KCSE 2025 Maths P2 Q12 — Compound Interest and Investment Duration

KCSE 2025 Form 4 Commercial Arithmetic

Published

The Question

“An investor deposited Ksh 20,000 in an account that paid compound interest at a rate of 2.5% every 6 months. At the end of the investment period, the interest earned was Ksh 5,600. Determine the duration in years of the investment period.”

1

Find the final amount

The total amount at the end is the original deposit plus the interest it earned. Add the principal of 20,000 to the interest of 5,600 to get the maturity amount that the compound interest formula must produce.

A=20000+5600=25600A = 20000 + 5600 = 25600
2

Set up the compound interest equation

Use the compound interest formula, where A is the final amount, P is the principal, R is the rate per period and n is the number of periods. The rate is 2.5% for every 6 months, so each period is a half year and n counts the number of half years. Substitute A = 25600, P = 20000 and R = 2.5.

A=P(1+R100)nA = P\left(1 + \frac{R}{100}\right)^{n}
25600=20000(1+2.5100)n25600 = 20000\left(1 + \frac{2.5}{100}\right)^{n}
25600=20000(1.025)n25600 = 20000(1.025)^{n}
3

Isolate the power term

Divide both sides by the principal 20,000 so that only the growth factor raised to the power n remains. This gives a clean equation with the unknown appearing only in the exponent.

2560020000=(1.025)n\frac{25600}{20000} = (1.025)^{n}
1.28=(1.025)n1.28 = (1.025)^{n}
4

Take logarithms to find n

Because the unknown n is in the exponent, take the logarithm of both sides. The power rule lets you bring n down as a multiplier in front of the log. Then divide to make n the subject and evaluate the logs.

log⁡(1.28)=nlog⁡(1.025)\log(1.28) = n\log(1.025)
n=log⁡(1.28)log⁡(1.025)=0.10720.0107≈10n = \frac{\log(1.28)}{\log(1.025)} = \frac{0.1072}{0.0107} \approx 10
5

Convert periods to years

There are 10 compounding periods, and each period is half a year. Multiply the number of half years by one half to convert the duration into years.

n=10 half-yearsn = 10 \text{ half-years}
10×12=5 years10 \times \tfrac{1}{2} = 5 \text{ years}

Final Result

The investment ran for 5 years.

Why this method works

Compound interest grows the balance by a fixed factor each period, so after n periods the amount is the principal multiplied by that factor n times, giving the power form of the formula. Dividing by the principal isolates the growth, and because the unknown lives in the exponent, logarithms are the natural tool: they turn a power into a product using the rule log(x^n) = n log(x), letting you solve for n directly. The final conversion is needed because n counts half-year periods, not years.

✓

With n = 10: 20000 × 1.025 to the power 10 is about 20000 × 1.28 = 25600, which is exactly the principal plus 5600 of interest, so the duration of 5 years is correct.