Question

Thanh invested $\$12,500$ in an account that pays $575$ simple interest. How much more could she have earned over a $13-\mathrm{year}$ period the interest had compounded annually?

Answer 1

Finding how much Thanh could have earned if the interest was compounded annually:

Step-1:Calculating the rate of interest:

Given,

$$\begin{gathered} \mathrm{P}=12500 \\ \mathrm{SI}=575 \\ \mathrm{T}=1\mathrm{year} \end{gathered}$$

therefore,

$\begin{array}{rcl}\mathrm{SI}& =& \frac{\mathrm{PRT}}{100}\\ & \Rightarrow & 575=\frac{12500\times \mathrm{R}}{100}\\ & \Rightarrow & \mathrm{R}=4.6\%\end{array}$

Step-2: Calculating the compound Interest:

Given,

$$\begin{gathered} \mathrm{P}=12500 \\ \mathrm{R}=4\% \\ \mathrm{T}=13\mathrm{years} \end{gathered}$$

Therefore,

$\begin{array}{rcl}\mathrm{A}& =& \mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n}}\\ & \Rightarrow & \mathrm{A}=12500{\left(1+\frac{46}{1000}\right)}^{13}\\ & \Rightarrow & \mathrm{A}=12500{\left(1.046\right)}^{13}\\ & =& 12500\times 1.794\\ & =& \$22430\end{array}$

So, the compound interest will be,

$\begin{array}{rcl}\mathrm{CI}& =& \mathrm{A}-\mathrm{P}\\ & =& 22430-12500\\ & =& \$9930\end{array}$

Step-3: Calculating the difference:

Hence, the difference between compound interest earned and simple interest earned will be,

$$\begin{gathered} =9930-13\times 575 \\ =\$2455 \end{gathered}$$

Therefore, she could have earned $\$2455$ more had the interest been compounded annually.