Question

1. You owe $2,348.62 on a credit card with an 8.75% APR. You pay $300.00 toward the card at the beginning of the month and another $300.00 in a savings account at a 3.25% APR. What is the difference in interest accrued if you had paid $600.00 toward the card instead of $300.00?
2. Using the information from Problem 1, how many months will it take you to pay off your debt if you pay at least $600.00 at the beginning of each month? Include the last month even if the payment is less than $600.00.

I don't understand these questions, please explain them to me?

Answer 1

OP was right about the second part, there is an easier way, and if you're learning on Connexus, like me,  (I believe) they teach us to do the calculation of how many months it takes to pay off the card like this:

Balance at the start of month 1: Interest accrued = ($2,348.62)(0.0875)(1/12) = $17.12

month 1 balance = $2,348.62 + $17.12 - $600.00 = $1,765.74

Balance at the start of month 2: Interest accrued = ($1,765.74)(0.0875)(1/12) = $12.87

month 2 balance = $1,765.74 + $12.87 - $600.00 = $1,178.61

Balance at the start of month 3: Interest accrued = ($1,178.61)(0.0875)(1/12) = $8.59

month 3 balance = $1,178.61 + $8.59 - $600.00 = $587.20

Balance at the start of month 4: Interest accrued = ($587.20)(0.0875)(1/12) = $4.28

month 4 balance = $587.20 + $4.28 = $591.48 paid on due date.

I'm not 100% sure, since it says at the beginning of the each month, and this calculation is usually used when paid at the end of each month, but the result was essentially the same, 4 months.

Answer 2

Part 1:

After payment of $300, remaining balance = $2,348.62 - $300 = $2,048.62.
Interest accrued is given by:

$I=Prt \\ \\ =2,048.62\times0.0875\times \frac{1}{12} \\ \\ =\ 14.94$

Had it been $600 was paid, remaining balance = $2,348.62 - $600 = $1748.62. Interest accrued is given by:

$I=1,748.62\times0.0875\times \frac{1}{12} \\ \\ = 12.75$

Difference in interest accrued = $14.94 - $12.75 = $2.19


Part 2:

The present value of an annuity is given by:

$PV= \frac{P\left[1-\left(1+ \frac{r}{12} \right)^{-12n}\right]}{ \frac{r}{12} }$

Where PV is the amount to be repaid, P is the equal monthly payment, r is the annual interest rate and n is the number of years.

Thus,

$2348.62= \frac{600\left[1-\left(1+ \frac{0.0875}{12}\right)^{-12n}\right]}{\frac{0.0875}{12}} \\ \\ \Rightarrow 1-(1+0.007292)^{-12n}= \frac{2348.62\times0.0875}{12\times600} =0.028542 \\ \\ \Rightarrow(1.007292)^{-12n}=1-0.028542=0.971458 \\ \\ \Rightarrow \log(1.007292)^{-12n}=\log0.971458 \\ \\ \Rightarrow-12n\log1.007292=\log0.971458 \\ \\ \Rightarrow-12n= \frac{\log0.971458}{\log1.007292} =-3.985559 \\ \\ \Rightarrow n= \frac{-3.985559}{-12} =0.332130$

Therefore, the number of months it will take to pay of the debt is 3.99 months which is approximately 4 months.