark scheme · Percentages
Total: 20 marks
Award the marks shown for each correct step, then add up the total out of 20. A method mark counts even if the final answer is wrong.
In a survey, $81\%$ of people said they liked a new drink.
What percentage of the people did NOT like the drink?[1 mark]
MethodThe two percentages must add up to $100\%$.
$100 - 81 = 19$, so the answer is $19\%$.
Answer: $19$
Marks✔1 mark100 − 81 = 19%
There are 360 chocolates in a tin. 85% of them have toffee centres.
Work out the number of chocolates with toffee centres.[2 marks]
Method$85\% = 0.85$ as a decimal multiplier.
$0.85 \times 360 = 306$.
Answer: $306$
Marks✔1 mark85% = 0.85 (or 1% = 3.6)
✔1 mark306
A class did a test. 200 students sat the test and 144 of them passed.
What percentage of the students passed?[2 marks]
Method$\dfrac{144}{200} \times 100 = 72\%$.
Answer: $72$
Marks✔1 markfraction $\dfrac{144}{200}$
✔1 mark72%
A train ticket costs £60.
The price goes up by 45%.
Work out the new price of the ticket.[2 marks]
MethodMultiplier $= 1 + 0.45 = 1.45$.
$1.45 \times 60 = 87$.
Answer: $87$
Marks✔1 markmultiplier 1.45
✔1 mark£87
In a survey, $68\%$ of people said they liked a new drink.
What percentage of the people did NOT like the drink?[1 mark]
MethodThe two percentages must add up to $100\%$.
$100 - 68 = 32$, so the answer is $32\%$.
Answer: $32$
Marks✔1 mark100 − 68 = 32%
A book has 120 pages. 55% of the pages have a picture.
Work out how many pages have a picture.[2 marks]
Method$55\% = 0.55$ as a decimal multiplier.
$0.55 \times 120 = 66$.
Answer: $66$
Marks✔1 mark55% = 0.55 (or 1% = 1.2)
✔1 mark66
A class did a test. 500 students sat the test and 175 of them passed.
What percentage of the students passed?[2 marks]
Method$\dfrac{175}{500} \times 100 = 35\%$.
Answer: $35$
Marks✔1 markfraction $\dfrac{175}{500}$
✔1 mark35%
A train ticket costs £60.
The price goes up by 24%.
Work out the new price of the ticket.[2 marks]
MethodMultiplier $= 1 + 0.24 = 1.24$.
$1.24 \times 60 = 74.40$.
Answer: $74.40$
Marks✔1 markmultiplier 1.24
✔1 mark£74.40
In a survey, $54\%$ of people said they liked a new drink.
What percentage of the people did NOT like the drink?[1 mark]
MethodThe two percentages must add up to $100\%$.
$100 - 54 = 46$, so the answer is $46\%$.
Answer: $46$
Marks✔1 mark100 − 54 = 46%
A team played 360 matches.
They won 85% of the matches.
Work out how many matches they won.[2 marks]
Method$85\% = 0.85$ as a decimal multiplier.
$0.85 \times 360 = 306$.
Answer: $306$
Marks✔1 mark85% = 0.85 (or 1% = 3.6)
✔1 mark306
A class did a test. 800 students sat the test and 480 of them passed.
What percentage of the students passed?[2 marks]
Method$\dfrac{480}{800} \times 100 = 60\%$.
Answer: $60$
Marks✔1 markfraction $\dfrac{480}{800}$
✔1 mark60%
A train ticket costs £150.
The price goes up by 25%.
Work out the new price of the ticket.[2 marks]
MethodMultiplier $= 1 + 0.25 = 1.25$.
$1.25 \times 150 = 187.50$.
Answer: $187.50$
Marks✔1 markmultiplier 1.25
✔1 mark£187.50
$12\%$ of the students in a school are boys.
What percentage of the students are girls?[1 mark]
MethodThe two percentages must add up to $100\%$.
$100 - 12 = 88$, so the answer is $88\%$.
Answer: $88$
Marks✔1 mark100 − 12 = 88%
A team played 80 matches.
They won 35% of the matches.
Work out how many matches they won.[2 marks]
Method$35\% = 0.35$ as a decimal multiplier.
$0.35 \times 80 = 28$.
Answer: $28$
Marks✔1 mark35% = 0.35 (or 1% = 0.8)
✔1 mark28
A class did a test. 200 students sat the test and 16 of them passed.
What percentage of the students passed?[2 marks]
Method$\dfrac{16}{200} \times 100 = 8\%$.
Answer: $8$
Marks✔1 markfraction $\dfrac{16}{200}$
✔1 mark8%
A shop’s sales this year were 130% of last year’s sales.
Last year the sales were 120 units.
Work out this year’s sales.[2 marks]
Method$130\% = 1.3$ as a decimal multiplier.
$1.3 \times 120 = 156$ units.
Answer: $156$
Marks✔1 mark130% = 1.3
✔1 mark156 units
A charity raised £60. 32% of the money goes to a local school.
Without a calculator, work out 32% of £60.[3 marks]
Method$10\% = 6$, so $30\% = 18$.
$1\% = 0.6$, so $2\% = 1.2$.
Add: $18 + 1.2 = 19.2$.
Answer: $19.2$
Marks✔1 mark10% = 6 (find 10% and 1%)
✔1 markbuild up the required parts
✔1 mark£19.2
A house was worth £20 thousand.
It is now worth £25 thousand.
Work out the percentage increase in the value of the house.[3 marks]
MethodChange $= 25 - 20 = 5$.
Percentage change $= \dfrac{5}{20} \times 100 = 25\%$.
Answer: $25$
Marks✔1 markchange = 5
✔1 markdivide by the ORIGINAL 20
✔1 mark25%
Ben bought a sofa for £250 and later sold it for £190.
Work out Ben’s percentage loss.[3 marks]
MethodLoss $= 250 - 190 = 60$.
Percentage loss $= \dfrac{60}{250} \times 100 = 24\%$.
Answer: $24$
Marks✔1 markloss = 60
✔1 markdivide by the COST price 250
✔1 mark24%
Sam invests £1200 in an account paying 2% simple interest per year for 5 years.
Work out the total interest earned after 5 years.[3 marks]
MethodInterest for one year $= 2\%$ of £1200 $= 24.00$.
For 5 years: $24.00 \times 5 = 120.00$.
Total interest $= 120$.
Answer: $120$
Marks✔1 markone year = 24.00
✔1 markx 5 years = 120.00
✔1 mark£120
In a sale, the sale price of every item is found by multiplying its normal price by $0.95$.
Work out the percentage reduction in the sale.[2 marks]
MethodThe sale price is $0.95$ of the normal price, i.e. $95\%$ of it.
Reduction $= 100\% - 95\% = 5\%$.
Answer: $5$
Marks✔1 markSale price is 95% of normal
✔1 mark5% reduction
A family buys 3 adult tickets at £22 each and 3 child tickets at £10 each.
A 10% discount is applied to the whole order.
Work out the total amount they pay.[3 marks]
MethodCost before discount $= 3 \times 22 + 3 \times 10 = £96$.
Apply the discount: $96 \times 0.9 = £86.40$.
Answer: $86.40$
Marks✔1 markTotal before discount £96
✔1 mark× 0.9
✔1 mark£86.40
A shop’s sales this year were 160% of last year’s sales.
Last year the sales were 200 units.
Work out this year’s sales.[2 marks]
Method$160\% = 1.6$ as a decimal multiplier.
$1.6 \times 200 = 320$ units.
Answer: $320$
Marks✔1 mark160% = 1.6
✔1 mark320 units
A jacket costs £80.
In a sale the discount works out as 65% of the price.
Without a calculator, work out 65% of £80.[3 marks]
Method$10\% = 8$, so $60\% = 48$.
$5\% = 4$.
Add: $48 + 4 = 52$.
Answer: $52$
Marks✔1 mark10% = 8 (find 10% and 1%)
✔1 markbuild up the required parts
✔1 mark£52
A coat cost £50.
In a sale it is now £30.
Work out the percentage decrease in the price of the coat.[3 marks]
MethodChange $= 50 - 30 = 20$.
Percentage change $= \dfrac{20}{50} \times 100 = 40\%$.
Answer: $40$
Marks✔1 markchange = 20
✔1 markdivide by the ORIGINAL 50
✔1 mark40%
For two positive numbers $x$ and $y$, $5y = x$.
Complete the statement: $y$ is $\square\%$ of $x$.[2 marks]
MethodRearrange for $y$: $y = \dfrac{1}{5}x = 0.2x$.
Multiplying $x$ by $0.2$ is the same as finding $20\%$ of $x$, so $y$ is $20\%$ of $x$.
Answer: $20$
Marks✔1 marky = 0.2x
✔1 mark20%
The area of rectangle B is 20% less than the area of rectangle A.
The area of rectangle C is 10% greater than the area of rectangle B.
Work out the percentage decrease from rectangle A to rectangle C.[3 marks]
MethodMultiplier A→B $= 0.8$; multiplier B→C $= 1.1$.
Overall multiplier $= 0.8 \times 1.1 = 0.88$.
That is a 12% decrease (since $0.88 = 88\%$).
Answer: $12$
Marks✔1 marktwo multipliers 0.8 and 1.1
✔1 markoverall multiplier 0.88
✔1 mark12% decrease
A garage gives a 20% discount for paying cash.
Gwen pays £200 cash for a car.
Work out the normal price of the car.[3 marks]
MethodThe £200 represents $80\%$ of the original.
Original $= 200 \div 0.8 = 250$.
Answer: $250$
Marks✔1 mark200 is 80% of the original
✔1 markdivide by 0.8
✔1 mark£250
In a sale, normal prices are reduced by 5%.
The normal price of a washing machine is reduced by £12.
Work out the sale price of the washing machine.[3 marks]
MethodThe £12 reduction is 5% of the normal price.
Normal price $= 12 \div 0.05 = 240.00$.
Sale price $= 240.00 - 12 = 228$.
Answer: $228$
Marks✔1 mark12 = 5% of normal price
✔1 marknormal price = 240.00
✔1 marksale price £228
Ben invests £2000 at $y\%$ simple interest per year.
After 3 years the total interest earned is £240.
Work out the value of $y$.[3 marks]
MethodSimple interest $= \dfrac{P \times y \times t}{100}$, so $240 = \dfrac{2000 \times y \times 3}{100}$.
$240 = 60y$, so $y = 240 \div 60 = 4$.
Answer: $4$
Marks✔1 markI = P y t / 100
✔1 marky = 240 ÷ 60
✔1 mark4%
For two positive numbers $x$ and $y$, $4y = 5x$.
Complete the statement: $y$ is $\square\%$ of $x$.[2 marks]
MethodRearrange for $y$: $y = \dfrac{5}{4}x = 1.25x$.
Multiplying $x$ by $1.25$ is the same as finding $125\%$ of $x$, so $y$ is $125\%$ of $x$.
Answer: $125$
Marks✔1 marky = 1.25x
✔1 mark125%
A garage gives a 20% discount for paying cash.
Aisha pays £200 cash for a car.
Work out the normal price of the car.[3 marks]
MethodThe £200 represents $80\%$ of the original.
Original $= 200 \div 0.8 = 250$.
Answer: $250$
Marks✔1 mark200 is 80% of the original
✔1 markdivide by 0.8
✔1 mark£250
In a sale, normal prices are reduced by 15%.
The normal price of a washing machine is reduced by £18.
Work out the sale price of the washing machine.[3 marks]
MethodThe £18 reduction is 15% of the normal price.
Normal price $= 18 \div 0.15 = 120.00$.
Sale price $= 120.00 - 18 = 102$.
Answer: $102$
Marks✔1 mark18 = 15% of normal price
✔1 marknormal price = 120.00
✔1 marksale price £102
Sharon invests £2000 at $y\%$ simple interest per year.
After 6 years the total interest earned is £360.
Work out the value of $y$.[3 marks]
MethodSimple interest $= \dfrac{P \times y \times t}{100}$, so $360 = \dfrac{2000 \times y \times 6}{100}$.
$360 = 120y$, so $y = 360 \div 120 = 3$.
Answer: $3$
Marks✔1 markI = P y t / 100
✔1 marky = 360 ÷ 120
✔1 mark3%
For two positive numbers $x$ and $y$, $4y = x$.
Complete the statement: $y$ is $\square\%$ of $x$.[2 marks]
MethodRearrange for $y$: $y = \dfrac{1}{4}x = 0.25x$.
Multiplying $x$ by $0.25$ is the same as finding $25\%$ of $x$, so $y$ is $25\%$ of $x$.
Answer: $25$
Marks✔1 marky = 0.25x
✔1 mark25%
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