Question

Paul wants to buy a car. He needs to take out a loan for $7000. The car salesman offers him a loan with an interest rate of 8%, compounded annually. Paul considers two options to repay the loan. Option 1: Pay $200 each month, until the loan is fully repaid Option 2: Make 24 equal monthly payments. Use option 1 to calculate the number of months it will take for Paul to repay the loan.

Answer 1

Given:

Paul wants to buy a car. He needs to take out a loan for $7000. The car salesman offers him a loan with an interest rate of 8%, compounded annually. Paul considers two options to repay the loan.

Option 1: Pay $200 each month, until the loan is fully repaid

Option 2: Make 24 equal monthly payments.

To Find:

Use option 1 to calculate the number of months it will take for Paul to repay the loan.

Solution:

To find the solution we need to know the formula for compound interest i.e.

                                             $A=P(1+r)^n$

we need to equate this value of compound interest with the monthly payment he is doing of $200 and see when $n$ the time period is coming equally

So,

$7000(1+0.08)^n=12*200*n\\7000(1.08)^n=2400n\\(1.08)^n=\frac{12n}{35}$

Now observing the equation or we can analyze the graph of both sides we come to know that for $n\approx4$ both sides are equal to be exact n=3.954

So at the end end of the 4th year in the last month, the loan is fully paid.

Hence, the number of months it takes for Paul to fully repay the loan is 48 months.