Problem 50 - Technical Mathematics with Calculus

Question

If an amount \(R\) is deposited once every year at a compound interest rate \(n,\) the number of years it will take to accumulate to an amount \(y\) is $$t=\frac{\log \left(\frac{n y}{R}+1\right)}{\log (1+n)} \quad \text { (years) }$$ How many years will it take an annual payment of \(\$ 1500\) to accumulate to \(\$ 13,800\) at \(9.0 \%\) per year?

Step-by-Step Solution

Answer

It will take approximately 7 years for the annual payment of \(1500 to accumulate to \)13,800 at 9.0% per year.

Step 1: Identify the given values

Identify the variables in the formula. In this scenario, the annual payment (R) is \(1500, the desired accumulated amount (y) is \)13,800, and the interest rate (n) is 9.0%, or 0.09 in its decimal form.

Step 2: Substitute the given values into the formula

Substitute the values of R, y, and n into the formula to find t: \[t=\frac{\log \left(\frac{0.09 \cdot 13800}{1500}+1\right)}{\log (1+0.09)}\]

Step 3: Simplify the formula

First calculate the fraction inside the logarithm, then the logarithm itself, and finally solve for t. \[t=\frac{\log \left(\frac{0.09 \cdot 13800}{1500}+1\right)}{\log (1+0.09)}\] \[t=\frac{\log \left(\frac{1242}{1500}+1\right)}{\log (1.09)}\] \[t=\frac{\log (1.828)}{\log (1.09)}\]

Step 4: Calculate the time (t) in years

Use a scientific calculator or a log function in a programming language to find the numerical value of t: \[t=\frac{\log (1.828)}{\log (1.09)}\approx\frac{0.2617}{0.0374}\approx 6.99\] Therefore, it will take approximately 7 years.

Key Concepts

Compound Interest Formula

Understanding the compound interest formula is crucial when dealing with investments or loans in which returns or charges are not simple but compound. Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. To put it simply, it's 'interest on interest' which can amplify the growth of your savings or the cost of your loans.