Percentages · applications

Applications: Percentages

23 problems

PSLE · O-Level · SAT · GCSE Higher

01

A Laptop with a Fair Discount, a Member's Discount and Sales Tax

heuristicTake Each Discount from the Price Before It / Add the Tax to the Price Paid

A laptop is marked at $1800. At a computer fair, every customer gets 20% off the marked price, and members get a further 5% off the price after that discount. A 9% sales tax is then added to the price a member pays. (a) What percentage of the marked price do a member's two discounts take off altogether? (b) How much does a member pay for the laptop, with the sales tax?

Marked100 units = $1800
The marked price, $1800, is 100 units, so 1 unit is 1800 ÷ 100 = $18.
Draw the marked price, $1800, as a bar of 100 units. Then 1 unit is 1800 ÷ 100 = $18.
step 1 of 6

Draw the marked price as 100 units, so that each percentage of it is a number of units. The member's 5% is a percentage of what is left after the first discount, so it is taken from those units, not from all 100.

All steps
  1. Draw the marked price, $1800, as a bar of 100 units. Then 1 unit is 1800 ÷ 100 = $18.
  2. The fair takes 20% off, which is 20 units. That leaves 80 units: 80 × $18 = $1440.
  3. The member's 5% is taken off the 80 units, not off the 100: 5100 × 80 = 4 units. So 80 − 4 = 76 units are left.
  4. (a) The two discounts take 20 + 4 = 24 units of the 100, which is 24% of the marked price.
  5. Before the tax, a member pays 76 units: 76 × $18 = $1368. The tax is 9% of this: 9100 × 1368 = $123.12.
  6. (b) A member pays 1368 + 123.12 = $1491.12. Check: 0.8 × 0.95 × 1800 = 1368, and 1.09 × 1368 = 1491.12.

answer(a) 24%; (b) $1491.12

techniqueTaking a Percent Off · Sales Tax

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Adding the two discounts to get 20% + 5% = 25%. The 5% is taken from the price after the first discount, $1440, which is less than $1800, so it is only 4 of the 100 units, and the two together are 24%.
  • Working out the tax on the marked price: 9% of $1800 is $162. The tax is added to the price the member actually pays, $1368, so it is $123.12.
02

A Camera Marked Up, Then Sold at a Discount: Working Back to the Cost

heuristicWork Backwards / Each Percentage Is of the Price Before It

A shop bought a camera and marked up its cost price by 25% to set the marked price. In a sale, the shop took 16% off the marked price, and a customer bought the camera for $840. (a) What was the marked price of the camera? (b) How much profit did the shop make on the camera?

Marked84 parts paid16 off100 parts
The 16% is off the marked price, so the marked price is 100 parts. The customer paid the other 84 parts: $840.
The 16% is off the marked price, so draw the marked price as 100 parts. The sale took 16 parts, and the customer paid the other 84 parts, so 84 parts are $840.
step 1 of 5

Two different wholes are used. The discount is a percentage of the marked price, so draw the marked price as 100 parts; the markup is a percentage of the cost, so draw the cost as 100 units.

All steps
  1. The 16% is off the marked price, so draw the marked price as 100 parts. The sale took 16 parts, and the customer paid the other 84 parts, so 84 parts are $840.
  2. 1 part is 840 ÷ 84 = $10.
  3. (a) The marked price is 100 parts: 100 × $10 = $1000.
  4. The 25% markup is of the cost price, so draw the cost as 100 units. The marked price is 100 + 25 = 125 units, and 125 units are $1000, so 1 unit is 1000 ÷ 125 = $8 and the cost is 100 × $8 = $800.
  5. (b) The profit is the price the camera sold for less its cost: 840 − 800 = $40. Check: 1.25 × 800 = 1000, and 0.84 × 1000 = 840.

answer(a) $1000; (b) $40

techniqueReverse Percentages · Percentage Change

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Adding 16% of $840 back on to get the marked price: 840 + 134.40 = $974.40. The 16% is a percentage of the marked price, not of the sale price, so $840 is 84% of the marked price, and the marked price is 840 ÷ 0.84 = $1000.
  • Taking the profit as the marked price less the cost, 1000 − 800 = $200. The shop was paid the sale price, $840, not the marked price, so the profit is 840 − 800 = $40.
03

Boys Come into a Hall as the Same Number of Girls Leave

heuristicThe Total Does Not Change / Both Percentages Are of the Same Total

In a hall, 40% of the audience were boys and the rest were girls. Then 48 boys came in and 48 girls left, and the boys became 60% of the audience. (a) How many people were in the hall altogether? (b) How many girls were in the hall at first?

Audience5 units
48 came in and 48 left, so the total did not change. It is 5 units, since 40% = 25 and 60% = 35.
48 people came in and 48 left, so the total did not change. Draw it as a bar of 5 equal units, since 40% = 25 and 60% = 35.
step 1 of 5

As many people came in as left, so the number in the hall did not change, and both percentages are of the same total. Draw that total as one bar of 5 units, because 40% and 60% are both fifths.

All steps
  1. 48 people came in and 48 left, so the total did not change. Draw it as a bar of 5 equal units, since 40% = 25 and 60% = 35.
  2. At first the boys were 25 of the audience, 2 units, and the girls were the other 3 units.
  3. In the end the boys were 35, 3 units, and the girls 2 units. The boys went from 2 units to 3 units, so 1 unit is the 48 boys who came in.
  4. (a) The audience is 5 units: 5 × 48 = 240 people.
  5. (b) At first the girls were 3 units: 3 × 48 = 144 girls. Check: in the end there were 144 − 48 = 96 girls, and 96240 = 40100 = 40%, so the boys were the other 60%.

answer(a) 240 people; (b) 144 girls

techniqueFinding a Percent · Ratios with a Constant Total

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Taking the change as 48 + 48 = 96 people, because 48 came in and 48 left. The boys' share grew only by the 48 boys who came in; the 48 girls who left changed the number of girls, not of boys. So the 20% rise in the boys is 48 people.
  • Answering (b) with 96. That is the 2 units of girls in the end, after 48 of them had left; the girls at first were 3 units, 144.
04

Mushrooms Dried in the Sun: Water Is Lost and the Pulp Stays

heuristicThe Pulp Does Not Change / Find It, Then the Whole It Is a Percentage Of

A crate of fresh mushrooms has a mass of 40 kg, and 90% of the mass is water. The mushrooms are dried in the sun until water makes up 60% of their mass. Only water is lost. (a) What is the mass of the dried mushrooms? (b) How much water was lost?

Freshwater 36 kg440 kgpulp: 10% = 4 kg
Only water is lost, so the pulp stays the same. Fresh, the pulp is 10% of 40 kg: 4 kg.
Only water is lost, so the pulp, the part that is not water, has the same mass before and after drying. Fresh, the pulp is 100% − 90% = 10% of the mass: 10100 × 40 = 4 kg.
step 1 of 5

The pulp is the same before and after drying, so it links the two bars. Find it in the fresh mushrooms, then use it to find the size of a unit in the dried ones.

All steps
  1. Only water is lost, so the pulp, the part that is not water, has the same mass before and after drying. Fresh, the pulp is 100% − 90% = 10% of the mass: 10100 × 40 = 4 kg.
  2. Dried, the water is 60% and the pulp is 40%. Draw the dried mushrooms as 5 equal units: water is 60% = 35, 3 units, and pulp is 40% = 25, 2 units.
  3. The pulp is still 4 kg, so 2 units are 4 kg, and 1 unit is 4 ÷ 2 = 2 kg.
  4. (a) The dried mushrooms are 5 units: 5 × 2 = 10 kg.
  5. (b) Only water was lost, so the water lost is 40 − 10 = 30 kg. Check: the fresh mushrooms held 0.9 × 40 = 36 kg of water and the dried ones 3 × 2 = 6 kg, and 36 − 6 = 30.

answer(a) 10 kg; (b) 30 kg

techniqueRatios with a Constant Part · Finding a Percent

examsPSLE · SAT

Common pitfalls

  • Taking 90% − 60% = 30% of the 40 kg away, to get 28 kg. The 60% is a percentage of the dried mass, which is less than 40 kg, so the two percentages are of different wholes and cannot be subtracted.
  • Taking 60% of 40 kg, 24 kg, as the dried mass. The 60% is the share of water in the dried mushrooms, not the share of the fresh mass that is left. The part that stays the same is the pulp, 4 kg, and it is 40% of the dried mass.
05

Two Friends Spend Different Percentages and Are Left with the Same Amount

heuristicWhat They Keep Is Equal / Make the Kept Parts the Same Number of Units

Alan had $60 more than Dylan. Alan spent 40% of his money and Dylan spent 25% of his. Then they had the same amount of money left. (a) How much money did Alan have at first? (b) How much money did each of them have left?

Alankeptkeptkeptspentspent5 units
Alan kept 60% = 35 of his money: 3 of 5 units.
Alan kept 100% − 40% = 60% = 35 of his money, so draw his money as 5 units and shade the 3 he kept.
step 1 of 5

The amounts they kept are equal. Alan kept 35 of his money and Dylan 34 of his, so with 3 units kept in each bar, every unit in both bars is the same size, and the two whole bars can be compared.

All steps
  1. Alan kept 100% − 40% = 60% = 35 of his money, so draw his money as 5 units and shade the 3 he kept.
  2. Dylan kept 100% − 25% = 75% = 34 of his money, so draw his money as 4 units and shade the 3 he kept.
  3. What they kept is the same amount, and it is 3 units in both bars, so all the units are the same size. Alan had 5 units and Dylan 4, so Alan had 1 unit more, and that unit is $60.
  4. (a) Alan had 5 × $60 = $300 at first.
  5. (b) Each of them had 3 units left: 3 × $60 = $180. Check: Dylan had 4 × $60 = $240; Alan spent 40% of $300, which is $120, and Dylan spent 25% of $240, which is $60, and 300 − 120 = 240 − 60 = 180.

answer(a) $300; (b) $180 each

techniqueFinding a Percent · Ratio, Fraction, Percent

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Taking 40% − 25% = 15% as the $60. The two percentages are of different amounts, Alan's money and Dylan's, so they cannot be subtracted. The $60 is the difference between their whole amounts, which is 1 unit.
  • Giving Alan 4 units and Dylan 5, because Alan spent the bigger percentage. Alan had $60 more, so his bar is the longer one: he kept 3 of his 5 units, and Dylan kept 3 of his 4.
06

A Scooter's Cost Found from Its Markup, Its Discount and the Profit

heuristicDraw the Cost as 100 Units / The Discount Is a Percentage of the Marked Price

A shop marked up the cost price of an electric scooter by 40%. In a sale, it took 15% off the marked price, and it sold the scooter at the sale price for a profit of $114. (a) What was the cost price of the scooter? (b) What was its marked price?

Cost100 unitsMarked100+40140 units
The cost is 100 units. The 40% markup makes the marked price 140 units.
The markup is a percentage of the cost price, so draw the cost as 100 units. The marked price is 100 + 40 = 140 units.
step 1 of 5

The markup is a percentage of the cost, so draw the cost as 100 units. The discount is a percentage of the marked price, so it is worked out on the marked price's units, not on the cost's 100.

All steps
  1. The markup is a percentage of the cost price, so draw the cost as 100 units. The marked price is 100 + 40 = 140 units.
  2. The discount is a percentage of the marked price: 15% of 140 units is 15100 × 140 = 21 units. The sale price is 140 − 21 = 119 units.
  3. The profit is the sale price less the cost: 119 − 100 = 19 units, and that is $114. So 1 unit is 114 ÷ 19 = $6.
  4. (a) The cost price is 100 × $6 = $600.
  5. (b) The marked price is 140 × $6 = $840. Check: 15% of $840 is $126, so the sale price is 840 − 126 = $714, and 714 − 600 = 114.

answer(a) $600; (b) $840

techniquePercentage Change · Taking a Percent Off

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Taking 40% − 15% = 25% as the profit, so that $114 is 25% of the cost. The 15% is of the marked price, 140 units, not of the cost, so it is 21 units, and the profit is only 19 units, which is 19% of the cost.
  • Taking the 15% off the cost, as 15 units. A discount is taken from the marked price, which here is 140 units, so the discount is 21 units and the sale price is 119 units, not 125.
07

Red Pens Restocked and Blue Pens Sold, with Only the New Total Known

heuristicSuppose Every Pen Fell by 30% / Each Red Pen Makes Up the Difference

A bookstore had 360 pens, some red and the rest blue. The number of red pens then went up by 20% and the number of blue pens went down by 30%. After that, the store had 327 pens. (a) How many red pens were there at first? (b) How many blue pens were there in the end?

At first360 pensIf all −30%252 left30% gone
Suppose every pen had gone down by 30%: 0.7 × 360 = 252 would be left.
Suppose every pen, red and blue, had gone down by 30%. Then 70% of the 360 pens would be left: 0.7 × 360 = 252 pens.
step 1 of 5

Suppose every pen had gone down by 30%, and compare that with the 327 pens there really are. The difference is made by the red pens, each of which went up instead of down.

All steps
  1. Suppose every pen, red and blue, had gone down by 30%. Then 70% of the 360 pens would be left: 0.7 × 360 = 252 pens.
  2. There are really 327 pens, which is 327 − 252 = 75 more than that.
  3. The 75 extra pens come from the red pens. Each red pen went up by 20% instead of down by 30%, so for every red pen there is 20% + 30% = 50% of a pen more than supposed. The 75 pens are 50% of the red pens.
  4. (a) There were 75 ÷ 0.5 = 150 red pens at first.
  5. (b) There were 360 − 150 = 210 blue pens at first, and 30% of them went: 210 − 0.3 × 210 = 210 − 63 = 147 blue pens. Check: the red pens went up to 1.2 × 150 = 180, and 180 + 147 = 327.

answer(a) 150 red pens; (b) 147 blue pens

techniqueFinding a Percent · Simultaneous by Elimination

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Taking the gap between a 20% rise and a 30% fall as 30% − 20% = 10%. A red pen was supposed to fall by 30% but rose by 20%. Getting back to where it started is 30% of a pen, and the rise is 20% more, so it ends 50% of a pen above what was supposed.
  • Answering (b) with 210. That is the number of blue pens at first; 30% of them then went, so in the end there were 0.7 × 210 = 147.
08

Money Spent on a Vacuum Cleaner, Then on a Rice Cooker from What Was Left

heuristicWork Backwards / Put Back the Extra Dollars, Then Undo Each Percentage

Mrs. Goh had some money. She spent 25% of it on a vacuum cleaner. Then she spent 40% of the money left and another $18 on a rice cooker. She had $126 left. (a) How much money did Mrs. Goh have at first? (b) How much did the rice cooker cost?

After vacuum2 units18$126 left5 units3 units, less $18, = $126
What was left after the vacuum cleaner is 5 units. The rice cooker took 2 units and $18 more, and $126 was left.
Draw the money left after the vacuum cleaner as 5 units, since 40% = 25. The rice cooker took 2 units and $18 more, so the other 3 units, less $18, are the $126 she had left.
step 1 of 5

Start from the end and work backwards. The 40% is of the money left after the vacuum cleaner, so draw that money as 5 units; the 25% is of all her money, so draw all of it as 4 parts. Put back each fixed amount before undoing a percentage.

All steps
  1. Draw the money left after the vacuum cleaner as 5 units, since 40% = 25. The rice cooker took 2 units and $18 more, so the other 3 units, less $18, are the $126 she had left.
  2. Put the $18 back: 3 units are 126 + 18 = $144, so 1 unit is 144 ÷ 3 = $48, and the money left after the vacuum cleaner was 5 × $48 = $240.
  3. Now draw all her money as 4 parts, since 25% = 14. The vacuum cleaner took 1 part, so the $240 is the other 3 parts, and 1 part is 240 ÷ 3 = $80.
  4. (a) She had 4 × $80 = $320 at first.
  5. (b) The rice cooker cost 2 units and $18: 2 × 48 + 18 = 96 + 18 = $114. Check: 320 − 80 = 240, and 240 − 114 = 126.

answer(a) $320; (b) $114

techniqueFinding a Percent · Multiplying and Dividing Undo Each Other

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Taking 126 − 18 = $108 as the 3 units. The $18 was spent, so working backwards it is put back: the 3 units are 126 + 18 = $144.
  • Taking 40% of all her money, $320, for the rice cooker. The 40% is of the money left after the vacuum cleaner, $240, so it is $96, and the cooker cost 96 + 18 = $114.
09

Visitors at a Science Exhibition: Adults and Children, Then Men and Boys

heuristicA Percentage of a Percentage / Draw Everyone as 100 Units

At a science exhibition, 60% of the visitors were adults and the rest were children. 70% of the adults were men, and 60% of the children were boys. There were 252 men. (a) How many children were at the exhibition? (b) What percentage of all the visitors were male?

Visitors100 unitsadults: 60children: 40
All the visitors are 100 units: the adults are 60 and the children 40.
Draw all the visitors as 100 units. The adults are 60 units and the children are the other 40 units.
step 1 of 5

Draw all the visitors as 100 units, so that every percentage of them is a number of units. The men are a percentage of the adults' units and the boys a percentage of the children's units.

All steps
  1. Draw all the visitors as 100 units. The adults are 60 units and the children are the other 40 units.
  2. The men are 70% of the adults: 70100 × 60 = 42 units. They are 252 people, so 1 unit is 252 ÷ 42 = 6 people.
  3. (a) The children are 40 units: 40 × 6 = 240 children.
  4. The boys are 60% of the children: 60100 × 40 = 24 units.
  5. (b) The males are the men and the boys: 42 + 24 = 66 units of the 100, which is 66%. Check: there were 100 × 6 = 600 visitors and 252 + 24 × 6 = 396 males, and 396600 = 66100.

answer(a) 240 children; (b) 66%

techniqueFinding a Percent · Ratio, Fraction, Percent

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Averaging the two percentages, 70% + 60%2 = 65%. There are more adults than children, so the men's 70% is of a bigger group: the males are 42 units and 24 units out of 100, which is 66%.
  • Taking the boys as 60% of all the visitors. The 60% is of the children only, 40 units, so the boys are 24 units, not 60.
10

A Father's Age as a Percentage of His Son's, in Three Different Years

heuristicThe Age Gap Never Changes / Turn Each Percentage into a Ratio and Match the Gaps

In 2016, Mr. Tan's age was 400% of his son Ken's age. In 2024, Mr. Tan's age was 250% of Ken's age. (a) How old was Ken in 2016? (b) In which year will Mr. Tan's age be 180% of Ken's age?

Tan 20164 unitsKen 20161 unitTan 20245 unitsKen 20242 units
400% is 4 times: in 2016, Mr. Tan : Ken = 4 : 1. 250% is 212 times: in 2024, 5 : 2.
400% means 4 times, so in 2016 Mr. Tan : Ken = 4 : 1. 250% means 212 times, so in 2024 Mr. Tan : Ken = 5 : 2.
step 1 of 6

Turn each percentage into a ratio of Mr. Tan's age to Ken's. The gap between their ages never changes, so when the gap is the same number of units in two ratios, the units are the same size.

All steps
  1. 400% means 4 times, so in 2016 Mr. Tan : Ken = 4 : 1. 250% means 212 times, so in 2024 Mr. Tan : Ken = 5 : 2.
  2. The gap between their ages never changes. It is 4 − 1 = 3 units in 2016 and 5 − 2 = 3 units in 2024, so the units are the same size in both years.
  3. Ken went from 1 unit to 2 units in the 2024 − 2016 = 8 years, so 1 unit is 8 years.
  4. (a) Ken was 1 unit, 8 years old, in 2016. Mr. Tan was 4 × 8 = 32, so the gap is 32 − 8 = 24 years.
  5. 180% = 95, so in that year Mr. Tan : Ken = 9 : 5. The gap is 9 − 5 = 4 parts, and it is still 24 years, so 1 part is 24 ÷ 4 = 6 years, and Ken will be 5 × 6 = 30.
  6. (b) Ken will be 30 in 30 − 8 = 22 years after 2016, which is 2038. Check: in 2038 Mr. Tan will be 32 + 22 = 54, and 5430 = 1.8 = 180%.

answer(a) 8 years old; (b) 2038

techniqueRatios with a Constant Difference · Percentage

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Writing 250% as the ratio 2.5 : 1 and comparing it unit for unit with 4 : 1. In 2.5 : 1 the gap is 1.5 units and in 4 : 1 it is 3 units, so the units are different sizes. Written as 5 : 2, the gap is 3 units in both, and the units can be compared.
  • Giving the year as 2016 + 30 = 2046, because Ken will be 30. Ken was already 8 in 2016, so he turns 30 only 30 − 8 = 22 years later, in 2038.
11

A Stationery Order Discounted Only on the Part Above a Set Amount

heuristicSplit the Order at the Threshold / Only the Part Above It Gets the Discount

An online stationery shop charges the first $60 of an order at full price, and takes 15% off the part of the order above $60. (a) Marcus places an order worth $140 at full price. How much does he pay? (b) Chloe pays $102.50 for her order. What was her order worth at full price?

Marcus$60 full price$80$140
Marcus's $140 splits into $60 at full price and $80 that gets the discount.
Split Marcus's $140 order at $60: the first $60 is at full price, and the other 140 − 60 = $80 gets the discount.
step 1 of 6

Draw each order as a bar split at $60. The first $60 is paid in full. Only the part above it gets the discount, so only that part is drawn as 100 parts.

All steps
  1. Split Marcus's $140 order at $60: the first $60 is at full price, and the other 140 − 60 = $80 gets the discount.
  2. 15% of $80 is 15100 × 80 = $12, so the $80 part costs 80 − 12 = $68.
  3. (a) Marcus pays 60 + 68 = $128.
  4. Chloe paid more than $60, so her first $60 was at full price, and the other 102.50 − 60 = $42.50 paid for the discounted part. With 15% off, that part costs 85 of its 100 parts, so 85 parts are $42.50.
  5. 1 part is 42.50 ÷ 85 = $0.50, so at full price the discounted part was worth 100 × $0.50 = $50.
  6. (b) Chloe's order was worth 60 + 50 = $110 at full price. Check: 15% of $50 is $7.50, and 60 + 50 − 7.50 = 102.50.

answer(a) $128; (b) $110

techniqueTaking a Percent Off · Finding a Percent

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Taking 15% off the whole $140: 0.85 × 140 = $119. The first $60 is charged at full price, so only the $80 above it gets the discount.
  • Dividing all of Chloe's $102.50 by 0.85. The $60 she paid for the first part had no discount, so it is taken away first; only the $42.50 is 85% of a full price.
12

A Savings Account Earning Simple Interest, with Money Taken Out After a Year

heuristicOne Year at a Time / Each Year's Interest Is on the Money in the Account That Year

Mr. Lim put $12000 into a savings account that pays simple interest at 3.5% a year. At the end of the first year, the year's interest was added to the account, and then Mr. Lim took $4420 out. The money left stayed in the account for a second year at the same rate. (a) How much interest did the account earn over the two years? (b) How much money was in the account at the end of the second year?

Year 1$12000+ $420year 1: 3.5% of $12000 = $420
The first year earns 3.5% of $12000: 3.5100 × 12000 = $420.
In the first year the account earns 3.5% of $12000: 3.5100 × 12000 = $420.
step 1 of 5

Follow the money in the account one year at a time. Each year's interest is 3.5% of the money in the account during that year, and the withdrawal makes that a smaller amount in the second year.

All steps
  1. In the first year the account earns 3.5% of $12000: 3.5100 × 12000 = $420.
  2. The interest is added, so the account holds 12000 + 420 = $12420. Mr. Lim takes $4420 out, which leaves 12420 − 4420 = $8000.
  3. In the second year the interest is worked out on the $8000 that is in the account: 3.5100 × 8000 = $280.
  4. (a) The interest over the two years is 420 + 280 = $700.
  5. (b) At the end of the second year the account holds 8000 + 280 = $8280. Check: $12000 was put in, $700 of interest was added and $4420 was taken out, and 12000 + 700 − 4420 = 8280.

answer(a) $700; (b) $8280

techniqueSimple Interest · Finding a Percent

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Working out the second year's interest on $12000, which gives $420 again and $840 in all. Interest is paid on the money in the account during the year, and after the withdrawal that is only $8000.
  • Taking the $4420 out of $12000 before adding the first year's interest, which leaves $7580. The interest was added at the end of the first year, before the money was taken out, so the $4420 comes out of $12420.
13

Cycling Shares Given as a Fraction, a Decimal and a Percentage

heuristicConvert to Percentages / Percentage of Each School's Total

Three schools recorded how many of their students cycle to school. At Northfield, which has 600 students, 720 of the students cycle. At Eastbrook, which has 750 students, the share of students who cycle is 0.32. At Westgate, which has 400 students, 36% of the students cycle. (a) Rank the three schools by the share of their students who cycle, largest share first. (b) Which school has the greatest number of students who cycle, and how many more cyclists does it have than the school with the largest share?

Northfield35 of 100do not cycle35%
Northfield: 720 = 35100, so 35 parts of 100 are shaded.
Northfield: 720 = 7 × 520 × 5 = 35100, so 35 parts out of 100 are shaded.
step 1 of 8

Draw each school as a bar of 100 equal parts and shade the parts that cycle, so that the three shares are out of the same whole. Then find the size of one part at each school to count the cyclists.

All steps
  1. Northfield: 720 = 7 × 520 × 5 = 35100, so 35 parts out of 100 are shaded.
  2. Eastbrook: 0.32 means 32 hundredths, so 32 parts out of 100 are shaded.
  3. Westgate: 36% means 36 out of 100, so 36 parts out of 100 are shaded.
  4. (a) Every bar has 100 parts, so the shaded parts can be compared directly. The order is Westgate (36%), Northfield (35%), Eastbrook (32%).
  5. Now count the cyclists. One part is 1% of a school. At Northfield, 1% of 600 is 6 students, so 35% is 35 × 6 = 210 students.
  6. At Eastbrook, 1% of 750 is 7.5 students, so 32% is 32 × 7.5 = 240 students.
  7. At Westgate, 1% of 400 is 4 students, so 36% is 36 × 4 = 144 students.
  8. (b) Eastbrook has the most cyclists: 240 − 144 = 96 more than Westgate. Eastbrook's share is the smallest, but it is a share of the most students. Check: 144400 = 0.36 and 240750 = 0.32.

answer(a) Westgate 36%, Northfield 35%, Eastbrook 32%; (b) Eastbrook, 96 more

techniqueComparing with Percent · Comparing Percents, Fractions and Decimals · Finding a Percent

examsO-Level · SAT · GCSE Higher

Common pitfalls

  • Comparing the numbers as they are written, so that 720 looks smaller than 0.32 or 36%. The three shares must be written in the same form, such as out of 100, before they can be compared.
  • Taking the school with the largest share to have the most cyclists. Each share is out of that school's own number of students, so a smaller share of a larger school can be more students.
14

A School's Enrollment Rise, Worked Back and Carried a Year On

heuristicPercentage Increase as a Multiplier / A New Base for the Next Rise

After a new wing opened, a school's enrollment rose by 25% from last year to this year, and the school now has 1000 students. The school expects its enrollment to rise next year by the same number of students as it rose this year. (a) How many students did the school have last year? (b) By what percentage of this year's enrollment is the school expected to grow next year?

Last year25%25%25%25%100%This year25%25%25%25%+25%125%
25% is a quarter: last year is 4 units and the rise is 1 unit.
25% = 14, so draw last year's enrollment as 4 units and this year's rise as 1 more unit.
step 1 of 6

Since 25% = 14, draw last year's enrollment as 4 units and the rise as 1 more unit. Next year's rise is one more unit, now compared with this year's 5 units.

All steps
  1. 25% = 14, so draw last year's enrollment as 4 units and this year's rise as 1 more unit.
  2. This year's enrollment is 4 + 1 = 5 units, and 5 units are 1000 students.
  3. 1 unit = 1000 ÷ 5 = 200 students.
  4. (a) Last year: 4 units = 4 × 200 = 800 students.
  5. Next year the school expects another rise of 1 unit, or 200 students, on top of this year's 5 units: 1000 + 200 = 1200 students.
  6. (b) The rise is 1 unit out of this year's 5 units: 15 = 20100 = 20%. Check: 2001000 × 100% = 20%.

answer(a) 800 students; (b) 20%

techniquePercentage Increase · Reverse Percentages · Percentage Change

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Taking 25% off 1000 to get 750 students for last year. The 25% was a percentage of last year's enrollment, not of this year's, so this year's 1000 students are 125% of last year's.
  • Answering 25% for next year because the number of new students is the same. The same 200 students are now compared with 1000 students instead of 800, so they make a smaller percentage.
15

A Bakery Oven Losing a Fixed Percentage of Its Value Each Year

heuristicRepeated Percentage Decrease / One Year at a Time

A bakery bought an oven for $25000. Each year, the oven loses 20% of the value it had at the start of that year. (a) What is the oven worth at the end of the third year? (b) By what percentage of the price the bakery paid has the oven's value fallen over the three years?

Year 15000500050005000lost$20000
Year 1: $25000 is 5 units of $5000. One is lost, so $20000 remains.
Year 1: $25000 is 5 units of $5000. One unit is lost, so 4 units remain: 4 × 5000 = $20000.
step 1 of 6

Since 20% = 15, draw each year's starting value as 5 equal units. One unit is lost, and the other 4 units carry on to the next year, where they are split into 5 new, smaller units.

All steps
  1. Year 1: $25000 is 5 units of $5000. One unit is lost, so 4 units remain: 4 × 5000 = $20000.
  2. Year 2: $20000 is 5 units of $4000. One unit is lost, so 4 × 4000 = $16000 remains.
  3. Year 3: $16000 is 5 units of $3200. One unit is lost, so 4 × 3200 = $12800 remains.
  4. (a) At the end of the third year, the oven is worth $12800.
  5. The value lost over the three years is 5000 + 4000 + 3200 = $12200, which agrees with 25000 − 12800 = 12200.
  6. (b) As a percentage of the price paid: 1220025000 × 100% = 48.8%. This is less than 3 × 20% = 60%, because each year's 20% is taken from a smaller value.

answer(a) $12800; (b) 48.8%

techniqueDepreciation · Percentage Change · Finding a Percent

examsSAT · GCSE Higher

Common pitfalls

  • Taking $5000, which is 20% of the price paid, off every year to reach $10000. Each year's 20% is of the value at the start of that year, and that value gets smaller every year.
  • Adding the three percentages to say that the value fell by 60%. The three 20% drops are of three different values, so they cannot be added as percentages of the price paid.
16

Charity Fair Money Shared Out as a Fraction, a Decimal and a Percentage

heuristicConvert to One Form / A Part of the Amount Given Away

A club raised $4800 at a charity fair. It gave 38 of the money to a children's hospital, 0.3 of the money to an animal shelter and 20% of the money to the town library, and kept the rest for next year's fair. (a) How much money did the club keep? (b) What fraction of the money that the club gave away went to the children's hospital?

Raised40 units = $4800Hospital15 units3/8
The $4800 is 40 units. Hospital: 38 = 1540, so 15 units.
Hospital: 38 = 3 × 58 × 5 = 1540, so the hospital gets 15 units.
step 1 of 6

The shares are in eighths, tenths and hundredths, and 40 is the smallest number of equal units that shows 38, 0.3 and 20% exactly. Draw the $4800 as 40 units and give each share its number of units.

All steps
  1. Hospital: 38 = 3 × 58 × 5 = 1540, so the hospital gets 15 units.
  2. Shelter: 0.3 = 310 = 1240, so the shelter gets 12 units.
  3. Library: 20% = 20100 = 15 = 840, so the library gets 8 units.
  4. Given away: 15 + 12 + 8 = 35 units, so the club kept 40 − 35 = 5 units.
  5. 1 unit = 4800 ÷ 40 = $120. (a) The club kept 5 × 120 = $600.
  6. (b) The club gave away 35 units, and 15 of them went to the hospital: 1535 = 37. Check: $1800 + $1440 + $960 + $600 = $4800.

answer(a) $600; (b) 37

techniqueFractions, Decimals and Percents · Comparing Percents, Fractions and Decimals · Finding a Percent

examsPSLE · SAT · GCSE Higher

Common pitfalls

  • Adding the numbers as they are written, 38 + 0.3 + 20, which mixes three forms. Each share must be written in the same form, such as fortieths or percentages, before the shares can be added.
  • Answering (b) with 38. That is the hospital's share of all the money raised, but (b) asks for its share of the $4200 given away, which is a smaller whole.
17

A Bank Balance Taken Back Through Two Withdrawals and a Deposit

heuristicWork Backwards / Undo the Deposit, Then Each Withdrawal in Turn

Mrs. Ang had a sum of money in her bank account. In January, she withdrew 30% of the money, and then another $180 to pay for a renovation. In February, she withdrew 40% of the money left in the account, and then another $120 to invest in shares. In March, she deposited $450 into the account, and her balance was then $1902. (a) How much money was in the account just after the February withdrawal? (b) How much money did Mrs. Ang have in the account at first? (c) What percentage of the sum she had at first did she spend on the renovation?

March$1452+450$1902
(a) Undo the last change first: before the $450 deposit the account held 1902 − 450 = $1452.
(a) The $450 deposit was the last change, so undo it first by taking it away: just after the February withdrawal the account held 1902 − 450 = $1452.
step 1 of 6

Work backwards from the final balance, undoing the last change first. Since 40% = 25, draw the balance at the start of February as 5 units; since 30% = 310, draw the sum at first as 10 parts.

All steps
  1. (a) The $450 deposit was the last change, so undo it first by taking it away: just after the February withdrawal the account held 1902 − 450 = $1452.
  2. In February she took 2 of the 5 units, and then $120 more. So the other 3 units, less $120, are $1452: 3 units = 1452 + 120 = $1572, and 1 unit = 1572 ÷ 3 = $524.
  3. At the start of February the account held all 5 units: 5 × 524 = $2620. This is what January left.
  4. In January she took 3 of the 10 parts, and then $180 more. So the other 7 parts, less $180, are $2620: 7 parts = 2620 + 180 = $2800, and 1 part = 2800 ÷ 7 = $400.
  5. (b) At first she had 10 parts: 10 × 400 = $4000.
  6. (c) The renovation took $180 of the $4000: 1804000 × 100% = 4.5%. Check: 4000 − 1200 − 180 = 2620, then 2620 − 1048 − 120 = 1452, and 1452 + 450 = 1902.

answer(a) $1452; (b) $4000; (c) 4.5%

techniqueReverse Percentages · Finding a Percent · Percentage

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Adding the $450 to $1902 to find the balance after February. The deposit put $450 into the account, so undoing it takes $450 out: the balance before it was $1452, not $2352.
  • Dividing by 60% before putting back the $120: 1452 ÷ 0.6 + 120 = $2540. The $120 was taken out after the 40%, so working backwards it is put back first: (1452 + 120) ÷ 0.6 = $2620.
18

A Coding Academy Whose Boys Fall and Girls Rise by Different Percentages

heuristicBefore and After / One Whole for Both Groups, Then the Gap Sizes a Part

At the start of a year, the ratio of the number of boys to the number of girls at a coding academy was 3 : 2. By the middle of the year, the number of boys had fallen by 15% and the number of girls had risen by 40%. There were then 45 more girls than boys. At the end of the year, 30 more boys joined the academy and some girls left, and the ratio of the number of boys to the number of girls became 1 : 1. (a) How many students were at the academy at the start of the year? (b) How many girls left at the end of the year?

Boys, start60 partsGirls, start40 parts
Boys : girls = 3 : 2 = 60 : 40, so the start is 100 parts: boys 60, girls 40.
Boys : girls = 3 : 2 = 60 : 40, so draw the students at the start as 100 parts: 60 parts boys and 40 parts girls.
step 1 of 6

The two percentages are of different groups, so first give both groups parts of one whole. Draw the students at the start as 100 parts, change each group by its own percentage, and let the gap of 45 students size one part.

All steps
  1. Boys : girls = 3 : 2 = 60 : 40, so draw the students at the start as 100 parts: 60 parts boys and 40 parts girls.
  2. By the middle of the year the boys fell by 15% of 60 parts, which is 9 parts, leaving 51 parts. The girls rose by 40% of 40 parts, which is 16 parts, making 56 parts.
  3. The girls now have 56 − 51 = 5 parts more than the boys, and that gap is 45 students. So 1 part is 45 ÷ 5 = 9 students.
  4. (a) At the start there were 100 parts: 100 × 9 = 900 students.
  5. In the middle of the year there were 51 × 9 = 459 boys and 56 × 9 = 504 girls. With 30 more boys there were 459 + 30 = 489 boys, and for a ratio of 1 : 1 the girls came down to 489.
  6. (b) 504 − 489 = 15 girls left. Check: at the start there were 540 boys and 360 girls, and 540 : 360 = 3 : 2; in the middle of the year, 504 − 459 = 45.

answer(a) 900 students; (b) 15 girls

techniquePercentage Increase · Taking a Percent Off · Finding a Percent

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Taking the 45 as one unit of the ratio 3 : 2, which gives 5 × 45 = 225 students. The 45 is the gap in the middle of the year, after the boys fell and the girls rose, and at the start the boys were the larger group.
  • Bringing the girls down to the boys' mid-year number, 459, and answering 504 − 459 = 45. The 30 new boys joined first, so the girls had to come down to 459 + 30 = 489.
19

Delegates from Three Continents, with Some Leaving and Others Arriving

heuristicThe Remaining Percentage Gives the Total / The Same Number Out and In

At an international debating tournament, 30% of the delegates were from Asia, 45% were from Europe and the remaining 75 delegates were from the Americas. All the delegates were in the auditorium. Halfway through the tournament, 25 of the European delegates left the auditorium and 25 more Asian delegates came in. (a) How many delegates were at the tournament at first? (b) At first, the number of delegates from the Americas was what percentage of the number of delegates from Europe? Give your answer correct to 1 decimal place. (c) In the end, what percentage of the delegates in the auditorium were from Asia? Give your answer correct to 1 decimal place.

At firstAsia 30Europe 45Americas 25100 parts
Asia and Europe are 75%, so the Americas are the other 25%: 100 parts, Asia 30, Europe 45, the Americas 25.
Asia and Europe together are 30% + 45% = 75%, so the Americas are the other 25%. Draw all the delegates as 100 parts: Asia 30, Europe 45 and the Americas 25.
step 1 of 6

Draw all the delegates as 100 parts, one part for each 1%. The part for the Americas is known, which sizes one part. The same number left as came in, so the whole stays at 100 parts.

All steps
  1. Asia and Europe together are 30% + 45% = 75%, so the Americas are the other 25%. Draw all the delegates as 100 parts: Asia 30, Europe 45 and the Americas 25.
  2. The 25 parts for the Americas are 75 delegates, so 1 part is 75 ÷ 25 = 3 delegates.
  3. (a) At first there were 100 × 3 = 300 delegates: 30 × 3 = 90 from Asia, 45 × 3 = 135 from Europe and 75 from the Americas.
  4. (b) Compare the Americas with Europe, 25 parts with 45 parts: 2545 × 100% = 59 × 100% = 5559%, which is 55.6% correct to 1 decimal place.
  5. 25 delegates left and 25 came in, so there were still 300 delegates in the auditorium. The delegates from Asia rose to 90 + 25 = 115.
  6. (c) 115300 × 100% = 3813%, which is 38.3% correct to 1 decimal place. Check: 115 + 110 + 75 = 300.

answer(a) 300 delegates; (b) 55.6%; (c) 38.3%

techniquePercentage · Finding a Percent

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Answering (b) with 75300 × 100% = 25%. That compares the Americas with all the delegates, but (b) compares them with the delegates from Europe, so the whole is 135.
  • Taking the total in (c) as 300 + 25 = 325 because 25 Asian delegates came in. The 25 who came in replaced the 25 who left, so there were still 300 delegates in the auditorium.
20

Sales Tax on a Pair of Sneakers, Then on Two Items Together

heuristicFind 10%, Then Halve It for 5% / The Tax Is Added to the Price, Never Taken Off

Leo buys a pair of sneakers marked $40. At the till, a sales tax of 5% of the price is added. (a) How much is the tax on the sneakers, and how much does Leo pay for them? (b) Leo also buys a cap marked $12. The till works out the 5% tax on the total of the two prices. How much tax does Leo pay on the two items, and how much does he pay in all?

Sneakers$4$4$4$4$4$4$4$4$4$4+ $2 tax10% = $45% = $210% of $40 is $4, so 5% is $4 ÷ 2 = $2
10% of $40 is $4, and 5% is half of that: $2.
10% of $40 is 40 ÷ 10 = $4, and 5% is half of that: 4 ÷ 2 = $2.
step 1 of 5

The price is 100%, and the till adds 5% of it on top. To find 5%, find 10% by dividing by 10, then halve it. The tax is 5 cents on every dollar of a price, so the tax on two prices together is the same as the tax on each price, added.

All steps
  1. 10% of $40 is 40 ÷ 10 = $4, and 5% is half of that: 4 ÷ 2 = $2.
  2. (a) The tax on the sneakers is $2. The till adds it to the price: 40 + 2 = 42, so Leo pays $42 for them.
  3. The two prices together come to 40 + 12 = $52. 10% of $52 is $5.20, so 5% is half of that, $2.60.
  4. Work the tax out item by item as well. 10% of $12 is $1.20, so the tax on the cap is $0.60. With the $2 on the sneakers, that is 2 + 0.60 = $2.60, the same amount.
  5. (b) Leo pays $2.60 in tax on the two items, and 52 + 2.60 = $54.60 in all. Check: $42 for the sneakers and 12 + 0.60 = $12.60 for the cap make $54.60.

answer(a) The tax is $2, and Leo pays $42; (b) the tax is $2.60, and Leo pays $54.60 in all

techniqueSales Tax · Finding a Percent

examsPSLE · O-Level

Common pitfalls

  • Taking the tax off the price: 40 − 2 = $38. A sales tax is added at the till, so Leo pays more than the price, not less: 40 + 2 = $42.
  • Adding the two rates and working out 10% of $52, which is $5.20. Each item is taxed at 5%, and so is the total of the two prices. The rate stays 5%, however many items there are.
21

Two Basketball Players' Shots, Out of Different Totals, Compared

heuristicWrite Both Scores Out of 100 / Simplify First When the Total Does Not Go into 100

At basketball practice, Maya takes 24 shots and 18 of them go in. Jo takes 30 shots and 21 of them go in. (a) Write each girl's score as a percentage of her shots. Which girl has the better record? (b) Ben takes 20 shots. How many of them must go in for him to match the better record?

Maya18 of 24Jo21 of 3018 of 24 and 21 of 30 are out of different totals
Each bar is one girl's shots, shaded where a shot went in. Jo has more in, but she took more shots.
Jo has more shots in, 21 against 18, but she also took more shots. Each score is out of a different total, so write both out of 100.
step 1 of 5

The two scores are out of different totals, so they cannot be compared as they stand. Write each one out of 100. Neither 24 nor 30 goes into 100 a whole number of times, so simplify each fraction first.

All steps
  1. Jo has more shots in, 21 against 18, but she also took more shots. Each score is out of a different total, so write both out of 100.
  2. Maya: divide 18 and 24 by 6, so 1824 = 34. Then multiply both by 25: 34 = 75100, which is 75%.
  3. Jo: divide 21 and 30 by 3, so 2130 = 710. Then multiply both by 10: 710 = 70100, which is 70%.
  4. (a) Maya's score is 75% and Jo's is 70%. Both are out of 100 now, so Maya has the better record, even though fewer of her shots went in.
  5. (b) To match Maya, 34 of Ben's shots must go in. 14 of 20 is 5, so 34 of 20 is 3 × 5 = 15. Check: 1520 = 75100 = 75%.

answer(a) Maya 75%, Jo 70%, so Maya has the better record; (b) 15 shots

techniqueComparing with Percent · Finding a Percent

examsO-Level · SAT · GCSE Higher

Common pitfalls

  • Saying that Jo has the better record because 21 is more than 18. Jo took 6 more shots than Maya, so her score is out of a bigger total. Only the two percentages, both out of 100, can be compared.
  • Answering 18 for part (b), the number of Maya's shots that went in. Ben takes 20 shots, not 24, so he needs 75% of 20, which is 15.
22

Two Bikes Losing Value, One by a Percentage and One by a Fixed Amount

heuristicOne Year at a Time / Each Year's 10% Is of That Year's Starting Value

Two bikes each cost $400 when new. Each year, the red bike loses 10% of the value it has at the start of that year. The blue bike loses $40 each year. (a) What is the red bike worth after 2 years? (b) After 2 years, which bike is worth more, and by how much?

Red, year 1404040404040404040lost$360year 1: 10% of $400 = $40, and 9 × $40 = $360
Year 1: $400 is 10 units of $40. One unit is lost, so 9 × 40 = $360 is left.
Year 1, red bike: $400 is 10 units of $40. One unit is lost, so 9 × 40 = $360 is left.
step 1 of 5

10% is one tenth. Draw the red bike's value at the start of each year as 10 equal units: 1 unit is lost and 9 units are left. The next year starts from what is left, so its units are smaller.

All steps
  1. Year 1, red bike: $400 is 10 units of $40. One unit is lost, so 9 × 40 = $360 is left.
  2. Year 2, red bike: the 10% is taken from $360 now. $360 is 10 units of $36. One unit is lost, so 9 × 36 = $324 is left.
  3. (a) After 2 years the red bike is worth $324.
  4. The blue bike loses the same $40 every year: 400 − 40 − 40 = $320.
  5. (b) The red bike is worth more, by 324 − 320 = $4. In the first year both bikes lose $40. In the second year the red bike loses only $36, because 10% of a smaller value is a smaller amount.

answer(a) $324; (b) the red bike, by $4

techniqueDepreciation · Finding a Percent

examsSAT · GCSE Higher

Common pitfalls

  • Saying that 10% a year for 2 years is 20% off, so the red bike is worth 400 − 80 = $320, the same as the blue bike. The second year's 10% is taken from $360, not from $400, so it is $36, not $40.
  • Taking 10% of $400 as $10. Percent means out of 100: 10% is 10 out of every 100, and $400 is 4 hundreds, so 10% of it is 4 × 10 = $40.
23

A Class Vote on Sports, Reported as a Fraction, a Decimal and a Percentage

heuristicWrite Every Result as a Percentage / The Rest Is What Is Left of 100%

The children in Class 5B each voted for one sport. 25% of the class voted for basketball, 0.3 of the class voted for swimming, 25 of the class voted for football, and the rest voted for cycling. (a) Put the four sports in order, from the most votes to the fewest. (b) There are 40 children in the class. How many more children voted for football than for cycling?

Basketball2525%Swimming300.3 = 30%
Each bar is the whole class, cut into 100 parts. Basketball is 25%, and 0.3 is 30 hundredths: 30%.
Basketball is already a percentage: 25%. Swimming: 0.3 is 3 tenths, and 3 tenths are 30 hundredths, so 0.3 = 30%.
step 1 of 6

The results are written in three different forms, so they cannot be compared as they stand. Write each one as a percentage, out of 100. The whole class is 100%, so cycling has what is left.

All steps
  1. Basketball is already a percentage: 25%. Swimming: 0.3 is 3 tenths, and 3 tenths are 30 hundredths, so 0.3 = 30%.
  2. Football: multiply the numerator and the denominator by 20: 25 = 40100 = 40%.
  3. Cycling has the rest: 100% − 25% − 30% − 40% = 5%.
  4. (a) From the most votes to the fewest: football (40%), swimming (30%), basketball (25%), cycling (5%).
  5. 10% of 40 children is 4 children. Football has 40%, which is 4 × 4 = 16 children. Cycling has 5%, which is half of 10%: 2 children.
  6. (b) 16 − 2 = 14 more children voted for football than for cycling. Check: swimming has 3 × 4 = 12 children and basketball has 2 × 4 + 2 = 10, and 16 + 12 + 10 + 2 = 40.

answer(a) Football (40%), swimming (30%), basketball (25%), cycling (5%); (b) 14 more children

techniqueFractions, Decimals and Percents · Finding a Percent

examsPSLE · SAT · GCSE Higher

Common pitfalls

  • Reading 0.3 as 3% and putting swimming last. 0.3 is 3 tenths, which is 30 hundredths, so it is 30%.
  • Putting basketball first because 25 is the biggest number written in the question. The other results are written as a decimal and a fraction. As percentages, swimming is 30% and football is 40%, and both are more than 25%.