Chapter-2 (Compound Interest) Notes

Compound Interest: Notes

Mathematics • Grade 10 • Complete notes, formulas and solved examples

Complete Notes

2.0 Review: Simple Interest

Money paid for using someone else’s money, calculated on the same principal every year, is simple interest. The total of principal and interest is the amount.

SI = P × T × R / 100    |    Amount = P + SI Example: Rs. 10,00,000 at 8% p.a. for 2 years → SI = 10,00,000 × 2 × 8 / 100 = Rs. 1,60,000, so amount = Rs. 11,60,000. For 5 years the SI would be Rs. 4,00,000.

2.1 What is Compound Interest?

If interest is added to the principal after each period (year, half-year or quarter) and the next interest is calculated on this new bigger principal, the interest is called compound interest (CI). The principal plus compound interest is the compound amount (CA).

Activity 1 — Teacher’s loan

Rs. 5,00,000 for 2 years at 12% p.a., interest not paid after the first year.

Year	Principal	Interest (12%)
1st	Rs. 5,00,000	Rs. 60,000
2nd	5,00,000 + 60,000 = Rs. 5,60,000	Rs. 67,200

Total interest = 60,000 + 67,200 = Rs. 1,27,200. The second-year interest is larger because it is charged on the first-year interest as well.

Activity 2 — Simple vs compound interest

Bishal (simple interest) and Badri (compound interest) each borrow Rs. 30,000 for 3 years at 10%.

Year	Bishal: principal → interest	Badri: principal → interest
1	30,000 → 3,000	30,000 → 3,000
2	30,000 → 3,000	33,000 → 3,300
3	30,000 → 3,000	36,300 → 3,630
Total	Rs. 9,000	Rs. 9,930

Conclusion: in SI the principal is the same every year; in CI the principal changes every year (it equals the previous year’s amount). For the same P, R and T, CI is more than SI (here by Rs. 930). For the first year only, CI = SI.

Activity 3 — Deriving the formula

Deposit P at R% per annum:

A₁ = P + PR/100 = P(1 + R/100) A₂ = P(1 + R/100) + P(1 + R/100)·R/100 = P(1 + R/100)² A₃ = P(1 + R/100)³  ⇒  A_T = P(1 + R/100)^T

Each year the principal is multiplied by the same factor (1 + R/100).

Yearly compound interest — key formulas

1. Compound amount: CA = P(1 + R/100)^T
2. Compound interest: CI = CA − P = P[(1 + R/100)^T − 1]
3. Principal of any year = amount at the end of the previous year.

Activity 4 — Other situations

Situation	Compound amount	Compound interest
(a) Different rates R₁, R₂, R₃ in successive years	P(1 + R₁/100)(1 + R₂/100)(1 + R₃/100)	CA − P
(b) T years and M months	P(1 + R/100)T(1 + MR/1200)	CA − P
(c) Half-yearly: rate R/2, time 2T	P(1 + R/200)2T	P[(1 + R/200)2T − 1]
(d) Quarterly (terminal): rate R/4, time 4T	P(1 + R/400)4T	P[(1 + R/400)4T − 1]

In (b), the extra M months are charged as simple interest on the amount at the end of T years, which gives the factor (1 + MR/1200).

Activity 5 — Which compounding is best?

Rs. 1,00,000 for 1 year at 15% p.a.

Compounded	Working	CI
Yearly	1,00,000[(1.15) − 1]	Rs. 15,000
Half-yearly	1,00,000[(1.075)² − 1]	Rs. 15,562.50
Quarterly	1,00,000[(1.0375)⁴ − 1]	Rs. 15,865.04

Conclusion: for the same P, R and T → quarterly CI > half-yearly CI > yearly CI. The more often interest is added, the more interest is earned (here quarterly gives Rs. 865.04 more than yearly and Rs. 302.54 more than half-yearly).

Quick results worth remembering

• For 1 year: CI (yearly) = SI.
• For 2 years (yearly): CI − SI = P(R/100)².
• If CA₁ and CA₂ are the amounts after n and n + 1 years: 1 + R/100 = CA₂ / CA₁, then P = CA₁ ÷ (1 + R/100)ⁿ.
• If CI for the 1st year is I₁ and for the first two years is I₂: rate R = (I₂ − I₁)/I₁ × 100, and P = I₁ × 100/R.
• To find time: write CA/P as a power of (1 + R/100), then compare the powers.
• When tax is deducted: tax-free interest = interest × (100 − tax%)/100, and the deposit grows by this amount only.

Solved Examples

Ex 1 Find the CI and CA of Rs. 2,000 at 12% p.a. for 2 years without using the formula.

I₁ = 2000 × 12/100 = Rs. 240. P₂ = 2000 + 240 = Rs. 2,240.

I₂ = 2240 × 12/100 = Rs. 268.80.

CI = 240 + 268.80 = Rs. 508.80   CA = 2000 + 508.80 = Rs. 2,508.80

Ex 2 Find the CI and CA of Rs. 25,000 for 3 years at 12% yearly compound interest.

CI = 25,000[(1.12)³ − 1] = 25,000 × 0.404928 = Rs. 10,123.20

CA = 25,000 + 10,123.20 = Rs. 35,123.20

Ex 3 A man borrowed Rs. 32,000 at 12.5% simple interest and lent it at the same rate of compound interest for 3 years. How much more does he receive?

SI = 32,000 × 3 × 12.5/100 = Rs. 12,000.

CI = 32,000[(1.125)³ − 1] = 32,000 × 0.423828125 = Rs. 13,562.50.

Gain = 13,562.50 − 12,000 = Rs. 1,562.50

Ex 4 Sameer invests Rs. 5,000 at 8% for 2 years: half-yearly CI or yearly CI — which is better?

Half-yearly: 5000[(1.04)⁴ − 1] = 5000 × 0.16985856 = Rs. 849.29.

Yearly: 5000[(1.08)² − 1] = 5000 × 0.1664 = Rs. 832.

Difference = Rs. 17.29. Choose half-yearly because it gives Rs. 17.29 more.

Ex 5 Rs. 10,000 is invested for 2 years at yearly CI. The amount after 1 year is Rs. 11,200. Find (a) the rate, (b) the amount after 2 years.

(a) 11,200 = 10,000(1 + R/100) ⇒ 1 + R/100 = 1.12 ⇒ R = 12%

(b) CA = 10,000(1.12)² = Rs. 12,544

Ex 6 Rs. 2,00,000 is invested for 3 years at 8%, 10% and 12% in the 1st, 2nd and 3rd years. Find CA and CI.

CA = 2,00,000 × 1.08 × 1.10 × 1.12 = Rs. 2,66,112

CI = 2,66,112 − 2,00,000 = Rs. 66,112

Ex 7 A sum amounts to Rs. 14,520 in 2 years and Rs. 15,972 in 3 years at yearly CI. Find the rate and the principal.

Divide the two amounts: P(1 + R/100)³ ÷ P(1 + R/100)² = 15,972/14,520 = 1.10.

So 1 + R/100 = 1.10 ⇒ R = 10%

14,520 = P(1.1)² = 1.21P ⇒ P = Rs. 12,000

Ex 8 Rs. 2,00,000 is deposited for 2 years at 10% p.a. half-yearly CI, with 5% tax on the interest. After one year the bank changes to quarterly compounding at the same rate. Find (a) first-year interest after tax, (b) second-year interest after tax, (c) their difference, (d) the percentage difference.

(a) CI₁ = 2,00,000[(1.05)² − 1] = 2,00,000 × 0.1025 = Rs. 20,500. Tax = 1,025. Interest = Rs. 19,475

(b) Principal for year 2 = 2,00,000 + 19,475 = Rs. 2,19,475. Quarterly: CI₂ = 2,19,475[(1.025)⁴ − 1] = 2,19,475 × 0.10381289 = Rs. 22,784.33. Tax = 1,139.21. Interest = Rs. 21,645.12

(c) Difference = 21,645.12 − 19,475 = Rs. 2,170.12

(d) 2,170.12 ÷ 19,475 × 100 = 11.14%

Ex 9 A bank lends Rs. 52,500 to two people at 10% yearly CI. Babulal’s amount after 2 years equals Jibanlal’s amount after 3 years. How much did each borrow?

Let Babulal borrow Rs. x, so Jibanlal borrows Rs. (52,500 − x).

x(1.1)² = (52,500 − x)(1.1)³ ⇒ 1.21x = 69,877.5 − 1.331x ⇒ 2.541x = 69,877.5

Babulal = Rs. 27,500 Jibanlal = Rs. 25,000

Compound Interest • Grade 10 Mathematics • Notes