Use the compound interest formula: \(\displaystyle{A}={P}{\left({1}+\frac{{r}}{{n}}\right)}^{{n}}{t}\)

where A is the final value, P is the present value, r is the rate (in decimal form), and n is compounding times per year, and t is the time in years.

Substitute A=52680, P=25000, n=4 for quarterly, and t=15 then solve for r: \(\displaystyle{52680}={25000}{\left({1}+\frac{{r}}{{4}}\right)}^{{4}}{\left({15}\right)}\)

\(\displaystyle{52680}={25000}{\left({1}+\frac{{r}}{{4}}\right)}^{{60}}\)

Divide both sides by 25000: \(\displaystyle{2.1072}={\left({1}+\frac{{r}}{{4}}\right)}^{{60}}\)

Raise both sides by \(\displaystyle\frac{{1}}{{60}}\): \(\displaystyle{2.1072}={1}+\frac{{r}}{{4}}\)

Subtract 1 from both sides: \(\displaystyle{\left(\frac{{2.1072}^{{1}}}{{60}}\right)}-{1}=\frac{{r}}{{4}}\)

Multiply both sides by 4: \(4((2.1072^1/60)-1)=r r=0.05->5%\)