GCSE Maths: Simple interest (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)
Simple interest
When you save money in a bank, the bank pays you for letting it use your money. This payment is called interest. When you borrow money you pay interest to the lender. Interest is usually given as a percentage rate per year, written “3% p.a.” (per annum is Latin for “each year”). The money you first put in or borrow is called the principal.
With simple interest the bank pays the same amount every year. It is always worked out on the original principal, never on the interest that has been added. So simple interest is just “a percentage of an amount” (Section 5.2), repeated for each year.
To find simple interest:
- Work out the interest for one year: the rate % of the principal.
- Multiply by the number of years.
- If you want the total, add the interest to the principal.
Short for the first two steps: interest = principal × rate × years ÷ 100 (with the rate written as a number, not as a decimal). If the time is given in months, change it to years first: 6 months = 0.5 years, 18 months = 1.5 years.
Working backwards
If you know the interest you can find the rate or the time. If a savings account earned £300 in 3 years from £2,000, the interest for one year was £300 ÷ 3 = £100, and £100 is 100 ÷ 2,000 × 100 = 5% of the principal. So the rate is 5% p.a.
Key ideas
- Principal = the money you start with. Rate = percentage per year (p.a.).
- Simple interest is the same each year: rate% of the original principal.
- Interest = P × R × T ÷ 100 (R = rate as a number, T = years).
- Total = principal + interest.
- Change months to years: 6 months = 0.5 years.
- Rate = (interest for one year) ÷ principal × 100.
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.
- 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 = £1,500, rate = 3% per year, time = 4 years
- Find the interest for ONE year. Simple interest is the same every year: it is 3% of the original amount each time.3% of £1,500 = 0.03 × £1,500 = £45
- Multiply by the number of years. The same interest is added every year, so multiply by 4.£45 × 4 = £180
- Add the interest to the original amount. The total is what the account holds at the end: the original money plus the interest.£1,500 + £180 = £1,680
Check. Formula check: I = P × R × T ÷ 100 = 1,500 × 3 × 4 ÷ 100 = 180. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
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.
- 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 = £800, rate = 2.5% per year, time = 3 years
- Find the interest for ONE year. Simple interest is the same every year: it is 2.5% of the original amount each time.2.5% of £800 = 0.025 × £800 = £20
- Multiply by the number of years. The same interest is added every year, so multiply by 3.£20 × 3 = £60
Check. Formula check: I = P × R × T ÷ 100 = 800 × 2.5 × 3 ÷ 100 = 60. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
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.
- Change the time to years. The rate is per year, so the time must be in years. 6 months is half a year.6 months = 6 ÷ 12 = 0.5 years
- 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 = £2,400, rate = 5% per year, time = 0.5 years
- Find the interest for ONE year. Simple interest is the same every year: it is 5% of the original amount each time.5% of £2,400 = 0.05 × £2,400 = £120
- Multiply by the number of years. The same interest is added every year, so multiply by 0.5.£120 × 0.5 = £60
Check. Formula check: I = P × R × T ÷ 100 = 2,400 × 5 × 0.5 ÷ 100 = 60. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
Method. Find the interest for one year, then write it as a percentage of the original amount.
- Write down what we know. The principal is the money at the start. The interest is the extra money earned.Principal = £2,000, interest = £300, time = 3 years
- Find the interest for one year. The total interest was earned over 3 years, so divide by 3.£300 ÷ 3 = £100
- Write one year’s interest as a percentage of the original amount. Rate = part over whole, times 100.£100 ÷ £2,000 × 100 = 5%
Check. 5% of £2,000 = £100, and 3 × £100 = £300. ✓
Watch out. Divide by the number of years first. The percentage asked for is the yearly rate.
Method. Find the interest for one year, then see how many years give the total interest.
- Write down what we know. The principal is the money at the start; the rate is the percentage per year.Principal = £2,000, rate = 5% per year, total interest wanted = £300
- Find the interest for one year. It is 5% of the original amount.5% of £2,000 = £100
- Divide the total interest by one year’s interest. This tells us how many years are needed.£300 ÷ £100 = 3
Check. 3 × £100 = £300. ✓
Method. Find the interest for one year, then use it to find what 1.5% of the principal is, and so the principal.
- Find the interest for one year. The interest is the same each year, so divide the total interest by the number of years.£72 ÷ 4 = £18
- Write what you know. 1.5% of the principal is the interest for one year.1.5% of the principal = £18
- Find 1%. Divide by 1.5.£18 ÷ 1.5 = £12
- Find 100%, the principal. Multiply by 100.£12 × 100 = £1,200
Check. 1.5% of £1,200 = £18, and 4 × £18 = £72. ✓
Watch out. This is a reverse-percentage step: the 1.5% is of the principal.
Watch out
Simple interest is worked out on the original amount every year. Do not add the interest on and then take a percentage of the bigger number: that is compound interest (next section). Also remember to add the interest to the principal when the question asks for the total in the account.Exercise 5L Simple interest
Non-calculatorKey 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.
- 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 = £400, rate = 3% per year, time = 2 years
- Find the interest for ONE year. Simple interest is the same every year: it is 3% of the original amount each time.3% of £400 = 0.03 × £400 = £12
- Multiply by the number of years. The same interest is added every year, so multiply by 2.£12 × 2 = £24
Check. Formula check: I = P × R × T ÷ 100 = 400 × 3 × 2 ÷ 100 = 24. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
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.
- 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 = £2,000, rate = 2% per year, time = 5 years
- Find the interest for ONE year. Simple interest is the same every year: it is 2% of the original amount each time.2% of £2,000 = 0.02 × £2,000 = £40
- Multiply by the number of years. The same interest is added every year, so multiply by 5.£40 × 5 = £200
Check. Formula check: I = P × R × T ÷ 100 = 2,000 × 2 × 5 ÷ 100 = 200. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
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.
- 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 = £650, rate = 10% per year, time = 3 years
- Find the interest for ONE year. Simple interest is the same every year: it is 10% of the original amount each time.10% of £650 = 0.1 × £650 = £65
- Multiply by the number of years. The same interest is added every year, so multiply by 3.£65 × 3 = £195
Check. Formula check: I = P × R × T ÷ 100 = 650 × 10 × 3 ÷ 100 = 195. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
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.
- 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 = £1,200, rate = 1.5% per year, time = 4 years
- Find the interest for ONE year. Simple interest is the same every year: it is 1.5% of the original amount each time.1.5% of £1,200 = 0.015 × £1,200 = £18
- Multiply by the number of years. The same interest is added every year, so multiply by 4.£18 × 4 = £72
Check. Formula check: I = P × R × T ÷ 100 = 1,200 × 1.5 × 4 ÷ 100 = 72. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
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.
- 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 = £5,000, rate = 4% per year, time = 3 years
- Find the interest for ONE year. Simple interest is the same every year: it is 4% of the original amount each time.4% of £5,000 = 0.04 × £5,000 = £200
- Multiply by the number of years. The same interest is added every year, so multiply by 3.£200 × 3 = £600
Check. Formula check: I = P × R × T ÷ 100 = 5,000 × 4 × 3 ÷ 100 = 600. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
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.
- 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 = £840, rate = 5% per year, time = 2 years
- Find the interest for ONE year. Simple interest is the same every year: it is 5% of the original amount each time.5% of £840 = 0.05 × £840 = £42
- Multiply by the number of years. The same interest is added every year, so multiply by 2.£42 × 2 = £84
Check. Formula check: I = P × R × T ÷ 100 = 840 × 5 × 2 ÷ 100 = 84. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
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.
- 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 = £1,500, rate = 3% per year, time = 4 years
- Find the interest for ONE year. Simple interest is the same every year: it is 3% of the original amount each time.3% of £1,500 = 0.03 × £1,500 = £45
- Multiply by the number of years. The same interest is added every year, so multiply by 4.£45 × 4 = £180
- Add the interest to the original amount. The total is what the account holds at the end: the original money plus the interest.£1,500 + £180 = £1,680
Check. Formula check: I = P × R × T ÷ 100 = 1,500 × 3 × 4 ÷ 100 = 180. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
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.
- 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 = £800, rate = 2.5% per year, time = 3 years
- Find the interest for ONE year. Simple interest is the same every year: it is 2.5% of the original amount each time.2.5% of £800 = 0.025 × £800 = £20
- Multiply by the number of years. The same interest is added every year, so multiply by 3.£20 × 3 = £60
- Add the interest to the original amount. The total is what the account holds at the end: the original money plus the interest.£800 + £60 = £860
Check. Formula check: I = P × R × T ÷ 100 = 800 × 2.5 × 3 ÷ 100 = 60. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
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.
- 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 = £3,000, rate = 4% per year, time = 5 years
- Find the interest for ONE year. Simple interest is the same every year: it is 4% of the original amount each time.4% of £3,000 = 0.04 × £3,000 = £120
- Multiply by the number of years. The same interest is added every year, so multiply by 5.£120 × 5 = £600
- Add the interest to the original amount. The total is what the account holds at the end: the original money plus the interest.£3,000 + £600 = £3,600
Check. Formula check: I = P × R × T ÷ 100 = 3,000 × 4 × 5 ÷ 100 = 600. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
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.
- 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 = £2,000, rate = 3% per year, time = 2 years
- Find the interest for ONE year. Simple interest is the same every year: it is 3% of the original amount each time.3% of £2,000 = 0.03 × £2,000 = £60
- Multiply by the number of years. The same interest is added every year, so multiply by 2.£60 × 2 = £120
- Add the interest to the original amount. The total is what the account holds at the end: the original money plus the interest.£2,000 + £120 = £2,120
Check. Formula check: I = P × R × T ÷ 100 = 2,000 × 3 × 2 ÷ 100 = 120. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
Method. If you know what a percentage of the whole is, find 1% first, then multiply by 100 to find the whole (100%).
- Write down what we know. Part of the whole has been given as a percentage.6% of the whole = £36
- Find 1%. Divide by 6. That takes us from 6% to 1%.£36 ÷ 6 = £6
- Find 100%. The whole is 100%, so multiply 1% by 100.£6 × 100 = £600
Check. 6% of £600 = 0.06 × £600 = £36. ✓
Watch out. The whole is 100%. Do not stop at 1%.
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.
- Change the time to years. The rate is per year, so the time must be in years. 6 months is half a year.6 months = 6 ÷ 12 = 0.5 years
- 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 = £2,400, rate = 5% per year, time = 0.5 years
- Find the interest for ONE year. Simple interest is the same every year: it is 5% of the original amount each time.5% of £2,400 = 0.05 × £2,400 = £120
- Multiply by the number of years. The same interest is added every year, so multiply by 0.5.£120 × 0.5 = £60
Check. Formula check: I = P × R × T ÷ 100 = 2,400 × 5 × 0.5 ÷ 100 = 60. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
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.
- 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 = 8% per year, time = 2 years
- Find the interest for ONE year. Simple interest is the same every year: it is 8% of the original amount each time.8% of £600 = 0.08 × £600 = £48
- Multiply by the number of years. The same interest is added every year, so multiply by 2.£48 × 2 = £96
- Add the interest to the original amount. The total is what the account holds at the end: the original money plus the interest.£600 + £96 = £696
Check. Formula check: I = P × R × T ÷ 100 = 600 × 8 × 2 ÷ 100 = 96. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
Method. Work out the interest from each bank, then find the difference.
- Bank A: interest for one year. 3% of £2,000.3% of £2,000 = £60
- Bank A: interest for 4 years. Multiply by 4.£60 × 4 = £240
- Bank B: interest for one year. 4% of £1,500.4% of £1,500 = £60
- Bank B: interest for 5 years. Multiply by 5.£60 × 5 = £300
- Compare and find the difference. 300 is more than 240.£300 − £240 = £60. Bank B pays £60 more.
Check. Formula: A = 2000 × 3 × 4 ÷ 100 = 240; B = 1500 × 4 × 5 ÷ 100 = 300. ✓
Watch out. Give the answer in the form the question asks for: which bank, and how much more.
Method. Find the interest for one year, then write it as a percentage of the original amount.
- Write down what we know. The principal is the money at the start. The interest is the extra money earned.Principal = £1,600, interest = £192, time = 3 years
- Find the interest for one year. The total interest was earned over 3 years, so divide by 3.£192 ÷ 3 = £64
- Write one year’s interest as a percentage of the original amount. Rate = part over whole, times 100.£64 ÷ £1,600 × 100 = 4%
Check. 4% of £1,600 = £64, and 3 × £64 = £192. ✓
Watch out. Divide by the number of years first. The percentage asked for is the yearly rate.
Method. Find the interest for one year, then see how many years give the total interest.
- Write down what we know. The principal is the money at the start; the rate is the percentage per year.Principal = £2,500, rate = 4% per year, total interest wanted = £500
- Find the interest for one year. It is 4% of the original amount.4% of £2,500 = £100
- Divide the total interest by one year’s interest. This tells us how many years are needed.£500 ÷ £100 = 5
Check. 5 × £100 = £500. ✓
Method. Work out the interest in year 1 and year 2 separately.
- Year 1. 5% of the principal £1,000.5% of £1,000 = £50
- Year 2. Simple interest is always worked out on the original £1,000, so it is the same.5% of £1,000 = £50
- Conclude. Fred is wrong: the interest is £50 every year.
Check. After 2 years the total is £1,000 + £100 = £1,100. ✓
Watch out. If the interest grew each year, that would be compound interest.
Method. Find one year’s interest, then work back to the principal (100%).
- Find the interest for one year. Divide the total interest by 4.£72 ÷ 4 = £18
- Find 1%. 1.5% is £18, so divide by 1.5.£18 ÷ 1.5 = £12
- Find 100%. Multiply by 100.£12 × 100 = £1,200
Check. 1.5% of £1,200 = £18; 4 years: £72. ✓
Watch out. The yearly interest is 1.5% of the principal; you must find 100%.
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.
- Change the time to years. 12 months make a year. 30 months is 30 ÷ 12 years.30 months = 30 ÷ 12 = 2.5 years
- 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 = £5,000, rate = 4.2% per year, time = 2.5 years
- Find the interest for ONE year. Simple interest is the same every year: it is 4.2% of the original amount each time.4.2% of £5,000 = 0.042 × £5,000 = £210
- Multiply by the number of years. The same interest is added every year, so multiply by 2.5.£210 × 2.5 = £525
Check. Formula check: I = P × R × T ÷ 100 = 5,000 × 4.2 × 2.5 ÷ 100 = 525. ✓
Watch out. Simple interest always uses the original amount, never the amount with interest added.
Method. Find the total percentage of the principal earned in each case.
- 6% for 5 years. The interest is 6% of the principal each year, 5 times.6 × 5 = 30, so the interest is 30% of the principal
- 5% for 6 years. The interest is 5% of the principal each year, 6 times.5 × 6 = 30, so the interest is 30% of the principal
- Conclude. Both are 30% of the same principal, so they are equal.The interest is the same.
Check. On £1,000: 6% for 5 years gives 1000 × 6 × 5 ÷ 100 = 300 and 5% for 6 years gives 1000 × 5 × 6 ÷ 100 = 300. ✓
Watch out. Multiplication can be done in any order.