How do you find the present value that will grow to $\$20, 000$ if interest is $7\%$ compounded quarterly for 15 quarters?

Hint: To solve the above problem, we need to calculate the principal value in the following formula: $Amount=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}$. In this formula, amount is the amount when compound interest applied on the original amount, P is the principal, r is the rate of interest, n is the number of quarters in a time period and t is the number of years.

Complete step by step solution:
In the above problem, we have given the amount when compound interest applied on the original amount which is equal to $\$20, 000$ and rate of interest is given as $7\%$. To find the present value (or principal), we are going to use the following formula:
$Amount=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}$ ………… (1)
In the above formula, amount is $\$20, 000$ and r is $7\%$, n is 4 because the principal is compounded quarterly. And we can calculate the value of “t” which is the number of years as follows:
We have to find the amount after 15 quarters, we know that 4 quarters make an year so 12 quarters will make 3 years and the remaining 3 quarters will make $\dfrac{3}{4}$ year so the total number of years are:
$\begin{align}
  & =\left( 3+\dfrac{3}{4} \right)years \\
 & =3.75years \\
\end{align}$
Hence, we got the value of t as 3.75 years.
Now, substituting all the values of amount, P, r, n and t in eq. (1) we will get the value of principal.
$\Rightarrow 20000=P{{\left( 1+\dfrac{7}{100\left( 4 \right)} \right)}^{4\left( 3.75 \right)}}$
We can write $\dfrac{7}{100}=0.07$ in the above equation and we get,
$\begin{align}
  & \Rightarrow 20000=P{{\left( 1+\dfrac{0.07}{\left( 4 \right)} \right)}^{15}} \\
 & \Rightarrow 20000=P{{\left( 1+0.0175 \right)}^{15}} \\
 & \Rightarrow 20000=P{{\left( 1.0175 \right)}^{15}} \\
 & \Rightarrow 20000=P\left( 1.297 \right) \\
\end{align}$
Dividing 1.297 on both the sides we get,
$\begin{align}
  & \Rightarrow \dfrac{20000}{1.297}=P \\
 & \Rightarrow 15420.2=P \\
\end{align}$

From the above, we have calculated the value of principal as $\$15420.2$ or the present value is equal to $\$15420.2$.

Note: The mistake that could be possible in the above problem is to convert the 15 quarters into years. You might write 15 quarters as 15 years in place of “t” in the above formula so make sure you won’t make this mistake and convert 15 quarters into years by taking into account that 4 quarters make an year.