About Compound Interest
Compound interest is simple interest applied again and again, because the interest already earned starts earning interest itself. That one idea is the whole topic. Nearly every question is settled by working out the amount first and subtracting the principal, rather than by trying to compute the interest directly.
What you need to understand
- The interest is added to the principal at the end of each period, and the next period is calculated on that larger sum.
- Compound interest = amount - principal. It is very rarely useful to work out the interest on its own.
- Over a single year compound interest and simple interest give exactly the same answer, because there is no accumulated interest yet for compounding to act on.
- For two years the gap between the two is P x (R/100)^2, which is the interest earned on the first year of interest.
- Depreciation behaves the same way with a factor of (1 - R/100), so a machine loses a fixed share of its current value each year rather than a fixed amount.
- A rise of R per cent followed by a fall of R per cent does not bring a price back to where it started.
Formulas to remember
How to work through these questions
- Decide first whether the question wants the amount or the interest. The two differ by the principal, and giving the wrong one is the commonest single mistake in the topic.
- Write the rate as a fraction, treating 10 per cent as 1/10, then multiply year by year instead of trying to raise anything to a power.
- For a two year question, work out the first year of interest, add it on, then work out the second year on the new total.
- For depreciation remember the value is falling, so use (1 - R/100) and expect a smaller number each year.
- Check the size of the answer, because compound interest is always slightly more than simple interest at the same rate for the same period.
Mistakes that cost marks
- Giving the amount when the question asked for the interest, or the other way round.
- Using the simple interest formula when the question says compounded.
- Adding the same interest figure each year, which is straight line depreciation rather than compounding.
- Applying depreciation to the original price instead of to the already reduced value.
- Assuming that a rise of 10 per cent followed by a fall of 10 per cent leaves a price unchanged.
Worked example
Find the compound interest on Rs. 8000 for 2 years at 10% per annum.
- Interest for the first year is 10 per cent of 8000, which is 800, so the amount after one year is 8800.
- Interest for the second year is 10 per cent of 8800, which is 880, so the amount after two years is 9680.
- Compound interest = amount - principal = 9680 - 8000 = 1680.
- Check: simple interest for the same period would be 8000 x 10 x 2 / 100 = 1600, and compound interest must be a little more. It is.
Answer: Rs. 1680
Practice questions with answers
A few Compound Interest questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
Calculate the compound interest on Rs. 700 for 2 years, the rate being 10% per annum.
Answer: Option A — with explanation
Amount = P(1 + R/100)^n = 700 x (1 + 10/100)^2 = 847.
Compound interest = amount - principal = 847 - 700 = 147.
Simple interest for the same period would have been only 140.
Common mistakes
- gave the interest for a single year: 70
- used simple interest instead of compound interest: 140
- gave the amount instead of the interest: 847
Question 2
The difference between the compound interest and the simple interest on a sum of Rs. 1600 at 5% per annum for 2 years is:
Answer: Option A — with explanation
For two years the two interests differ only on the interest earned in the first year.
Difference = P x \(\frac{R}{100}\)^2 = 1600 x \(\frac{5}{100}\)^2 = 4.
Simple interest for two years is 160, and the compound interest is 164.
Common mistakes
- gave the whole simple interest for the two years: 160
- halved the difference because the period is two years: 2
- gave one year of simple interest: 80
Question 3
The value of a machine depreciates by 20% every year. If its present value is Rs. 1000, what will it be after 3 years?
Answer: Option D — with explanation
Depreciation compounds exactly like interest, except that the factor is (1 - R/100).
Value = P(1 - R/100)^n = 1000 x (1 - 20/100)^3 = 512.
Straight-line depreciation would have taken off the same amount each year and left 400, which is not what compounding does.
Common mistakes
- took off the same amount every year instead of compounding the fall: 400
- gave the total amount lost instead of the value that remains: 488
- applied the depreciation only once: 800
Question 4
Calculate the compound interest on Rs. 4000 for 3 years, the rate being 10% per annum.
-
A
1200
-
B
1324
-
C
400
-
D
5324
Answer: Option B — with explanation
Amount = P(1 + R/100)^n = 4000 x (1 + 10/100)^3 = 5324.
Compound interest = amount - principal = 5324 - 4000 = 1324.
Simple interest for the same period would have been only 1200.
Common mistakes
- gave the interest for a single year: 400
- gave the amount instead of the interest: 5324
- used simple interest instead of compound interest: 1200
Question 5
The value of a machine depreciates by 10% every year. If its present value is Rs. 9000, what will it be after 3 years?
-
A
2439
-
B
8100
-
C
6300
-
D
6561
Answer: Option D — with explanation
Depreciation compounds exactly like interest, except that the factor is (1 - R/100).
Value = P(1 - R/100)^n = 9000 x (1 - 10/100)^3 = 6561.
Straight-line depreciation would have taken off the same amount each year and left 6300, which is not what compounding does.
Common mistakes
- applied the depreciation only once: 8100
- gave the total amount lost instead of the value that remains: 2439
- took off the same amount every year instead of compounding the fall: 6300
Frequently asked questions
What is the difference between simple and compound interest?
Simple interest is charged on the original principal every year. Compound interest is charged on the principal plus the interest already added, so it grows faster and overtakes simple interest from the second year onwards.
How do I find the difference between the two for two years?
Use P x (R/100)^2. On Rs. 5000 at 10 per cent for two years the difference is 5000 x 0.1 x 0.1 = Rs. 50.
How does depreciation differ from compound interest?
Only in the sign of the rate. Interest multiplies the value by (1 + R/100) each year, while depreciation multiplies it by (1 - R/100), so the value shrinks by a fixed proportion each year.
Why is compound interest equal to simple interest for one year?
Because the first year of interest is calculated on the original principal in both cases. There is no accumulated interest yet for compounding to work on.
