Finding an unknown principal, rate or time

Learning outcome

Find an unknown principal, annual simple-interest rate or time by reversing the correct relationship.

Concepts and assumptions

Simple interest satisfies I = P × r × t ÷ 100, and amount A = P + I. Here P is principal, r is the numerical annual percentage rate, and t is time in years.

A missing factor is found by undoing multiplication:

P = 100 × I ÷ (r × t).

r = 100 × I ÷ (P × t), expressed as a percentage per year.

t = 100 × I ÷ (P × r), expressed in years.

These are rearrangements, not separate rules to memorise without understanding. Substitute the recovered value into the original interest equation to check it.

When both principal and amount are known, first obtain I = A - P. Do not put the amount into the interest position. When principal is unknown but amount is given, use A = P × (1 + r × t/100), so P = A ÷ (1 + r × t/100).

Assume no extra deposits, withdrawals, interim payments or fees, and constant annual simple interest on the original principal. Convert months using 12 months per year; for stated day-based calculations use a 365-day year. Keep intermediate values exact and round money only at the end to ₹0.01, rounding half a paise upward.

The denominator used in a reverse calculation must be nonzero. In the interest-based formulas, finding P requires positive rate and time; finding r requires positive principal and time; finding t requires positive principal and rate. Zero interest can arise from zero rate or zero time, so it does not automatically identify a unique missing quantity.

Worked examples

Example 1 — Unknown principal. Simple interest is ₹1,296 at 9% for 2 years. P = 100 × 1296 ÷ (9 × 2) = 129600 ÷ 18 = ₹7,200. Check: 7200 × 9 × 2 ÷ 100 = ₹1,296.

Example 2 — Unknown rate. ₹9,600 becomes ₹10,608 in 15 months. Interest = 10608 - 9600 = ₹1,008; time = 15/12 = 1.25 years. Therefore r = 100 × 1008 ÷ (9600 × 1.25) = 100800 ÷ 12000 = 8.4% per year.

Example 3 — Unknown time. ₹7,500 becomes ₹8,400 at 8% simple interest. I = 8400 - 7500 = ₹900. Time = 100 × 900 ÷ (7500 × 8) = 90000 ÷ 60000 = 1.5 years, or 18 months. Check: annual interest is ₹600, and 1.5 × 600 = ₹900.

Common mistakes

Do not confuse amount with interest or return a time in years labelled as months. A fractional principal is possible; a negative time in a positive-interest problem signals an error.

Practice questions

1. Interest is ₹1,320 at 11% simple interest for 2 years. Find the principal.
2. ₹14,400 earns ₹1,296 simple interest in 9 months. Find the annual rate.
3. ₹15,625 earns ₹2,500 simple interest at 8%. Find the time.
4. An amount of ₹17,025 is reached after 18 months at 9% simple interest. Find the principal.

Worked answers

1. P = 1320 × 100 ÷ (11 × 2) = ₹6,000. Checking gives 6000 × 11 × 2 ÷ 100 = ₹1,320.
2. Time = 9/12 = 0.75 year. r = 1296 × 100 ÷ (14400 × 0.75) = 129600 ÷ 10800 = 12% per year.
3. t = 2500 × 100 ÷ (15625 × 8) = 250000 ÷ 125000 = 2 years.
4. Time = 18/12 = 1.5 years. Amount multiplier = 1 + 9 × 1.5/100 = 1.135. P = 17025 ÷ 1.135 = ₹15,000; its interest is ₹2,025.