Simple and Compound Interest for CSAT: Formulae and PYQs

Interest is the price paid for using someone else's money: the principal (P) is the amount borrowed or invested, the rate (R) is the percent charged per period, and the time (T or n) is the number of periods. Simple interest (SI) is calculated only on the original principal every period, so it grows linearly. Compound interest (CI) is calculated on the principal plus all interest accumulated so far, so it grows exponentially and always exceeds SI for the same rate and time beyond one period. CSAT tests both through 1 to 2 short numerical questions a year.

Simple interest: linear and predictable

SI = P x R x T / 100. Because the base never changes, each year's interest is identical, and the total is just the yearly interest multiplied by the years.

Worked example 1. What is the simple interest on Rs 10,000 at 20 percent per annum for 2 years? SI = 10000 x 20 x 2/100 = Rs 4,000. The amount (principal plus interest) is Rs 14,000.

Worked example 2. A sum at simple interest for 2 years would have fetched Rs 24 more had the rate been 1 percent higher. Find the sum. The extra interest = P x 1 x 2/100 = 24, so P = 24 x 100/2 = Rs 1,200. Rate-difference questions are always SI = P x (rate difference) x time / 100.

Worked example 3. The simple interest on a sum is one-fourth of the sum; the number of years and the rate are numerically equal. Find the years (CDS 2020). Let both be r: P x r x r/100 = P/4, so r squared = 25 and r = 5. The answer is 5 years at a 5 percent rate. Cancelling P first is the key move.

Compound interest: interest on interest

CI uses the amount formula A = P x (1 + R/100)^n, where n is the number of compounding periods, and CI = A - P. When interest compounds more than once a year, divide the rate by the frequency and multiply the periods: half-yearly compounding at 20 percent annual means 10 percent per half-year for twice as many periods.

Worked example 4. Rs 10,000 at 20 percent per annum compounded half-yearly for 2 years. Rate per half-year = 10 percent, periods = 4. A = 10000 x (1.1)^4 = 10000 x 1.4641 = Rs 14,641. CI = Rs 4,641, while SI for the same 2 years was only Rs 4,000. More frequent compounding always earns more at the same nominal rate.

Worked example 5. In how many complete years will a sum more than treble at 40 percent annual compound interest (CDS 2019)? Need (1.4)^n > 3. 1.4^2 = 1.96, 1.4^3 = 2.744, 1.4^4 = 3.8416. So 4 years. Test successive powers; CSAT options are built for this.

The CI minus SI difference: a favourite shortcut

For 2 years, the entire gap between CI and SI is the interest earned on the first year's interest: difference = P x (R/100)^2. This single formula solves a whole family of CSAT questions.

Worked example 6. The difference between CI and SI at 5 percent for 2 years is Rs 250 (CAPF 2021). Find the sum. 250 = P x (5/100)^2 = P/400, so P = Rs 1,00,000.

Worked example 7. The CI-SI difference for 2 years is Rs 60 and the SI for 2 years is Rs 1,440. Find the rate (CAPF 2017). Divide the two: difference/SI = (P x R^2/10000)/(2PR/100) = R/200. So R/200 = 60/1440 = 1/24, giving R = 200/24 = 8.33 percent. Dividing the difference by the SI cancels P elegantly.

Effective rates and the doubling family

For quick mental work, convert CI into an effective 2-year or 3-year rate. At rate R, the 2-year effective rate is (2R + R^2/100) percent: at 10 percent, two years compound to 21 percent, not 20. At 20 percent, two years give 44 percent. For doubling questions, the Rule of 72 estimates doubling time as 72/R years: at 8 percent, money doubles in about 9 years. CSAT occasionally frames 'double in n years' questions where this checks your exact calculation.

Worked example 8. What is the effective 2-year CI rate at 10 percent per annum? (1.1)^2 - 1 = 0.21 = 21 percent. SI would give 20 percent; the extra 1 percent is interest on the first year's interest.

Traps that cost marks

First, compounding frequency: 'compounded half-yearly' halves the rate and doubles the periods; using the annual rate directly is the designed error. Second, the first-year identity: CI equals SI for exactly one year, so any stem asking for the difference over 1 year has answer zero. Third, rate-time confusion: R is percent per period and T counts those same periods; a monthly rate with yearly time must be aligned first. Fourth, 'amount' versus 'interest': the amount includes the principal, CI does not; read which one the question asks for. Fifth, in difference questions, squaring the rate as a decimal versus as a percent: (5/100)^2 = 1/400, not 25.

Speed tips for the exam hall

Memorise three instant tools: SI = PRT/100, the 2-year CI-SI difference = P(R/100)^2, and the 2-year effective CI rate = 2R + R^2/100. For rate-from-difference questions, divide difference by SI to cancel P. For 'more than double/treble' questions, raise (1 + R/100) to successive powers until you cross the target. And always check the compounding frequency before writing any number down.

Key Terms

• Principal (P) is the original sum of money borrowed or invested, before any interest is added. Both SI and CI are ultimately computed from it.
• Rate of interest (R) is the percent charged per period, usually per annum. In CI with half-yearly compounding, the per-period rate is half the annual rate.
• Time (T or n) is the number of interest periods. It must use the same period as the rate: annual rate pairs with years, half-yearly rate with half-years.
• Amount (A) is the principal plus all interest at the end of the term: A = P + SI, or A = P(1 + R/100)^n for compound interest.
• Simple interest (SI) is interest calculated only on the original principal each period, so it grows linearly: SI = P x R x T / 100.
• Compound interest (CI) is interest calculated on the principal plus previously accumulated interest, so it grows exponentially: CI = P(1 + R/100)^n - P.
• Compounding frequency is how often interest is added to the principal within a year: annually, half-yearly, quarterly or monthly. Higher frequency raises the effective return.
• Effective rate is the actual percent growth over a multi-period term under compounding, for example 21 percent over two years at a 10 percent annual CI rate.