Chapter 12.4, Problem 28E

Interest Rate A man purchases a $2000 diamond ring for a down payment of $200 and monthly installments of $88 for 2 years. Assuming that interest is compounded monthly, what interest rate is he paying?

To determine

To find: The interest rate on the diamond ring.

Answer to Problem 28E

The interest rate on the diamond ring is $15.6\%$ per year.

Explanation of Solution

Given:

The price of a diamond ring is $\$2000$.

The down payment is $\$200$.

Monthly installment of the diamond ring is $\$88$ for $2$ years.

Formula used:

The principal amount is,

$A_p=A-A_D$ (1)

Where $A_p$ is the principal amount, $A$ is the actual amount and $A_D$ is the amount of down payment.

The monthly amount payment is,

$R=\frac{i{A}_p}{1-{\left(1+i\right)}^{-n}}$

Where R is the monthly amount, $i$ is the interest rate and $n$ is the number of payments.

Calculation:

The total number of payments is,

$\begin{array}{c}n=2\times 12\\ =24\end{array}$

Substitute $\$2000$ for $A$ and $\$200$ $A_D$ in equation (1) to find $A$.

$\begin{array}{c}{A}_p=\$2000-\$200\\ =\$1800\end{array}$

Substitute $24$ for $n$ and $\$1800$ for $A_p$ in $R=\frac{i{A}_p}{1-{\left(1+i\right)}^{-n}}$ to find $i$.

$R=\frac{i\left(1800\right)}{1-{\left(1+i\right)}^{-24}}$

The value of $i$ cannot be solved algebraically.

So, plot the graph by taking $R$ as a function of $i$ as shown in the figure below.

From the graph it can be observed that the corresponding value of the monthly installment of $88 is 0.013.

Multiply the value of interest rate by a factor $(100\times 12)$ to convert it into percentage per year.

$\begin{array}{c}i=\left(0.013\right)\times 100\times 12\\ =15.6\%\end{array}$

Thus, the interest rate on the diamond ring is $15.6\%$ per year.