[Math] Finding the Compound Interest on 7500 Dollars at 4% per annum for 2 Years

problem solvingword problem

Find compound interest on $\$7500$ at $4\%$ per annum for $2$ years, compounded annually.

The choices are as follow: $\$512$, $\$552$, $\$612$, $\$622$.

I tried to solve this problem by:

C.I. $= 7,500(.02) = 150$

C.I. $= 7650 (.02) = 153$

So, $150 + 153 = 303$.

The answer key given to us states that the answer is $\$612$. Here is its solution:

Amount $=7500(1 + 4/100)2$

i.e. $= 7500(26/25)(26/25) \rightarrow 8112$

C.I. $= 8112 – 7500$

C.I. $= 612$

Please help me reconcile this solution with the one I have. Where Were I mistaken?

PS I am a college student having troubles with word problems.

Best Answer

You wrote

C.I. = 7,500(.02) = 150

C.I. = 7650 (.02) = 153

In particular you used (.02) there. The problem stated however "4% per annum", so you should have used (.04) instead.

7500(.04) = 300

7800(.04) = 312

300+312 = 612

Your answer and method would have been correct if the problem read "Find compound interest on 7500 dollars at 2% per annum for 2 years, compounded annually." or if it read as "Find compound interest on 7500 dollars at 4% per annum for 1 year, compounded semi-annually."

Edit: In response to your question on how to interpret the answer key's solution.

[7500(1 + 4/100)2], although poorly typed is meant to be $7500\cdot(1+\frac{4}{100})^2$

Interest rate formula: $$F = P(1+\frac{r}{n})^{ny}$$ where $F$ = future value, $P$ = initial value (principle), $r$ = APR (interest rate per year), $n$ = number of times per year it is compounded, $y$ = number of years.

In this case, $P=7500$, $r=.04$, $n=1$, and $y=2$, and $F$ is unknown. So, by the formula above, $F=7500(1+.04)^2$. With a bit of arithmetical simplifications, $=7500(1.04)^2 = 7500(\frac{26}{25}\cdot\frac{26}{25})$

Often times you will see the formula rewritten as $F = P(1+i)^t$ where $i$= interest rate per payment period = $\frac{r}{n}$, and $t$ = number of payment periods = $n\cdot y$.

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