Compound Interest [Applications]
ICSE Class 9 · Mathematics · Chapter 3
What this chapter covers
- From year-by-year working to a formula
- Interest compounded half-yearly
- Finding the principal
- Finding the rate of interest
- Finding the time
- Problems using both simple and compound interest
- The difference between compound interest and simple interest
- Rate of growth
- Depreciation
- Common mistakes and how to avoid them
Key points
- Compound interest is interest calculated on a principal that grows by the addition of each period's interest.
- Compounded annually: A = P(1 + (r)/(100))n and CI = A - P.
- Compounded half-yearly: use the rate (r)/(2)% for each of 2n half-years, so A = P(1 + (r)/(200))2n.
- Any one of A, P, r and n can be found when the other three are known; for r and n, recognise (A)/(P) as an exact power.
- For the first year, compound interest equals simple interest; from the second year onwards compound interest is greater.
- For 2 years, CI - SI = P((r)/(100))2; for 3 years, CI - SI = P((r)/(100))2(3 + (r)/(100)).
- Growth at r% per annum multiplies a quantity by (1 + (r)/(100)) each year.
- Depreciation at r% per annum multiplies a value by (1 - (r)/(100)) each year: V = V0(1 - (r)/(100))n.
- Keep exact values until the last step and round only the final answer.
Questions answered
Write the formula for the amount when a principal P is lent at r% per annum compound interest for n years, compounded annually, and the formula for the compound interest.
A = P(1 + (r)/(100))n and CI = A - P = P[(1 + (r)/(100))n - 1].
Find the compound interest on ₹5,000 for 2 years at 12% per annum, compounded annually.
A = 5000 × (1.12)2 = 5000 × 1.2544 = ₹6,272, so CI = ₹1,272.
Find the compound interest on ₹20,000 for 1 year at 8% per annum, compounded half-yearly. Why is it more than the simple interest for the same year?
A = 20000 × (1.04)2 = ₹21,632, so CI = ₹1,632. It exceeds the simple interest of ₹1,600 because the ₹800 added after six months earns interest for the second half-year.
On what sum will the compound interest for 2 years at 10% per annum, compounded annually, be ₹2,100?
P[(1.1)2 - 1] = 0.21P = 2100, so P = ₹10,000.
At what rate per cent per annum, compounded annually, will ₹6,250 amount to ₹7,290 in 2 years?
(7290)/(6250) = 1.1664 = (1.08)2, so r = 8%.
In how many years will ₹12,500 amount to ₹13,520 at 4% per annum, compounded annually?
(13520)/(12500) = 1.0816 = (1.04)2, so n = 2 years.
Find the difference between the compound interest and the simple interest on ₹12,000 for 2 years at 5% per annum.
12000 × ((5)/(100))2 = 12000 × 0.0025 = ₹30.
The difference between the compound interest and the simple interest on a sum for 2 years at 4% per annum is ₹48. Find the sum.
P × 0.0016 = 48, so P = ₹30,000.
Past board exam questions
ICSE 2019 sample paper · 4 marks
A sum of ₹ 12,500 is deposited for 1½ years, compounded half yearly. It amounts to ₹ 13,000/- at the end of first half year. Find: (i) The rate of interest (ii) The final amount. Give your answer correct to the nearest rupee.
Show answer
(i) Interest for the first half year = 13,000 − 12,500 = ₹500.
Rate per half year = (500 × 100)/(12500) = 4%, so rate of interest = 8% per annum.
(ii) Number of half years = 3.
A = 12500(1 + (4)/(100))3 = 12500 × 1.124864 = ₹14,060.80
Final amount = ₹14,061 (to the nearest rupee).