Independent solution

How to solve this Compound and Simple Interest Comparison question

Setup

Setup

Write i for the annual nominal rate convertible semiannually. Each half-year credits i/2, and the comparison involves fifteen completed half-year accumulation periods.

100(1+i/2)15(i/2)=200(i/2)100(1+i/2)^{15}(i/2)=200(i/2)

Model

Model

After canceling the common half-year interest-rate factor, equality of the two interest amounts leaves the accumulation equation shown. This isolates the effect of compound growth rather than treating i as an annual effective rate.

(1+i/2)15=2(1+i/2)^{15}=2

Compute

Compute

Taking the fifteenth root gives 1 + i/2 = 2 to the power 1/15. Hence i = 0.09459, or 9.46% as an annual nominal rate convertible semiannually.

i=2(21/15−1)=0.09459i=2(2^{1/15}-1)=0.09459

Answer

Answer

The calculation gives 9.46% for compound and simple interest comparison, matching published choice C.

i=9.46%(C)\boxed{i=9.46\%\quad\text{(C)}}