Question

How much time will be needed for $37,000 to grow to $40,871.02 if deposited at 4% compounded quarterly?Do not round until the final answer. Then round to the nearest tenth as needed.

Answer 1

We are given the following information

Deposit amount = P = $37,000

Accumulated amount (or ending balance) = A = $40,871.02

Interest rate = r = 4% = 0.04

Compounding interval = n = 4 (quarterly)

We are asked to find the number of years (t)

Recall that the compound interest formula is given by

$A=P(1+\frac{r}{n})^{n\cdot t}$

A = Accumulated amount (or ending balance)

P = Principle amount (or deposit amount)

r = Interest rate in decimal

n = Number of compounding in a year

t = Number of years

Now let us substitute the given values into the above formula and solve for (t)

$\begin{gathered} 40,871.02=37,000(1+\frac{0.04}{4})^{4\cdot t} \\ \frac{40,871.02}{37,000}=(1+0.01)^{4\cdot t} \\ \frac{40,871.02}{37,000}=(1+0.01)^{4\cdot t} \\ 1.1046=(1.01)^{4\cdot t} \end{gathered}$

Now take log on both sides

$\begin{gathered} \ln (1.1046)=\ln (1.01^{4\cdot t}) \\ \ln (1.1046)=4t\ln (1.01)^{} \\ 4t=\frac{\ln (1.1046)}{\ln (1.01)^{}} \\ 4t=9.997 \\ t=\frac{9.997}{4} \\ t=2.49 \\ t=2.5\: \text{years} \end{gathered}$

Therefore, it would take 2.5 years