# Suppose that $ 3000 is borrowed at an interest rate at 4 %. (a) Estimate the monthly payment on...

## Question:

Suppose that {eq}\$ 3000 {/eq} is borrowed at an interest rate at {eq}4 \%.{/eq}

(a) Estimate the monthly payment on the loan.

(b) If the interest rate increases to {eq}7 \% {/eq} estimate how much less would need to be borrowed so as not to increase the monthly payment.

## Fixed Payment of a Loan

Given a loan with:

- n number of periods.
- r interest rate per period.
- C initial loan.

Then the fixed payment for each period is:

{eq}P = \frac{r C }{ 1 - (1+r)^{-n}} {/eq}

## Answer and Explanation:

Assume the time for the loan is 10 years.

a) We have:

n = 120 months.

r = 0.04/12 interest rate per month.

C = $ 3000 the principal.

Let's use the formula for fixed payments for loans:

{eq}P = \frac{r C }{ 1 - (1+r)^{-n}} \\ = \frac{3000 \times 0.04/12 }{ 1 - (1+0.04/12)^{-120}} \\ = $ 30.37 {/eq} per month.

b) Using the formula for the payment, we can solve for C as follows:

{eq}C = \frac{P}{r}( 1 - (1+r)^{-n}) \\ {/eq}

Now, we can substitute the values with the new interest rate, in order to find C:

{eq}C = \frac{30.37}{0.07/12}( 1 - (1+0.07/12)^{-120}) = $ 2615.65 {/eq}

So, we need to borrow $2615.65 to keep same monthly payment.

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