GCSE Maths: Compound interest, growth and decay (Higher)
Part of the GCSE Maths percentages chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Compound interest, growth and decay
Simple interest pays the same amount every year. Most savings accounts work differently. With compound interest the interest is added to the account each year, and next year’s interest is worked out on the new, bigger amount. So you earn interest on your interest. Look at £1,000 at 5% a year.
Year 1: 5% of £1,000 = £50, so you have £1,050. Year 2: 5% of £1,050 = £52.50, so you have £1,102.50. With simple interest you would have had only £1,100.
| Years | Simple interest total | Compound interest total |
|---|---|---|
| 0 | £1,000.00 | £1,000.00 |
| 1 | £1,050.00 | £1,050.00 |
| 2 | £1,100.00 | £1,102.50 |
| 3 | £1,150.00 | £1,157.63 |
| 4 | £1,200.00 | £1,215.51 |
| 5 | £1,250.00 | £1,276.28 |
The multiplier shortcut
Each year the amount is multiplied by 1.05. After 2 years it has been multiplied by 1.05 twice: 1.05 × 1.05 = 1.052. After n years it is multiplied by 1.05n. So
final amount = starting amount × multipliern
where n is the number of years. On a calculator use the power key (xy or ^). Do not round in the middle of a calculation; round the final money answer to the nearest penny.
Repeated percentage change: growth and decay
The same idea works for anything that changes by a fixed percentage again and again. A population that grows 3% each year, bacteria that increase 20% every hour: use a multiplier above 1 (growth). A car that loses 15% of its value each year, or a medicine that falls by 30% each hour in the blood: use a multiplier below 1 (decay). A fall in value is called depreciation.
Higher tier. Because the amount is multiplied each time, you can work backwards or solve for the unknown:
- How many years? Try 1, 2, 3, … years in the multiplier power until the target is passed (trial and improvement).
- What rate? If 3000 grows to 5184 in 3 years, then m3 = 5184 ÷ 3000 = 1.728. Take the cube root: m = 1.2, so the rate is 20%.
- What was the starting amount? Divide the final amount by the multiplier to the power n.
- Different rates in different years: multiply the separate multipliers: 3% then 2.5% is 1.03 × 1.025.
- Shorter periods: if interest is added every 6 months, n counts the number of 6-month periods and the rate is for 6 months.
Key ideas
- Compound interest: interest is added each year and then earns interest itself.
- Final amount = start × multipliern, with n = number of years (or other periods).
- Growth by p%: multiplier = 1 + p/100. Decay by p%: multiplier = 1 − p/100.
- Compound interest earned = final amount − starting amount.
- Do not round until the end. Money: nearest penny. Population: nearest whole number.
- Compound interest is always at least as much as simple interest, and more after year 1.
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 1.05 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £2,000, increase = 5% per year, number of years = 3
- Find the multiplier for one year. Each year the amount grows by 5%, so it becomes 105% of the previous amount. The multiplier is 105% as a decimal.105% = 105 ÷ 100 = 1.05
- Apply the multiplier once for each year. Each year we multiply the amount we have by 1.05.After year 1: £2,000 × 1.05 = £2,100
After year 2: £2,100 × 1.05 = £2,205
After year 3: £2,205 × 1.05 = £2,315.25
Check. Estimate: with simple interest the total would be £2,300. Compound interest is a little more because the interest earns interest, and £2,315.25 is a little above £2,300. ✓
Watch out. Use the new amount each year as the starting point for the next year.
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 1.03 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £5,000, increase = 3% per year, number of years = 4
- Find the multiplier for one year. Each year the amount grows by 3%, so it becomes 103% of the previous amount. The multiplier is 103% as a decimal.103% = 103 ÷ 100 = 1.03
- Raise the multiplier to the power 4. The same increase happens 4 times, so we multiply by 1.03 4 times. That is the multiplier to the power 4.1.034 = 1.125509…
- Multiply the starting amount by 1.034. This gives the amount after all the changes in one calculation.£5,000 × 1.125509… = £5,627.54405
- Round the answer. Money is rounded to the nearest penny (2 decimal places). Round only at the very end.£5,627.54405 = £5,627.54
Check. Estimate: with simple interest the total would be £5,600. Compound interest is a little more because the interest earns interest, and £5,627.54 is a little above £5,600. ✓
Watch out. Do not multiply by 1.03 and then by 4. Use the multiplier to the power 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 1.025 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £4,000, increase = 2.5% per year, number of years = 2
- Find the multiplier for one year. Each year the amount grows by 2.5%, so it becomes 102.5% of the previous amount. The multiplier is 102.5% as a decimal.102.5% = 102.5 ÷ 100 = 1.025
- Raise the multiplier to the power 2. The same increase happens 2 times, so we multiply by 1.025 2 times. That is the multiplier to the power 2.1.0252 = 1.050625
- Multiply the starting amount by 1.0252. This gives the amount after all the changes in one calculation.£4,000 × 1.050625 = £4,202.50
- Subtract the original amount to find the interest. The interest is only the extra money, not the total.£4,202.50 − £4,000 = £202.50
Check. Estimate: with simple interest the total would be £4,200. Compound interest is a little more because the interest earns interest, and £4,202.50 is a little above £4,200. ✓
Watch out. Do not multiply by 1.025 and then by 2. Use the multiplier to the power 2.
Method. Work out each total, then subtract.
- Simple interest: interest for one year. 4% of £1,000.4% of £1,000 = £40
- Simple interest: total. Three years of £40 added to the £1,000.£1,000 + 3 × £40 = £1,120
- Compound interest: the multiplier. 104% as a decimal.1.04
- Compound interest: total. Multiply by 1.04 three times.£1,000 × 1.043 = £1,000 × 1.124864 = £1,124.864 = £1,124.86
- Find the difference. Subtract the simple total from the compound total.£1,124.86 − £1,120 = £4.86
Check. Compound interest in year 2 is 4% of £1,040 = £41.60 (more than £40), and in year 3 it is bigger again. ✓
Watch out. Compound interest only beats simple interest from year 2 onwards.
Key words. Depreciation means the value falls each year. Decay is a repeated percentage decrease.
Method. Repeated percentage decrease: multiply by 0.85 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £14,000, decrease = 15% per year, number of years = 3
- Find the multiplier for one year. Each year the amount falls by 15%, so it becomes 85% of the previous amount. The multiplier is 85% as a decimal.85% = 85 ÷ 100 = 0.85
- Raise the multiplier to the power 3. The same decrease happens 3 times, so we multiply by 0.85 3 times. That is the multiplier to the power 3.0.853 = 0.614125
- Multiply the starting amount by 0.853. This gives the amount after all the changes in one calculation.£14,000 × 0.614125 = £8,597.75
Check. Estimate: losing 15% of the original each time would leave £7,700. Each cut is smaller than that because it is taken from a smaller amount, so the answer should be a bit more than £7,700. ✓
Watch out. Do not multiply by 0.85 and then by 3. Use the multiplier to the power 3.
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 1.04 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = 12,500, increase = 4% per year, number of years = 5
- Find the multiplier for one year. Each year the amount grows by 4%, so it becomes 104% of the previous amount. The multiplier is 104% as a decimal.104% = 104 ÷ 100 = 1.04
- Raise the multiplier to the power 5. The same increase happens 5 times, so we multiply by 1.04 5 times. That is the multiplier to the power 5.1.045 = 1.216653…
- Multiply the starting amount by 1.045. This gives the amount after all the changes in one calculation.12,500 × 1.216653… = 15,208.16128
- Round the answer. Round to the nearest whole number.15,208.16128 = 15,208
Check. Estimate: with simple interest the total would be 15,000. Compound interest is a little more because the interest earns interest, and 15,208 is a little above 15,000. ✓
Watch out. Do not multiply by 1.04 and then by 5. Use the multiplier to the power 5.
Method. Use the multiplier to a power and try more and more years until the target is reached.
- Find the multiplier for one year. Write the percentage as a decimal.104% = 1.04
- 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: £1,000 × 1.041 = £1,040
After 2 years: £1,000 × 1.042 = £1,081.60
After 3 years: £1,000 × 1.043 = £1,124.864 = £1,124.86
After 4 years: £1,000 × 1.044 = £1,169.85856 = £1,169.86
After 5 years: £1,000 × 1.045 = £1,216.6529… = £1,216.65
After 6 years: £1,000 × 1.046 = £1,265.3190… = £1,265.32
After 7 years: £1,000 × 1.047 = £1,315.9318… = £1,315.93
After 8 years: £1,000 × 1.048 = £1,368.5691… = £1,368.57
After 9 years: £1,000 × 1.049 = £1,423.3118… = £1,423.31
After 10 years: £1,000 × 1.0410 = £1,480.2443… = £1,480.24
After 11 years: £1,000 × 1.0411 = £1,539.4541… = £1,539.45 - Find the first time the target is passed. We want the amount to be more than £1,500.After 11 years the amount is £1,539.45, which is above £1,500. After 10 it was not.
Check. After 10 years the amount is £1,480.24, which has not yet passed £1,500, so 11 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.
- Write the equation. Let the yearly multiplier be m. After 3 years the start amount has been multiplied by m3.£3,000 × m3 = £5,184
- Divide both sides by the start amount. This isolates m3.m3 = £5,184 ÷ £3,000 = 1.728
- Take the cube root of both sides. The cube root undoes the power 3. On a calculator use the x1/3 or the n-th root key.m = cube root of 1.728 = 1.2
- Turn the multiplier into a percentage. 1.2 = 120%, and the original is 100%.120% − 100% = 20% increase
Check. £3,000 × 1.23 = £3,000 × 1.728 = £5,184. ✓
Watch out. Do not divide the total percentage change by the number of years. The change compounds.
Method. Find the overall multiplier by multiplying the separate multipliers, then use it on the starting amount.
- Write a multiplier for each change. Each percentage change has its own multiplier (100% plus or minus the change, as a decimal).increase of 3%: 103% = 1.03
increase of 2.5%: 102.5% = 1.025 - Multiply the multipliers together. One change is applied after the other, so the multipliers are multiplied.1.03 × 1.025 = 1.05575
- Multiply the starting amount by the overall multiplier. This gives the amount after both changes.£8,000 × 1.05575 = £8,446
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%.
Watch out
Do not work out the percentage once and multiply it by the number of years. 5% for 3 years is not 15%: the multiplier is 1.053 = 1.157625, which is a 15.76% increase. Also take care with the power: 1.053 means 1.05 × 1.05 × 1.05, not 1.05 × 3.Exercise 5M Compound interest, growth and decay
Calculator allowedKey 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 1.05 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £1,000, increase = 5% per year, number of years = 2
- Find the multiplier for one year. Each year the amount grows by 5%, so it becomes 105% of the previous amount. The multiplier is 105% as a decimal.105% = 105 ÷ 100 = 1.05
- Raise the multiplier to the power 2. The same increase happens 2 times, so we multiply by 1.05 2 times. That is the multiplier to the power 2.1.052 = 1.1025
- Multiply the starting amount by 1.052. This gives the amount after all the changes in one calculation.£1,000 × 1.1025 = £1,102.50
Check. Estimate: with simple interest the total would be £1,100. Compound interest is a little more because the interest earns interest, and £1,102.50 is a little above £1,100. ✓
Watch out. Do not multiply by 1.05 and then by 2. Use the multiplier to the power 2.
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 1.03 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £2,000, increase = 3% per year, number of years = 3
- Find the multiplier for one year. Each year the amount grows by 3%, so it becomes 103% of the previous amount. The multiplier is 103% as a decimal.103% = 103 ÷ 100 = 1.03
- Raise the multiplier to the power 3. The same increase happens 3 times, so we multiply by 1.03 3 times. That is the multiplier to the power 3.1.033 = 1.092727
- Multiply the starting amount by 1.033. This gives the amount after all the changes in one calculation.£2,000 × 1.092727 = £2,185.454
- Round the answer. Money is rounded to the nearest penny (2 decimal places). Round only at the very end.£2,185.454 = £2,185.45
Check. Estimate: with simple interest the total would be £2,180. Compound interest is a little more because the interest earns interest, and £2,185.45 is a little above £2,180. ✓
Watch out. Do not multiply by 1.03 and then by 3. Use the multiplier to the power 3.
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 1.04 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £5,000, increase = 4% per year, number of years = 2
- Find the multiplier for one year. Each year the amount grows by 4%, so it becomes 104% of the previous amount. The multiplier is 104% as a decimal.104% = 104 ÷ 100 = 1.04
- Raise the multiplier to the power 2. The same increase happens 2 times, so we multiply by 1.04 2 times. That is the multiplier to the power 2.1.042 = 1.0816
- Multiply the starting amount by 1.042. This gives the amount after all the changes in one calculation.£5,000 × 1.0816 = £5,408
Check. Estimate: with simple interest the total would be £5,400. Compound interest is a little more because the interest earns interest, and £5,408 is a little above £5,400. ✓
Watch out. Do not multiply by 1.04 and then by 2. Use the multiplier to the power 2.
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.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £800, increase = 2% per year, number of years = 3
- 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
- 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
- Multiply the starting amount by 1.023. This gives the amount after all the changes in one calculation.£800 × 1.061208 = £848.9664
- Round the answer. Money is rounded to the nearest penny (2 decimal places). Round only at the very end.£848.9664 = £848.97
Check. Estimate: with simple interest the total would be £848. Compound interest is a little more because the interest earns interest, and £848.97 is a little above £848. ✓
Watch out. Do not multiply by 1.02 and then by 3. Use the multiplier to the power 3.
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 1.06 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £1,500, increase = 6% per year, number of years = 2
- Find the multiplier for one year. Each year the amount grows by 6%, so it becomes 106% of the previous amount. The multiplier is 106% as a decimal.106% = 106 ÷ 100 = 1.06
- Raise the multiplier to the power 2. The same increase happens 2 times, so we multiply by 1.06 2 times. That is the multiplier to the power 2.1.062 = 1.1236
- Multiply the starting amount by 1.062. This gives the amount after all the changes in one calculation.£1,500 × 1.1236 = £1,685.40
Check. Estimate: with simple interest the total would be £1,680. Compound interest is a little more because the interest earns interest, and £1,685.40 is a little above £1,680. ✓
Watch out. Do not multiply by 1.06 and then by 2. Use the multiplier to the power 2.
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 1.015 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £10,000, increase = 1.5% per year, number of years = 4
- Find the multiplier for one year. Each year the amount grows by 1.5%, so it becomes 101.5% of the previous amount. The multiplier is 101.5% as a decimal.101.5% = 101.5 ÷ 100 = 1.015
- Raise the multiplier to the power 4. The same increase happens 4 times, so we multiply by 1.015 4 times. That is the multiplier to the power 4.1.0154 = 1.061364…
- Multiply the starting amount by 1.0154. This gives the amount after all the changes in one calculation.£10,000 × 1.061364… = £10,613.6355…
- Round the answer. Money is rounded to the nearest penny (2 decimal places). Round only at the very end.£10,613.6355… = £10,613.64
Check. Estimate: with simple interest the total would be £10,600. Compound interest is a little more because the interest earns interest, and £10,613.64 is a little above £10,600. ✓
Watch out. Do not multiply by 1.015 and then by 4. Use the multiplier to the power 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 1.025 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £4,000, increase = 2.5% per year, number of years = 2
- Find the multiplier for one year. Each year the amount grows by 2.5%, so it becomes 102.5% of the previous amount. The multiplier is 102.5% as a decimal.102.5% = 102.5 ÷ 100 = 1.025
- Raise the multiplier to the power 2. The same increase happens 2 times, so we multiply by 1.025 2 times. That is the multiplier to the power 2.1.0252 = 1.050625
- Multiply the starting amount by 1.0252. This gives the amount after all the changes in one calculation.£4,000 × 1.050625 = £4,202.50
- Subtract the original amount to find the interest. The interest is only the extra money, not the total.£4,202.50 − £4,000 = £202.50
Check. Estimate: with simple interest the total would be £4,200. Compound interest is a little more because the interest earns interest, and £4,202.50 is a little above £4,200. ✓
Watch out. Do not multiply by 1.025 and then by 2. Use the multiplier to the power 2.
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 1.03 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £6,000, increase = 3% per year, number of years = 2
- Find the multiplier for one year. Each year the amount grows by 3%, so it becomes 103% of the previous amount. The multiplier is 103% as a decimal.103% = 103 ÷ 100 = 1.03
- Raise the multiplier to the power 2. The same increase happens 2 times, so we multiply by 1.03 2 times. That is the multiplier to the power 2.1.032 = 1.0609
- Multiply the starting amount by 1.032. This gives the amount after all the changes in one calculation.£6,000 × 1.0609 = £6,365.40
- Subtract the original amount to find the interest. The interest is only the extra money, not the total.£6,365.40 − £6,000 = £365.40
Check. Estimate: with simple interest the total would be £6,360. Compound interest is a little more because the interest earns interest, and £6,365.40 is a little above £6,360. ✓
Watch out. Do not multiply by 1.03 and then by 2. Use the multiplier to the power 2.
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 1.04 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £2,500, increase = 4% per year, number of years = 3
- Find the multiplier for one year. Each year the amount grows by 4%, so it becomes 104% of the previous amount. The multiplier is 104% as a decimal.104% = 104 ÷ 100 = 1.04
- Raise the multiplier to the power 3. The same increase happens 3 times, so we multiply by 1.04 3 times. That is the multiplier to the power 3.1.043 = 1.124864
- Multiply the starting amount by 1.043. This gives the amount after all the changes in one calculation.£2,500 × 1.124864 = £2,812.16
- Subtract the original amount to find the interest. The interest is only the extra money, not the total.£2,812.16 − £2,500 = £312.16
Check. Estimate: with simple interest the total would be £2,800. Compound interest is a little more because the interest earns interest, and £2,812.16 is a little above £2,800. ✓
Watch out. Do not multiply by 1.04 and then by 3. Use the multiplier to the power 3.
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 1.04 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £3,000, increase = 4% per year, number of years = 3
- Find the multiplier for one year. Each year the amount grows by 4%, so it becomes 104% of the previous amount. The multiplier is 104% as a decimal.104% = 104 ÷ 100 = 1.04
- Raise the multiplier to the power 3. The same increase happens 3 times, so we multiply by 1.04 3 times. That is the multiplier to the power 3.1.043 = 1.124864
- Multiply the starting amount by 1.043. This gives the amount after all the changes in one calculation.£3,000 × 1.124864 = £3,374.592
- Round the answer. Money is rounded to the nearest penny (2 decimal places). Round only at the very end.£3,374.592 = £3,374.59
Check. Estimate: with simple interest the total would be £3,360. Compound interest is a little more because the interest earns interest, and £3,374.59 is a little above £3,360. ✓
Watch out. Do not multiply by 1.04 and then by 3. Use the multiplier to the power 3.
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 1.04 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £3,000, increase = 4% per year, number of years = 3
- Find the multiplier for one year. Each year the amount grows by 4%, so it becomes 104% of the previous amount. The multiplier is 104% as a decimal.104% = 104 ÷ 100 = 1.04
- Raise the multiplier to the power 3. The same increase happens 3 times, so we multiply by 1.04 3 times. That is the multiplier to the power 3.1.043 = 1.124864
- Multiply the starting amount by 1.043. This gives the amount after all the changes in one calculation.£3,000 × 1.124864 = £3,374.592
- Round the answer. Money is rounded to the nearest penny (2 decimal places). Round only at the very end.£3,374.592 = £3,374.59
- Subtract the original amount to find the interest. The interest is only the extra money, not the total.£3,374.59 − £3,000 = £374.59
Check. Estimate: with simple interest the total would be £3,360. Compound interest is a little more because the interest earns interest, and £3,374.59 is a little above £3,360. ✓
Watch out. Do not multiply by 1.04 and then by 3. Use the multiplier to the power 3.
Method. Work out the simple interest and the compound interest separately, then subtract.
- Simple interest. 4% of £2,000 = £80 each year, for 3 years.3 × £80 = £240
- Compound interest: the total. Multiplier 1.04 to the power 3.£2,000 × 1.043 = £2,000 × 1.124864 = £2,249.728 = £2,249.73
- Compound interest: the interest. Subtract the amount invested.£2,249.73 − £2,000 = £249.73
- Find the difference. Compound is bigger.£249.73 − £240 = £9.73
Check. Year 2 compound interest = 4% of £2,080 = £83.20, which is £3.20 more than £80; year 3 adds even more. ✓
Watch out. The question asks how much more, so end with a subtraction.
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 1.03 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = 25,000, increase = 3% per year, number of years = 4
- Find the multiplier for one year. Each year the amount grows by 3%, so it becomes 103% of the previous amount. The multiplier is 103% as a decimal.103% = 103 ÷ 100 = 1.03
- Raise the multiplier to the power 4. The same increase happens 4 times, so we multiply by 1.03 4 times. That is the multiplier to the power 4.1.034 = 1.125509…
- Multiply the starting amount by 1.034. This gives the amount after all the changes in one calculation.25,000 × 1.125509… = 28,137.72025
- Round the answer. Round to the nearest whole number.28,137.72025 = 28,138
Check. Estimate: with simple interest the total would be 28,000. Compound interest is a little more because the interest earns interest, and 28,138 is a little above 28,000. ✓
Watch out. Do not multiply by 1.03 and then by 4. Use the multiplier to the power 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 1.2 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = 500, increase = 20% per year, number of years = 5
- Find the multiplier for one year. Each year the amount grows by 20%, so it becomes 120% of the previous amount. The multiplier is 120% as a decimal.120% = 120 ÷ 100 = 1.2
- Raise the multiplier to the power 5. The same increase happens 5 times, so we multiply by 1.2 5 times. That is the multiplier to the power 5.1.25 = 2.48832
- Multiply the starting amount by 1.25. This gives the amount after all the changes in one calculation.500 × 2.48832 = 1,244.16
- Round the answer. Round to the nearest whole number.1,244.16 = 1,244
Check. Estimate: with simple interest the total would be 1,000. Compound interest is a little more because the interest earns interest, and 1,244 is a little above 1,000. ✓
Watch out. Do not multiply by 1.2 and then by 5. Use the multiplier to the power 5.
Key words. Depreciation means the value falls each year. Decay is a repeated percentage decrease.
Method. Repeated percentage decrease: multiply by 0.85 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £14,000, decrease = 15% per year, number of years = 3
- Find the multiplier for one year. Each year the amount falls by 15%, so it becomes 85% of the previous amount. The multiplier is 85% as a decimal.85% = 85 ÷ 100 = 0.85
- Raise the multiplier to the power 3. The same decrease happens 3 times, so we multiply by 0.85 3 times. That is the multiplier to the power 3.0.853 = 0.614125
- Multiply the starting amount by 0.853. This gives the amount after all the changes in one calculation.£14,000 × 0.614125 = £8,597.75
Check. Estimate: losing 15% of the original each time would leave £7,700. Each cut is smaller than that because it is taken from a smaller amount, so the answer should be a bit more than £7,700. ✓
Watch out. Do not multiply by 0.85 and then by 3. Use the multiplier to the power 3.
Key words. Depreciation means the value falls each year. Decay is a repeated percentage decrease.
Method. Repeated percentage decrease: multiply by 0.85 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £14,000, decrease = 15% per year, number of years = 5
- Find the multiplier for one year. Each year the amount falls by 15%, so it becomes 85% of the previous amount. The multiplier is 85% as a decimal.85% = 85 ÷ 100 = 0.85
- Raise the multiplier to the power 5. The same decrease happens 5 times, so we multiply by 0.85 5 times. That is the multiplier to the power 5.0.855 = 0.443705…
- Multiply the starting amount by 0.855. This gives the amount after all the changes in one calculation.£14,000 × 0.443705… = £6,211.874375
- Round the answer. Money is rounded to the nearest penny (2 decimal places). Round only at the very end.£6,211.874375 = £6,211.87
Check. Estimate: losing 15% of the original each time would leave £3,500. Each cut is smaller than that because it is taken from a smaller amount, so the answer should be a bit more than £3,500. ✓
Watch out. Do not multiply by 0.85 and then by 5. Use the multiplier to the power 5.
Key words. Depreciation means the value falls each year. Decay is a repeated percentage decrease.
Method. Repeated percentage decrease: multiply by 0.75 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £800, decrease = 25% per year, number of years = 2
- Find the multiplier for one year. Each year the amount falls by 25%, so it becomes 75% of the previous amount. The multiplier is 75% as a decimal.75% = 75 ÷ 100 = 0.75
- Apply the multiplier once for each year. Each year we multiply the amount we have by 0.75.After year 1: £800 × 0.75 = £600
After year 2: £600 × 0.75 = £450
Check. Estimate: losing 25% of the original each time would leave £400. Each cut is smaller than that because it is taken from a smaller amount, so the answer should be a bit more than £400. ✓
Watch out. Use the new amount each year as the starting point for the next year.
Method. Work out both totals and compare.
- Compound interest, year 1. 10% of £1,000 = £100.£1,000 + £100 = £1,100
- Compound interest, year 2. The interest is 10% of the new amount £1,100.10% of £1,100 = £110, so £1,100 + £110 = £1,210
- 20% simple interest. 20% of £1,000 = £200 added once.£1,000 + £200 = £1,200
- Compare. £1,210 is not £1,200.Joe is wrong. Compound interest gives £10 more.
Check. 1.12 = 1.21, and £1,000 × 1.21 = £1,210. ✓
Watch out. The second 10% is of a bigger amount, which is why compound beats simple.
Method. Find the correct multiplier and compare it with Dan’s.
- Dan’s method. He added 5% three times to get 15%.3 × 5% = 15%, so he used 1.15
- Explain the mistake. Each year the multiplier is 1.05 and the amounts multiply: the change in year 2 is 5% of a bigger amount.He should use 1.053, not 1 + 3 × 0.05.
- Work out the correct value. £1,000 × 1.053 = £1,000 × 1.157625 = £1,157.625 = £1,157.63
Check. Year by year: 1,000 → 1,050 → 1,102.50 → 1,157.625. ✓
Watch out. Use the multiplier raised to a power; do not multiply the percentage by the number of years.
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 1.028 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £8,500, increase = 2.8% per year, number of years = 6
- Find the multiplier for one year. Each year the amount grows by 2.8%, so it becomes 102.8% of the previous amount. The multiplier is 102.8% as a decimal.102.8% = 102.8 ÷ 100 = 1.028
- Raise the multiplier to the power 6. The same increase happens 6 times, so we multiply by 1.028 6 times. That is the multiplier to the power 6.1.0286 = 1.180208…
- Multiply the starting amount by 1.0286. This gives the amount after all the changes in one calculation.£8,500 × 1.180208… = £10,031.7711…
- Round the answer. Money is rounded to the nearest penny (2 decimal places). Round only at the very end.£10,031.7711… = £10,031.77
Check. Estimate: with simple interest the total would be £9,928. Compound interest is a little more because the interest earns interest, and £10,031.77 is a little above £9,928. ✓
Watch out. Do not multiply by 1.028 and then by 6. Use the multiplier to the power 6.
Key words. Depreciation means the value falls each year. Decay is a repeated percentage decrease.
Method. Repeated percentage decrease: multiply by 0.968 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = 14,500, decrease = 3.2% per year, number of years = 7
- Find the multiplier for one year. Each year the amount falls by 3.2%, so it becomes 96.8% of the previous amount. The multiplier is 96.8% as a decimal.96.8% = 96.8 ÷ 100 = 0.968
- Raise the multiplier to the power 7. The same decrease happens 7 times, so we multiply by 0.968 7 times. That is the multiplier to the power 7.0.9687 = 0.796393…
- Multiply the starting amount by 0.9687. This gives the amount after all the changes in one calculation.14,500 × 0.796393… = 11,547.7003…
- Round the answer. Round to the nearest whole number.11,547.7003… = 11,548
Check. Estimate: losing 3.2% of the original each time would leave 11,252. Each cut is smaller than that because it is taken from a smaller amount, so the answer should be a bit more than 11,252. ✓
Watch out. Do not multiply by 0.968 and then by 7. Use the multiplier to the power 7.
Key words. Depreciation means the value falls each year. Decay is a repeated percentage decrease.
Method. Repeated percentage decrease: multiply by 0.815 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £640, decrease = 18.5% per year, number of years = 4
- Find the multiplier for one year. Each year the amount falls by 18.5%, so it becomes 81.5% of the previous amount. The multiplier is 81.5% as a decimal.81.5% = 81.5 ÷ 100 = 0.815
- Raise the multiplier to the power 4. The same decrease happens 4 times, so we multiply by 0.815 4 times. That is the multiplier to the power 4.0.8154 = 0.441195…
- Multiply the starting amount by 0.8154. This gives the amount after all the changes in one calculation.£640 × 0.441195… = £282.3647…
- Round the answer. Money is rounded to the nearest penny (2 decimal places). Round only at the very end.£282.3647… = £282.36
Check. Estimate: losing 18.5% of the original each time would leave £166.40. Each cut is smaller than that because it is taken from a smaller amount, so the answer should be a bit more than £166.40. ✓
Watch out. Do not multiply by 0.815 and then by 4. Use the multiplier to the power 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 1.02 once for each half-year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = £10,000, increase = 2% per half-year, number of half-years = 4
- Find the multiplier for one half-year. Each half-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
- Raise the multiplier to the power 4. The same increase happens 4 times, so we multiply by 1.02 4 times. That is the multiplier to the power 4.1.024 = 1.082432…
- Multiply the starting amount by 1.024. This gives the amount after all the changes in one calculation.£10,000 × 1.082432… = £10,824.3216
- Round the answer. Money is rounded to the nearest penny (2 decimal places). Round only at the very end.£10,824.3216 = £10,824.32
Check. Estimate: with simple interest the total would be £10,800. Compound interest is a little more because the interest earns interest, and £10,824.32 is a little above £10,800. ✓
Watch out. Do not multiply by 1.02 and then by 4. Use the multiplier to the power 4.
Method. Work out the amount at the end of year 1, then find 5% of that.
- Amount after year 1. 5% of £1,000 = £50, added to the principal.£1,000 + £50 = £1,050
- Interest in year 2. It is 5% of the new amount £1,050.5% of £1,050 = 0.05 × £1,050 = £52.50
- Explain. In year 1 the 5% is of £1,000. In year 2 it is of £1,050, which includes the first year’s interest, so it is bigger.£52.50 is £2.50 more than £50.
Check. 1,000 × 1.052 = 1,102.50 and 1,102.50 − 1,050 = 52.50. ✓
Watch out. This is the key difference: compound interest earns interest on the interest.
Exercise 5N Growth and decay: years, rates and starting amounts
Calculator allowedMethod. Use the multiplier to a power and try more and more years until the target is reached.
- Find the multiplier for one year. Write the percentage as a decimal.104% = 1.04
- 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: £1,000 × 1.041 = £1,040
After 2 years: £1,000 × 1.042 = £1,081.60
After 3 years: £1,000 × 1.043 = £1,124.864 = £1,124.86
After 4 years: £1,000 × 1.044 = £1,169.85856 = £1,169.86
After 5 years: £1,000 × 1.045 = £1,216.6529… = £1,216.65
After 6 years: £1,000 × 1.046 = £1,265.3190… = £1,265.32
After 7 years: £1,000 × 1.047 = £1,315.9318… = £1,315.93
After 8 years: £1,000 × 1.048 = £1,368.5691… = £1,368.57
After 9 years: £1,000 × 1.049 = £1,423.3118… = £1,423.31
After 10 years: £1,000 × 1.0410 = £1,480.2443… = £1,480.24
After 11 years: £1,000 × 1.0411 = £1,539.4541… = £1,539.45 - Find the first time the target is passed. We want the amount to be more than £1,500.After 11 years the amount is £1,539.45, which is above £1,500. After 10 it was not.
Check. After 10 years the amount is £1,480.24, which has not yet passed £1,500, so 11 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. Use the multiplier to a power and try more and more years until the target is reached.
- Find the multiplier for one year. Write the percentage as a decimal.103% = 1.03
- 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: £5,000 × 1.031 = £5,150
After 2 years: £5,000 × 1.032 = £5,304.50
After 3 years: £5,000 × 1.033 = £5,463.635 = £5,463.64
After 4 years: £5,000 × 1.034 = £5,627.54405 = £5,627.54
After 5 years: £5,000 × 1.035 = £5,796.3704… = £5,796.37
After 6 years: £5,000 × 1.036 = £5,970.2615… = £5,970.26
After 7 years: £5,000 × 1.037 = £6,149.3693… = £6,149.37 - Find the first time the target is passed. We want the amount to be more than £6,000.After 7 years the amount is £6,149.37, which is above £6,000. After 6 it was not.
Check. After 6 years the amount is £5,970.26, which has not yet passed £6,000, so 7 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. Use the multiplier to a power and try more and more years until the target is reached.
- Find the multiplier for one year. Write the percentage as a decimal.88% = 0.88
- 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: £20,000 × 0.881 = £17,600
After 2 years: £20,000 × 0.882 = £15,488
After 3 years: £20,000 × 0.883 = £13,629.44
After 4 years: £20,000 × 0.884 = £11,993.9072 = £11,993.91
After 5 years: £20,000 × 0.885 = £10,554.638336 = £10,554.64
After 6 years: £20,000 × 0.886 = £9,288.0817… = £9,288.08 - Find the first time the target is passed. We want the amount to be below £10,000.After 6 years the amount is £9,288.08, which is below £10,000. After 5 it was not.
Check. After 5 years the amount is £10,554.64, which has not yet passed £10,000, so 6 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.
- Write the equation. Let the yearly multiplier be m. After 3 years the start amount has been multiplied by m3.£3,000 × m3 = £5,184
- Divide both sides by the start amount. This isolates m3.m3 = £5,184 ÷ £3,000 = 1.728
- Take the cube root of both sides. The cube root undoes the power 3. On a calculator use the x1/3 or the n-th root key.m = cube root of 1.728 = 1.2
- Turn the multiplier into a percentage. 1.2 = 120%, and the original is 100%.120% − 100% = 20% increase
Check. £3,000 × 1.23 = £3,000 × 1.728 = £5,184. ✓
Watch out. Do not divide the total percentage change by the number of years. The change compounds.
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.
- Write the equation. Let the yearly multiplier be m. After 2 years the start amount has been multiplied by m2.£2,500 × m2 = £3,025
- Divide both sides by the start amount. This isolates m2.m2 = £3,025 ÷ £2,500 = 1.21
- Take the square root of both sides. The square root undoes the power 2. On a calculator use the √ key.m = square root of 1.21 = 1.1
- Turn the multiplier into a percentage. 1.1 = 110%, and the original is 100%.110% − 100% = 10% increase
Check. £2,500 × 1.12 = £2,500 × 1.21 = £3,025. ✓
Watch out. Do not divide the total percentage change by the number of years. The change compounds.
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.
- Write the equation. Let the yearly multiplier be m. After 2 years the start amount has been multiplied by m2.£16,000 × m2 = £12,960
- Divide both sides by the start amount. This isolates m2.m2 = £12,960 ÷ £16,000 = 0.81
- Take the square root of both sides. The square root undoes the power 2. On a calculator use the √ key.m = square root of 0.81 = 0.9
- Turn the multiplier into a percentage. 0.9 = 90%, and the original is 100%.90% − 100% = 10% decrease
Check. £16,000 × 0.92 = £16,000 × 0.81 = £12,960. ✓
Watch out. Do not divide the total percentage change by the number of years. The change compounds.
Method. The final amount is the original multiplied by the multiplier to the power 3. To go backwards, divide by 1.13.
- Find the multiplier for one year. A increase of 10% means the new amount is 110% of the old amount.110% = 1.1
- Raise it to the power 3. The change happened 3 times.1.13 = 1.331
- Write what we know. original × multiplier3 = final amount.original × 1.331 = £6,655
- Divide to find the original. To undo a multiplication, divide.£6,655 ÷ 1.331 = £5,000
Check. £5,000 × 1.331 = £6,655. ✓
Watch out. Do not take the percentage off the final amount and repeat. Divide by the multiplier to the power n.
Method. The final amount is the original multiplied by the multiplier to the power 2. To go backwards, divide by 0.952.
- Find the multiplier for one year. A decrease of 5% means the new amount is 95% of the old amount.95% = 0.95
- Raise it to the power 2. The change happened 2 times.0.952 = 0.9025
- Write what we know. original × multiplier2 = final amount.original × 0.9025 = £9,025
- Divide to find the original. To undo a multiplication, divide.£9,025 ÷ 0.9025 = £10,000
Check. £10,000 × 0.9025 = £9,025. ✓
Watch out. Do not take the percentage off the final amount and repeat. Divide by the multiplier to the power n.
Method. The final amount is the original multiplied by the multiplier to the power 4. To go backwards, divide by 1.24.
- Find the multiplier for one year. A increase of 20% means the new amount is 120% of the old amount.120% = 1.2
- Raise it to the power 4. The change happened 4 times.1.24 = 2.0736
- Write what we know. original × multiplier4 = final amount.original × 2.0736 = £10,368
- Divide to find the original. To undo a multiplication, divide.£10,368 ÷ 2.0736 = £5,000
Check. £5,000 × 2.0736 = £10,368. ✓
Watch out. Do not take the percentage off the final amount and repeat. Divide by the multiplier to the power n.
Method. Find the overall multiplier by multiplying the separate multipliers, then use it on the starting amount.
- Write a multiplier for each change. Each percentage change has its own multiplier (100% plus or minus the change, as a decimal).increase of 4%: 104% = 1.04
increase of 3%: 103% = 1.03
increase of 2.5%: 102.5% = 1.025 - Multiply the multipliers together. One change is applied after the other, so the multipliers are multiplied.1.04 × 1.03 × 1.025 = 1.09798
- Multiply the starting amount by the overall multiplier. This gives the amount after both changes.£5,000 × 1.09798 = £5,489.90
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%.
Key words. Depreciation means the value falls each year. Decay is a repeated percentage decrease.
Method. Repeated percentage decrease: multiply by 0.85 once for each year.
- Write down what we know. The starting amount, the percentage change each time, and how many times it happens.Start = 80 g, decrease = 15% per year, number of years = 4
- Find the multiplier for one year. Each year the amount falls by 15%, so it becomes 85% of the previous amount. The multiplier is 85% as a decimal.85% = 85 ÷ 100 = 0.85
- Raise the multiplier to the power 4. The same decrease happens 4 times, so we multiply by 0.85 4 times. That is the multiplier to the power 4.0.854 = 0.522006…
- Multiply the starting amount by 0.854. This gives the amount after all the changes in one calculation.80 g × 0.522006… = 41.7605 g
- Round the answer. Round to 2 decimal places.41.7605 g = 41.76 g
Check. Estimate: losing 15% of the original each time would leave 32 g. Each cut is smaller than that because it is taken from a smaller amount, so the answer should be a bit more than 32 g. ✓
Watch out. Do not multiply by 0.85 and then by 4. Use the multiplier to the power 4.
Method. The money is multiplied by 1.0810. Show that this is greater than 2.
- Write the multiplier for 10 years. 8% growth has multiplier 1.08, applied 10 times.1.0810
- Work it out on a calculator. Use the power key.1.0810 = 2.1589…
- Compare with 2. Doubling means a multiplier of 2.2.1589… > 2, so the money more than doubles.
Check. After 9 years: 1.089 = 1.9990… < 2, so 10 years is the first time. ✓
Watch out. Always state the comparison and your conclusion.
Method. Doubling means the multiplier to the power n is at least 2. Test n = 35 and n = 36.
- Write the multiplier. 2% growth: 100% + 2% = 102% = 1.02.1.02
- Try 35 years. Use the power key.1.0235 = 1.99989…
- Compare with 2. 1.99989… is a tiny bit less than 2.After 35 years it has not quite doubled.
- Try 36 years. 1.0236 = 2.0399… > 2
- Conclude. Priya is nearly right: it takes 36 years to pass double.
Check. 1.0236 = 1.0235 × 1.02, and 1.9999 × 1.02 is about 2.04. ✓
Watch out. Do not round 1.9999… up to 2 when the question is about being more than 2.