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Chapter 3: Problem 24

Suppose a bank account paying \(6 \%\) interest per year, compounded four times per year, contains \(\$ 27,707\) at the end of 20 years. What was the initial amount deposited in the bank account?

Short Answer

Expert verified
The initial amount deposited in the bank account is approximately $ \$4,513.76.

Step by step solution

01

Write down the given information

We are given: - A (final amount): $27,707 - r (annual interest rate): 6% = 0.06 (as a decimal) - n (times interest compounded per year): 4 - t (number of years): 20
02

Rewrite the formula to solve for P

We need to find the initial deposited amount (P). To do so, we'll rewrite the compound interest formula to isolate P: \(A = P(1 + r/n)^{nt}\) \(P = \frac{A}{(1 + r/n)^{nt}}\)
03

Substitute the given values into the formula

Now we will substitute the given values into our formula: \(P = \frac{\$27,707}{(1 + 0.06/4)^{4 \cdot 20}}\)
04

Evaluate the expression

Now, we will evaluate the expression: \(P = \frac{\$27,707}{(1 + 0.015)^{80}}\) \(P = \frac{\$27,707}{(1.015)^{80}}\) \(P = \frac{\$27,707}{6.1329}\) \(P \approx \$4,513.76\)
05

Give the final answer

The initial amount deposited in the bank account is approximately $ \$4,513.76.

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