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Helga [31]
1 year ago
15

Altred is saving up money for a down payment on a house. He currently has $4739, but knows he can get a loan at a lower interest

rate if he can put down $5336. If heinvests the $4739 in an account that earns 5.1 % annually, compounded monthly, how long will it take Alfred to accumulate the $5336? Round your answer to two decimal places, if necessary.
Mathematics
1 answer:
Stella [2.4K]1 year ago
5 0

Given the word problem, we can deduce the following information:

Principal amount = $4739

Future amount = $5336

Interest = 5.1 % =0.051

To determine the time to accumulate the $5336, we use the compound interest formula:

A=P(1+\frac{r}{n})^{nt}

A=Future amount=$5336

P=Present amount=$4739

r=interest rate = 0.051

n=number of compounding periods =12

t=time in years

We also note that the number of compounding periods must be 12 since the investment is compounded monthly.

We plug in what we know:

\begin{gathered} A=P(1+\frac{r}{n})^{nt} \\ 5336=4739(1+\frac{0.051}{12})^{12t} \\ Simplify\text{ and rearrange} \\ 5336=4739(\frac{12.051}{12})^{12t} \\ 12t=\frac{\ln\frac{5336}{4739}}{\ln\frac{12.051}{12}} \\ t=\frac{\operatorname{\ln}\frac{5,336}{4,739}}{12\operatorname{\ln}\frac{12.051}{12}} \\ Calculate \\ t=2.33 \end{gathered}

Therefore, the answer is 2.33 years.

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