Simple and compound interest for LIC AAO

Interest questions are the arithmetic closest to the work of an insurer: money is invested, it earns a return, and the return itself earns a return. For the exam, the whole chapter rests on one difference. Simple interest is paid on the original sum only. Compound interest is paid on the sum plus the interest already earned. Every formula and shortcut below follows from that one line.

The formulas and where they come from

Simple interest (SI)

SI = P × R × T ÷ 100, where P is the principal, R the rate per cent a year and T the time in years.

The interest is the same every year, R% of P, so over T years it is T times that.

Compound interest (CI)

Amount A = P × (1 + R/100)ᵀ, and CI = A − P.

Each year the amount is multiplied by (1 + R/100), because the interest for the year is added to the balance and earns interest from then on.

Year-by-year view

For ₹10,000 at 10% a year:

In year 1, SI and CI are equal. From year 2, CI pulls ahead because of interest on interest. This table explains the two difference formulas below.

The CI–SI difference

• Two years: CI − SI = P × (R/100)². The only extra is interest on the first year's interest: (P × R/100) × R/100.
• Three years: CI − SI = P × (R/100)² × (3 + R/100). For the table above: 10,000 × 0.01 × 3.1 = 310.

Compounding more than once a year

Half-yearly: halve the rate and double the number of periods. Quarterly: divide the rate by four and multiply the periods by four. So 10% a year compounded half-yearly is 5% for each half-year, and in a year it gives 1.05² − 1 = 10.25%, the effective annual rate.

Doubling

At SI, a sum doubles when the interest equals the principal, so R × T = 100. At 12.5%, that is 8 years. At CI there is no exact mental formula; the "rule of 72" (years ≈ 72 ÷ rate) is only an approximation, useful for eliminating options.

Useful powers

1.05³ = 9,261/8,000, which is why ₹8,000 at 5% grows to ₹9,261 in three years.

Worked examples

Example 1. Find the SI on ₹15,000 at 6% a year for 4 years.
15,000 × 6 × 4 ÷ 100 = ₹3,600.

Example 2. Find the CI on ₹8,000 at 5% a year for 2 years.
8,000 × 1.1025 = 8,820. CI = ₹820. Check by years: 400 in year 1, then 5% of 8,400 = 420 in year 2.

Example 3. Find CI − SI on ₹25,000 at 4% for 2 years.
25,000 × (0.04)² = 25,000 × 0.0016 = ₹40.

Example 4. A sum becomes ₹9,261 after 3 years at 5% compound interest. Find the principal.
P = 9,261 ÷ 1.157625 = ₹8,000.

Example 5. Find CI − SI on ₹10,000 at 10% for 3 years.
10,000 × 0.01 × 3.1 = ₹310, matching the year-by-year table.

Example 6. A sum doubles in 8 years at simple interest. Find the rate, and the time it takes to become three times itself.
R × 8 = 100, so R = 12.5%. To triple, the interest must be 2P: R × T = 200, so T = 16 years.

Example 7. Find the effective annual rate for 10% compounded half-yearly.
1.05 × 1.05 = 1.1025. Effective rate = 10.25%.

Example 8. On a sum, the SI for 2 years is ₹2,000 and the CI for 2 years at the same rate is ₹2,100. Find the rate and the sum.
SI for one year = 1,000. The extra ₹100 in CI is interest on the first year's ₹1,000.
So R = 100 ÷ 1,000 × 100 = 10%, and P = 1,000 ÷ 0.10 = ₹10,000.
Check: 10,000 × 1.21 − 10,000 = 2,100.

Common mistakes

• Giving the amount when the question asks for the interest, or the reverse. Read the last line.
• Using the CI–SI two-year formula for three years.
• Treating the rule of 72 as exact.
• Dividing an amount by (1 + R/100) once when it has been compounded for several years.

Practice set

1. SI on ₹12,000 at 7.5% for 2 years? ₹1,800. 12,000 × 7.5 × 2 ÷ 100.
2. CI on ₹10,000 at 20% for 2 years? ₹4,400. 10,000 × 0.44.
3. CI on ₹20,000 at 10% a year, compounded half-yearly, for 1 year? ₹2,050. 20,000 × 1.05² = 22,050.
4. CI − SI on ₹8,000 at 10% for 3 years? ₹248. 8,000 × 0.01 × 3.1. Check: CI = 2,648 and SI = 2,400.
5. At what simple interest rate does ₹6,000 become ₹7,920 in 4 years? 8%. Interest 1,920 ÷ (6,000 × 4) × 100.
6. At compound interest, a sum becomes ₹12,100 in 2 years and ₹13,310 in 3 years. Find the rate and the sum. 10% and ₹10,000. 1,210 ÷ 12,100 = 10%; 12,100 ÷ 1.21 = 10,000.
7. CI on ₹5,000 at 8% a year, compounded quarterly, for 6 months? ₹202. 2% per quarter for 2 quarters: 5,000 × 1.0404 = 5,202.
8. The CI–SI difference on a sum for 2 years at 5% is ₹15. Find the sum. ₹6,000. 15 ÷ 0.0025.

What to do next

• Learn the powers table so 1.21, 1.331 and 1.1025 are instant.
• Solve five CI–SI difference questions using the formula, then check one by the year-by-year method.
• Revise successive change in percentage for LIC AAO, since CI is successive change with equal steps.
• Practise interest inside data sufficiency questions next.