GCSE Maths Percentages (Higher): check-up and exam-style questions

Method. To find a percentage of an amount without a calculator, find 10% (divide by 10) or 1% (divide by 100) first. Then build up to the percentage you need.

1. Split the percentage into easy parts. We can find 10% and 1% very quickly, so we build the percentage out of them.
   15% = 10% + 5%
2. Find 10%. 10% is one tenth, so divide the amount by 10 (each digit moves one place to the right).
   10% of £240 = £240 ÷ 10 = £24
3. Work out each part. Use the 10% or 1% you found.
   10% = £24
   5% = half of 10% = £24 ÷ 2 = £12
4. Add the parts together. The parts together make up the whole percentage.
   £24 + £12 = £36

Check. Another way: 15% = 0.15 as a decimal, and 0.15 × £240 = £36. ✓

Watch out. 10% means divide by 10 once. Do not divide by 10 and then by 10 again.

Method. To write one quantity as a percentage of another: write it as a fraction (part over whole), then change the fraction into a percentage.

1. Write the part over the whole as a fraction. A percentage compares a part with the whole. Both numbers must be in the same units.
   1824
2. Divide the top by the bottom. This changes the fraction into a decimal.
   18 ÷ 24 = 0.75
3. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
   0.75 × 100 = 75%

Check. Check: 75% of 24 = 0.75 × 24 = 18, which is the part we started with. ✓

Watch out. Divide by the whole (the amount you are comparing with), not by the part.

Method. Find 20% of the amount, then add it to the original.

1. Find 10%. 10% is one tenth, so divide the amount by 10 (each digit moves one place to the right).
   10% of £60 = £60 ÷ 10 = £6
2. Work out each part. Multiply to build up to the percentage you want.
   20% = 2 × 10% = 2 × £6 = £12
3. Write the answer. Include the units.
   20% of £60 = £12
4. Add it to the original amount. We want the amount after the increase.
   £60 + £12 = £72

Check. Multiplier check: 120% = 1.2, and 1.2 × £60 = £72. ✓

Watch out. Remember the last step. Finding the percentage is not the final answer: add it on or take it off.

Method. Find 35% of the amount, then subtract it from the original.

1. Split the percentage into easy parts. We can find 10% and 1% very quickly, so we build the percentage out of them.
   35% = 30% + 5%
2. Find 10%. 10% is one tenth, so divide the amount by 10 (each digit moves one place to the right).
   10% of 80 kg = 80 kg ÷ 10 = 8 kg
3. Work out each part. Use the 10% or 1% you found.
   30% = 3 × 10% = 3 × 8 kg = 24 kg
   5% = half of 10% = 8 kg ÷ 2 = 4 kg
4. Add the parts together. The parts together make up the whole percentage.
   24 kg + 4 kg = 28 kg
5. Subtract it from the original amount. We want the amount after the decrease.
   80 kg − 28 kg = 52 kg

Check. Multiplier check: 65% = 0.65, and 0.65 × 80 kg = 52 kg. ✓

Watch out. Remember the last step. Finding the percentage is not the final answer: add it on or take it off.

Method. Percentage change = change ÷ original × 100.

1. Work out the change. Subtract the smaller value from the bigger one. The change is the difference.
   £30 − £25 = £5 (increase)
2. Write the change as a fraction of the ORIGINAL. Percentage change always compares the change with the starting (original) value.
   525
3. Make the denominator 100. “Per cent” means out of 100. Multiply the top and the bottom by the same number: 25 × 4 = 100.
   525 = 20100 = 20%
4. Say whether it is an increase or a decrease.
   20% increase

Check. Check: 20% of £25 = £5, and £25 + £5 = £30. ✓

Watch out. Divide by the original value, not the new one.

Method. In a reverse percentage question the amount we are given is NOT the original. It is a percentage of the original. Work backwards to find 100%.

1. Find the percentage that the new amount represents. The original amount is 100%. After an increase of 20% the new amount is 100% + 20% of the original.
   100% + 20% = 120%
2. Write down what we know. The new amount is a known percentage of the unknown original.
   120% = £84
3. Find 1%. Divide by 120 to go from 120% down to 1%.
   £84 ÷ 120 = £0.70
4. Find 100%. The original is 100%, so multiply 1% by 100.
   £0.70 × 100 = £70

Check. Check by going forwards: £70 × 1.2 = £84, which matches the new amount. ✓

Watch out. Do not work out 20% of the new amount and subtract it. The percentage was taken from the original, not from the new amount.

Key words. Interest is money paid for saving or borrowing. Simple interest is a fixed percentage of the original amount, paid each year.

Method. Simple interest is the same amount every year. Find one year’s interest, then multiply by the number of years.

1. Write down what we know. The principal is the money at the start; the rate is the percentage per year; the time is in years.
   Principal = £600,  rate = 4% per year,  time = 3 years
2. Find the interest for ONE year. Simple interest is the same every year: it is 4% of the original amount each time.
   4% of £600 = 0.04 × £600 = £24
3. Multiply by the number of years. The same interest is added every year, so multiply by 3.
   £24 × 3 = £72

Check. Formula check: I = P × R × T ÷ 100 = 600 × 4 × 3 ÷ 100 = 72. ✓

Watch out. Simple interest always uses the original amount, never the amount with interest added.

Method. To compare fractions, decimals and percentages, write them all as decimals. Then put the decimals in order.

1. Change every number into a decimal. Decimals are easy to compare digit by digit, so we use them as a common language.
   0.6 is already a decimal: 0.6
   55% = 55 ÷ 100 = 0.55
   58 = 5 ÷ 8 = 0.625
2. Put the decimals in order. We want smallest first. Compare the digits from the left, starting with the tenths.
   0.55 , 0.6 , 0.625
3. Write the original numbers in that order. The answer should use the forms given in the question.
   55% ,  0.6 ,  58

Check. Read the final list again as decimals: it goes up each time. ✓

Watch out. Do not compare the digits of different forms directly. 3/5 is not smaller than 0.4 just because 3 is smaller than 4.

Key words. Compound interest is interest paid on the original amount and on the interest already added. Depreciation is a fall in value.

Method. Compound interest means the interest is added each year, so next year’s interest is worked out on the bigger amount. Use the multiplier 1.1 once for each year.

1. Write down what we know. The starting amount, the percentage change each time, and how many times it happens.
   Start = £1,000,  increase = 10% per year,  number of years = 2
2. Find the multiplier for one year. Each year the amount grows by 10%, so it becomes 110% of the previous amount. The multiplier is 110% as a decimal.
   110% = 110 ÷ 100 = 1.1
3. Apply the multiplier once for each year. Each year we multiply the amount we have by 1.1.
   After year 1: £1,000 × 1.1 = £1,100
   After year 2: £1,100 × 1.1 = £1,210

Check. Estimate: with simple interest the total would be £1,200. Compound interest is a little more because the interest earns interest, and £1,210 is a little above £1,200. ✓

Watch out. Use the new amount each year as the starting point for the next year.

Method. To change a fraction to a percentage, find an equivalent fraction with denominator 100, or divide the top by the bottom and multiply by 100.

1. Write the fraction. A fraction already compares a part with a whole.
   720
2. Make the denominator 100. “Per cent” means out of 100. Multiply the top and the bottom by the same number: 20 × 5 = 100.
   720 = 35100 = 35%

Check. 35% = 35 ÷ 100 = 0.35, and 7 ÷ 20 = 0.35. ✓

Method. To find a percentage of an amount without a calculator, find 10% (divide by 10) or 1% (divide by 100) first. Then build up to the percentage you need.

1. Split the percentage into easy parts. We can find 10% and 1% very quickly, so we build the percentage out of them.
   35% = 30% + 5%
2. Find 10%. 10% is one tenth, so divide the amount by 10 (each digit moves one place to the right).
   10% of £80 = £80 ÷ 10 = £8
3. Work out each part. Use the 10% or 1% you found.
   30% = 3 × 10% = 3 × £8 = £24
   5% = half of 10% = £8 ÷ 2 = £4
4. Add the parts together. The parts together make up the whole percentage.
   £24 + £4 = £28

Check. Another way: 35% = 0.35 as a decimal, and 0.35 × £80 = £28. ✓

Watch out. 10% means divide by 10 once. Do not divide by 10 and then by 10 again.

Method. Percentages are all out of 100, so they let us compare amounts that have different totals.

1. Write Test A as a percentage. To compare fairly we need the same total, so we use percentages (out of 100).
   34 ÷ 40 = 0.85
   0.85 × 100 = 85%
2. Write Test B as a percentage. Do exactly the same for the second one.
   4150 = 82100 = 82%
3. Compare the percentages. The bigger percentage is the better result.
   85% and 82%, so Test A is higher.

Check. Cross-check with decimals: 0.85 and 0.82; Test A is bigger. ✓

Watch out. Compare the percentages, not the raw scores. A bigger score can be a smaller percentage.

Method. To decrease an amount by a percentage, find the multiplier and multiply. The multiplier for a 30% decrease is 0.7.

1. Find the percentage the new amount will be. A decrease of 30% means we subtract 30% of the original. The original is 100%, so the new amount is 100% − 30%.
   100% − 30% = 70% of the original
2. Write that percentage as a decimal: this is the multiplier. Divide by 100.
   70% = 70 ÷ 100 = 0.7
3. Multiply the original amount by the multiplier. One multiplication gives the new amount straight away.
   £90 × 0.7 = £63

Check. Other way: 30% of £90 = 27; £90 − 27 = 63. ✓

Watch out. Do not multiply by 0.3 only: that finds the 30% but not the new amount. The multiplier is what is left after the decrease.

Method. To increase an amount by a percentage, find the multiplier and multiply. The multiplier for a 20% increase is 1.2.

1. Find the percentage the new amount will be. An increase of 20% means we add 20% of the original. The original is 100%, so the new amount is 100% + 20%.
   100% + 20% = 120% of the original
2. Write that percentage as a decimal: this is the multiplier. Divide by 100.
   120% = 120 ÷ 100 = 1.2
3. Multiply the original amount by the multiplier. One multiplication gives the new amount straight away.
   £420 × 1.2 = £504

Check. Other way: 20% of £420 = 84; £420 + 84 = 504. ✓

Watch out. Do not multiply by 0.2 only: that finds the 20% but not the new amount. The multiplier includes the original 100%.

Method. Percentage change = change ÷ original × 100.

1. Work out the change. Subtract the smaller value from the bigger one. The change is the difference.
   £264,000 − £240,000 = £24,000 (increase)
2. Write the change as a fraction of the ORIGINAL. Percentage change always compares the change with the starting (original) value.
   24,000240,000
3. Divide the top by the bottom. This changes the fraction into a decimal.
   24,000 ÷ 240,000 = 0.1
4. Multiply by 100. To change a decimal into a percentage multiply by 100, because a percentage is a number out of 100.
   0.1 × 100 = 10%
5. Say whether it is an increase or a decrease.
   10% increase

Check. Check: 10% of £240,000 = £24,000, and £240,000 + £24,000 = £264,000. ✓

Watch out. Divide by the original value, not the new one.

Method. In a reverse percentage question the amount we are given is NOT the original. It is a percentage of the original. Work backwards to find 100%.

1. Find the percentage that the new amount represents. The original amount is 100%. After a decrease of 15% the new amount is 100% − 15% of the original.
   100% − 15% = 85%
2. Write down what we know. The new amount is a known percentage of the unknown original.
   85% = £68
3. Find 1%. Divide by 85 to go from 85% down to 1%.
   £68 ÷ 85 = £0.80
4. Find 100%. The original is 100%, so multiply 1% by 100.
   £0.80 × 100 = £80

Check. Check by going forwards: £80 × 0.85 = £68, which matches the new amount. ✓

Watch out. Do not work out 15% of the new amount and add it. The percentage was taken from the original, not from the new amount.

Key words. Interest is money paid for saving or borrowing. Simple interest is a fixed percentage of the original amount, paid each year.

Method. Simple interest is the same amount every year. Find one year’s interest, then multiply by the number of years.

1. Write down what we know. The principal is the money at the start; the rate is the percentage per year; the time is in years.
   Principal = £2,500,  rate = 3.2% per year,  time = 4 years
2. Find the interest for ONE year. Simple interest is the same every year: it is 3.2% of the original amount each time.
   3.2% of £2,500 = 0.032 × £2,500 = £80
3. Multiply by the number of years. The same interest is added every year, so multiply by 4.
   £80 × 4 = £320
4. Add the interest to the original amount. The total is what the account holds at the end: the original money plus the interest.
   £2,500 + £320 = £2,820

Check. Formula check: I = P × R × T ÷ 100 = 2,500 × 3.2 × 4 ÷ 100 = 320. ✓

Watch out. Simple interest always uses the original amount, never the amount with interest added.

Key words. Compound interest is interest paid on the original amount and on the interest already added. Depreciation is a fall in value.

Method. Compound interest means the interest is added each year, so next year’s interest is worked out on the bigger amount. Use the multiplier 1.02 once for each year.

1. Write down what we know. The starting amount, the percentage change each time, and how many times it happens.
   Start = £5,000,  increase = 2% per year,  number of years = 3
2. Find the multiplier for one year. Each year the amount grows by 2%, so it becomes 102% of the previous amount. The multiplier is 102% as a decimal.
   102% = 102 ÷ 100 = 1.02
3. Raise the multiplier to the power 3. The same increase happens 3 times, so we multiply by 1.02 3 times. That is the multiplier to the power 3.
   1.02³ = 1.061208
4. Multiply the starting amount by 1.02³. This gives the amount after all the changes in one calculation.
   £5,000 × 1.061208 = £5,306.04

Check. Estimate: with simple interest the total would be £5,300. Compound interest is a little more because the interest earns interest, and £5,306.04 is a little above £5,300. ✓

Watch out. Do not multiply by 1.02 and then by 3. Use the multiplier to the power 3.

Method. Find the overall multiplier by multiplying the separate multipliers, then use it on the starting amount.

1. Write a multiplier for each change. Each percentage change has its own multiplier (100% plus or minus the change, as a decimal).
   decrease of 20%: 80% = 0.8
   decrease of 15%: 85% = 0.85
2. Multiply the multipliers together. One change is applied after the other, so the multipliers are multiplied.
   0.8 × 0.85 = 0.68
3. Multiply the starting amount by the overall multiplier. This gives the amount after both changes.
   £16,000 × 0.68 = £10,880

Check. Apply the changes one at a time and you reach the same amount. ✓

Watch out. Do not add the percentages. A 10% rise then a 10% rise is a 21% rise, not 20%.

Method. Find the overall multiplier for the two increases, then turn it into a percentage.

1. Write the multiplier for one year. 5% increase: 105% = 1.05.
   1.05
2. Find the overall multiplier. The increase happens twice, so multiply by 1.05 twice.
   1.05 × 1.05 = 1.05² = 1.1025
3. Change to a percentage increase. Subtract 1 (the original) and multiply by 100.
   (1.1025 − 1) × 100 = 10.25%
4. Conclude.
   The total increase over 2 years is 10.25%.

Check. Start at 100: up 5% gives 105; up 5% of 105 = 5.25 gives 110.25, which is a 10.25% increase. ✓

Watch out. Marks: 1 for the multiplier 1.05, 1 for 1.05² or equivalent, 1 for the conclusion.

Method. If you know what a percentage of the whole is, find 1% first, then multiply by 100 to find the whole (100%).

1. Write down what we know. Part of the whole has been given as a percentage.
   150% of the whole = £60,000
2. Find 1%. Divide by 150. That takes us from 150% to 1%.
   £60,000 ÷ 150 = £400
3. Find 100%. The whole is 100%, so multiply 1% by 100.
   £400 × 100 = £40,000

Check. 150% of £40,000 = 1.5 × £40,000 = £60,000. ✓

Watch out. The whole is 100%. Do not stop at 1%.

Method. Use the multiplier to a power and try more and more years until the target is reached.

1. Find the multiplier for one year. Write the percentage as a decimal.
   105% = 1.05
2. Try years one at a time. We cannot solve this directly yet, so we use trial and improvement: work out the amount after 1, 2, 3, … years using the multiplier to a power.
   After 1 year: £2,000 × 1.05¹ = £2,100
   After 2 years: £2,000 × 1.05² = £2,205
   After 3 years: £2,000 × 1.05³ = £2,315.25
   After 4 years: £2,000 × 1.05⁴ = £2,431.0125 = £2,431.01
   After 5 years: £2,000 × 1.05⁵ = £2,552.563125 = £2,552.56
3. Find the first time the target is passed. We want the amount to be more than £2,500.
   After 5 years the amount is £2,552.56, which is above £2,500. After 4 it was not.

Check. After 4 years the amount is £2,431.01, which has not yet passed £2,500, so 5 is the first. ✓

Watch out. The answer is a whole number of years: say the first year in which the target is reached or passed.

Method. The multiplier to the power n gives the overall change. To find the yearly multiplier, divide the end amount by the start amount and take the n-th root.

1. Write the equation. Let the yearly multiplier be m. After 3 years the start amount has been multiplied by m³.
   £15,000 × m³ = £10,935
2. Divide both sides by the start amount. This isolates m³.
   m³ = £10,935 ÷ £15,000 = 0.729
3. Take the cube root of both sides. The cube root undoes the power 3. On a calculator use the x^1/3 or the n-th root key.
   m = cube root of 0.729 = 0.9
4. Turn the multiplier into a percentage. 0.9 = 90%, and the original is 100%.
   90% − 100% = 10% decrease

Check. £15,000 × 0.9³ = £15,000 × 0.729 = £10,935. ✓

Watch out. Do not divide the total percentage change by the number of years. The change compounds.

Method. Combine the two changes into one multiplier, then divide the final population by it.

1. Multiplier for the 8% increase. 108% = 1.08.
   1.08
2. Multiplier for the 5% decrease. 95% = 0.95.
   0.95
3. Overall multiplier. Multiply.
   1.08 × 0.95 = 1.026
4. Divide to find the starting population. original × 1.026 = 12,312.
   12,312 ÷ 1.026 = 12,000

Check. 12,000 × 1.08 = 12,960 and 12,960 × 0.95 = 12,312. ✓

Watch out. Marks: 1 each for the two multipliers, 1 for their product, 1 for the division.

Method. Separate the difference in the rates from the percentage change in the rate.

1. Work out the difference between the rates. This is the change measured in percentage points.
   4% − 3% = 1 percentage point
2. Work out the percentage change of the rate. Divide the change by the original rate (4%).
   1 ÷ 4 × 100 = 25%
3. Write the two correct statements. Alex did not say whether he meant percentage points or a percentage change.
   The rate fell by 1 percentage point, and this is a 25% decrease in the rate.

Check. 25% of 4% = 1%, and 4% − 1% = 3%. ✓

Watch out. Marks: 1 for 1 percentage point, 1 for 25%, 1 for the explanation.