Question

Time Required to Reach a Goal

If Angela has \(100 to invest at 10% per annum compounded monthly, how long will it be before she has \)175? If the compounding is continuous, how long will it be?

Short answer

If the amount is compounded monthly, it becomes $175 in approximately 5.619 years.

If the amount is compounded continuously, it becomes $175 in approximately 5.596 years.

Step-by-step solution

Step 1. Given Information

Given, if Angela has $100 to invest at 10% per annum compounded monthly, the time taken for the amount to become $175 is to be determined.

If the compounding is continuous, the time taken for the amount to become $175 is to be determined.

Step 2. Calculation for monthly compounding

According to the compound interest formula, the amount A after t years for the principal P with an annual rate of interest r compounded n times per year is $A=P{\left(1+\frac{r}{n}\right)}^{nt}$

Here, the amount invested is $100 i.e. $P=100$

Rate of interest is 10% i.e. $r=0.1$

Since the amount is compounded monthly, $n=12$

Finally the amount becomes $175 i.e. $A=175$

Plugging the values:

$$\begin{gathered} A=P{\left(1+\frac{r}{n}\right)}^{nt} \\ 175=100{\left(1+\frac{0.1}{12}\right)}^{12t} \\ {\left(1+\frac{1}{120}\right)}^{12t}=\frac{175}{100} \\ {\left(\frac{121}{120}\right)}^{12t}=1.75 \\ 12t=\frac{\ln 1.75}{\ln 121-\ln 120} \\ 12t\cong 67.433 \\ t\cong 5.619 \end{gathered}$$

So if the amount is compounded monthly, it becomes $175 in approximately 5.619 years.

Step 3. Calculation for continuous compounding

According to the compound interest formula, the amount A after t years for the principal P with an annual rate of interest r compounded continuously is $A=P{e}^{rt}$

Plugging the values:

role="math" localid="1646705968180" $$\begin{gathered} A=P{e}^{rt} \\ 175=100e^{0.1t} \\ {e}^{0.1t}=\frac{175}{100} \\ 0.1t=\ln 1.75 \\ t=\frac{\ln 1.75}{0.1} \\ t\cong 5.596 \end{gathered}$$

So if the amount is compounded continuously, it becomes $175 in approximately 5.596 years.

Step 4. Conclusion

Hence, if $100 is compounded monthly, it becomes $175 in approximately 5.619 years and if compounded continuously, it becomes $175 in approximately 5.596 years.