Q.₹8,000 amounts to ₹9,261 in 1 year 6 months when compounded half-yearly. Find the rate of interest per annum.

Find the ratio $A/P$, express it as a power matching the number of half-year periods, and identify the per-period rate.

Periods $= 2 \times 1.5 = 3$. $A/P = 9{,}261/8{,}000 = 1.157625 = (1.05)^{3}$, so $1+r/200 = 1.05 \Rightarrow r = 10$.

Here $P = ₹8{,}000$, $A = ₹9{,}261$, $t = 1.5$ years, compounded half-yearly, so the number of half-year periods is $2t = 3$.

Step 1 — Form the ratio $A/P$.

Step 2 — Match this to $\left(1+\dfrac{r}{200}\right)^{3}$. Since $(1.05)^{3} = 1.157625$, as found in an earlier worked example, it follows that

Step 3 — Solve for $r$.

Check: $8{,}000(1.05)^3 = 8{,}000 \times 1.157625 = ₹9{,}261$, matching the given amount.

Forgetting that the exponent 3 in $(1+r/200)^3$ is the number of half-year periods, not the number of years, leads students to divide the exponent by 2 again. Also, since this chapter has not yet introduced logarithms, the per-period growth factor must be recognised directly here as a perfect cube, rather than solved by taking logs.