Question

"If a loan of P dollars, at an interest rate of r percent per year compounded monthly, is payable in n monthly installments of m dollars each, then m is determined by the formula \small m=\frac{\frac{rP}{1,200}}{1-\left (1+\frac{r}{1,200}\right)^{-n}}. John and Sue took out loans whose monthly installments were determined using the formula above. Both loans had the same interest rate and the same number of monthly installments. John's monthly installment was what percent of Sue's monthly installment? (1) The amount of Sue's loan was 4 times the amount of John's loan. (2) Sue's monthly installment was 4 times John's monthly installment. "

Answer 1

Answer:

John's monthly installment was 25% of Sue's monthly installment.

Explanation:

The problem gives us the following data:

Sue's monthly installment was 4 times John's monthly installment.

if we wrote this as an equation we would get that:

$m_{S}=4m_{J}$

where:

$m_{S}$= Sue's monthly installment

$m_{J}$= John's monthly installment

in order to find what percent of Johns monthly installment represented regarding Sue's monthly installment, we would need to find the following ratio.

$percent=\frac{m_{J}}{m_{S}}*100\percent$

Therefore:

$percent=\frac{m_{J}}{4m_{J}}*100\percent$

or:

percent=25%

If you wanted to use the given formula, you could also do that. Notice that the formula tells us that the monthly installments are proportional to the loan in dollars, so we get the following:

$m=\frac{\frac{rP}{1,200}}{(1-(1+\frac{r}{1200})^{-n}}$

so we can use it to build our ration:

$\frac{m_{J}}{m_{S}}*100\percent=\frac{\frac{\frac{rP_{J}}{1,200}}{(1-(1+\frac{r}{1200})^{-n}}}{\frac{\frac{rP_{S}}{1,200}}{(1-(1+\frac{r}{1200})^{-n}}}*100\percent$

Since most of the data remains constant and the monthly installments are proportional to the original amount of the loan, that ratio can be simplified to:

$\frac{m_{J}}{m_{S}}*100\percent=\frac{P_{J}}{P_{S}}*100\percent$

So, the problem tells us that the amount of Sue's loan was 4 times the amount of John's loan, so:

$P_{S}=4P_{J}$

so:

$percent=\frac{P_{J}}{4P_{J}}*100\percent$

or:

percent=25%