AP DSC School Assistant Math Model Test 2

Concept:This question is based on the compound interest formula: $A = P \left(1 + \frac{r}{100}\right)^n$, where $P$ is the principal, $r$ is the annual interest rate, and $n$ is the number of years. The key idea is to relate the doubling time to the time required for the sum to become eight times by using powers of the growth factor.Explanation:Let the principal be $P$. The sum doubles in 15 years, so after 15 years the amount is $2P$. Thus:$$2P = P \left(1 + \frac{r}{100}\right)^{15} \quad \Rightarrow \quad 2 = \left(1 + \frac{r}{100}\right)^{15} \quad \text{(i)}$$Now we want the time $n$ such that the sum becomes eight times, i.e., $8P$. Write $8$ as $2^3$:$$8P = P \left(1 + \frac{r}{100}\right)^{n} \quad \Rightarrow \quad 8 = \left(1 + \frac{r}{100}\right)^{n} \quad \Rightarrow \quad 2^3 = \left(1 + \frac{r}{100}\right)^{n}$$From equation (i), $2 = \left(1 + \frac{r}{100}\right)^{15}$, so $2^3 = \left[\left(1 + \frac{r}{100}\right)^{15}\right]^3 = \left(1 + \frac{r}{100}\right)^{45}$. Therefore:$$\left(1 + \frac{r}{100}\right)^{45} = \left(1 + \frac{r}{100}\right)^{n}$$Comparing exponents gives $n = 45$ years.Answer:The sum will become 8 times in 45 years. Hence the correct option is C (45 years).