Simple interest

1. Meaning, components and the basic equation

Interest is the payment made for using borrowed money, or the return received for lending or investing it. Under simple interest, this payment is calculated on the original principal throughout the specified period. Interest already earned does not itself earn interest. This is the defining difference from compound interest, where accumulated interest becomes part of the base for subsequent calculations.

Let P denote principal, R the rate expressed as a percentage per year, T the time in years, SI the interest, and A the final amount. Then SI = PRT/100 and A = P + SI. For example, ₹8,000 lent at 7.5% per annum for two years earns ₹1,200 as simple interest. The amount repayable is ₹9,200, not ₹1,200.

The same relationship can be rearranged according to the unknown: P = 100SI/(RT), R = 100SI/(PT), and T = 100SI/(PR). These are not separate concepts to memorise. Start with the main equation, identify the missing quantity and isolate it. If the question supplies the amount, first subtract the principal to obtain interest.

Simple interest describes a calculation method, not every feature of a financial product. Actual contracts may specify fees, repayment schedules, day-count rules or compounding. In CSAT, use the stated assumptions rather than importing banking practices that the question does not mention.

• Principal: the original sum lent or invested.
• Interest: the additional sum earned or payable.
• Amount: principal plus accumulated interest.
• Per annum means per year; it does not by itself mean annual compounding.

2. Time units and proportional reasoning

The rate and time must use compatible units. With an annual rate, m months becomes m/12 years. Thus, ₹12,000 at 8% per annum for nine months earns 12,000 × 8 × 9/(100 × 12) = ₹720. Convert 18 months to 1.5 years, not 1.8 years. Likewise, two years and three months equals 2.25 years.

For a duration expressed in days, use the year length specified in the question. Under a 365-day convention, time is d/365 years; under an explicitly stated 360-day convention, it is d/360. If actual calendar dates are given, check whether a leap year or an inclusive-day instruction affects the calculation. Do not silently substitute a different convention.

Because SI is directly proportional to each of P, R and T when the other two remain fixed, many questions need only ratios. Doubling the principal doubles the interest. Reducing time to one-third reduces interest to one-third. If both principal and time double while the rate is halved, interest becomes twice its original value.

For two investments, SI₁/SI₂ = (P₁R₁T₁)/(P₂R₂T₂). Cancel common terms before calculating. If two sums earn equal interest over the same period at 6% and 9%, their principals are in the inverse ratio 9:6, or 3:2. This approach is faster and less error-prone than calculating each principal separately.

• At a simple rate of 2% per month, interest over 12 months is 24% of principal.
• A rise from 8% to 10% is two percentage points, but a 25% increase in the rate.
• Use fractions such as 12.5% = 1/8 and 6.25% = 1/16 to simplify arithmetic.

3. Amount differences, doubling and divided investments

When amounts at two dates are given, subtract them. The principal cancels, leaving the interest earned during the intervening period. Suppose a sum amounts to ₹6,200 after three years and ₹7,000 after five years at an unchanged simple rate. The difference of ₹800 is interest for two years, so annual interest is ₹400. Principal is therefore ₹6,200 − 3 × ₹400 = ₹5,000, and the rate is 8% per annum.

If a sum becomes k times its principal, the interest earned is (k − 1)P. Substitution gives T = 100(k − 1)/R. A sum doubles when interest equals the principal, so doubling time is 100/R years. If it doubles in eight years, the rate is 12.5%; it will triple in 16 years because tripling requires interest equal to twice the principal.

Divided-investment questions combine a total-principal equation with an interest equation. Suppose ₹10,000 is split between 6% and 9% for one year and earns ₹780. Let x be invested at 6%. Then 0.06x + 0.09(10,000 − x) = 780, giving x = ₹4,000. The remaining ₹6,000 earns 9%.

Alternatively, the overall rate is 7.8%. For equal investment periods, alligation gives the principal ratio as (9 − 7.8):(7.8 − 6), or 2:3. This shortcut uses a principal-weighted average rate. If the investment periods differ, incorporate time into each return before comparing them.

• Annual interest = difference between amounts ÷ difference between their times.
• Principal = known amount − interest earned up to that date.
• At a fixed simple rate, doubling and tripling times are in the ratio 1:2.

4. Changing rates, repayments and compound-interest comparisons

If the rate changes but the principal remains unchanged, calculate interest separately for each interval and add it. On ₹5,000 at 6% for two years and 8% for the next three years, total interest is 5,000 × (6 × 2 + 8 × 3)/100 = ₹1,800. The final amount is ₹6,800. Do not apply the final rate retrospectively to the entire period.

For unchanged principal, the effective annual simple rate across several intervals is the time-weighted average: Σ(RᵢTᵢ)/ΣTᵢ. In the preceding example it is 36/5 = 7.2%. An ordinary arithmetic average of 6% and 8% would be wrong because the rates apply for unequal durations.

Repayment questions require special care. A payment may discharge accrued interest first, reduce principal, or follow another rule stated in the question. Future interest falls only to the extent that the interest-bearing principal is reduced. Draw a timeline showing each payment date and identify the outstanding principal for every interval. Do not subtract a payment from the original principal unless the stated arrangement permits this.

For the same principal and annual rate, one year's simple interest equals one year's compound interest with annual compounding. Over two years, the excess of compound interest is P(R/100)². Thus, on ₹10,000 at 10%, two-year simple interest is ₹2,000 and compound interest is ₹2,100. The ₹100 difference represents interest earned on the first year's interest.

• Split a problem wherever the rate or interest-bearing principal changes.
• The two-year SI–CI difference shortcut assumes annual compounding.
• A flat rate on the original loan and a rate on a declining outstanding balance are not directly comparable.

5. CSAT problem-solving and error control

Begin by marking what the question actually asks: interest, amount, principal, rate or time. Next, translate its wording into an equation or ratio. Estimate the likely answer before carrying out arithmetic. At 10% simple interest for three years, the amount must be 130% of principal; an option near 300% can be rejected immediately.

In data-sufficiency questions, ask whether the information determines a unique requested quantity. Knowing interest and time identifies the product PR, but not P and R separately. Two amounts at distinct times determine annual interest and principal when the same simple-interest arrangement applies throughout. Always test sufficiency using the stated conditions rather than an assumed rate.

Common traps include treating amount as interest, interpreting tripling as earning three times the principal, and calculating interest on previously earned simple interest. A final dimensional check is useful: rate should be a percentage per stated period, time should match that period, and interest should be expressed in money.

For timed practice, prefer cancellation, ratios and amount differences to lengthy decimal multiplication. If the repayment convention or day-count basis is unclear, re-read the wording before proceeding. Speed comes from selecting the right model, not from applying a memorised formula without checking its assumptions.

• Write percentages consistently: use either R/100 or its decimal equivalent, not both.
• Keep intermediate values exact where possible.
• Check the chosen option by substitution into the original conditions.

Previous year questions

No UPSC question has been asked directly on this micro-topic yet. Use the practice questions below.

Practice questions