(a) \(\$6000\) is invested at \(r\%\) per year compound interest. After 5 years the investment is worth \(\$7500\). Calculate the value of \(r\), correct to...

Part (a) reverses the usual compound interest calculation: the start value, the end value and the number of years are known, and the rate is the unknown. That requires a root, not a division.

1. (a) Write the standard compound interest equation with \(r\) as the unknown: \[6000\times\left(1+\frac{r}{100}\right)^{5}=7500\] [M1] Divide by 6000 to isolate the multiplier: \(\left(1+\dfrac{r}{100}\right)^{5}=\dfrac{7500}{6000}=1.25\). Undo the power of 5 by taking the fifth root, which on a calculator is the power \(0.2\): \[1+\frac{r}{100}=1.25^{0.2}=1.045639\ldots\] [M1] So \(\dfrac{r}{100}=0.045639\ldots\) and \(r=4.56\) correct to 2 decimal places. [A1]
2. (b) Here the rate is known, so apply the multiplier \(1.045\) five times: \[6000\times 1.045^{5}=6000\times 1.246182\ldots=7477.09\ldots\] [M1] giving \(\$7477.09\). [A1]

A tempting but wrong method in part (a) is \(\dfrac{7500-6000}{6000}\times 100\div 5=5\%\). That is the simple interest rate; it is too high because it ignores the interest that itself earns interest. Part (b) confirms the reasoning: 4.5% compound over five years gives \(\$7477.09\), slightly less than \(\$7500\), so the rate needed must be slightly above 4.5%, exactly as part (a) found.