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+(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+(0.04)/(4))^(4\cdot t) \\ (40,871.02)/(37,000)=(1+0.01)^(4\cdot t) \\ (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=(9.997)/(4) \\ t=2.49 \\ t=2.5\: \text{years} \end{gathered}`

Therefore, it would take 2.5 years