Chapter 0.5, Problem 47E

To determine

To find: The time doubling the money value when interest is compounded annually.

Answer to Problem 47E

The amount $ 500 will gets double in 14 years and $342\text{days}$ .

Explanation of Solution

Given data:

The principal amount is $\$500$ and interest rate $4.75\%\text{per}\text{year}$ .

Method/Formula used:

Suppose a principal amount P is invested in a scheme at the interest rate $r\%\text{per}\text{year}$ , then the amount A received after t years when interest is compounded annually is given by

  $A=P{\left(\frac{100+r}{100}\right)}^t$ … (1)

Calculation:

The principal amount P is $\$500$ and interest rate $4.75\%\text{per}\text{year}$ .

Let in time t years the amount P gets double, that is the principal amount A equals 2P .

Substitute 2P for A and $4.75$ for r in the formula (1),

  $\begin{array}{c}2P=P{\left(\frac{100+4.75}{100}\right)}^t\\ \frac{2P}{P}={\left(\frac{100+4.75}{100}\right)}^t\\ 2={\left(\frac{100+4.75}{100}\right)}^t\\ 2={\left(\frac{104.75}{100}\right)}^t\end{array}$

Taking log (to the base 10) of both sides

  $\begin{array}{c}\log 2=\log {\left(\frac{104.75}{100}\right)}^t\\ \log 2=t\left(\log 104.75-\log 100\right)\\ t=\frac{\log 2}{\left(\log 104.75-\log 100\right)}\\ t=14.93646537\end{array}$

Thus, the amount $ 500 will gets double in 14 years and $0.93646537\times 365=342\text{days}$ .