Find the Cheapest $2500 Monthly Compounded Personal Loan for Judy

Question

Judy needs to take out a personal loan for $2,500 for tuition assistance. Her bank has offered her one of the four loan packages outlined in the chart below. Determine which of the four loans will be cheapest for Judy in the long run. All interest rates are compounded monthly.
|Loan|Duration (Months)|Interest Rate|
|----|-----------------|-------------|
|A|12|9.50%|
|B|24|8.75%|
|C|36|7.75%|
|D|48|6.60%|
Options: a. loan A, b. loan B, c. loan C, d. loan D

Accepted Answer

a. loan A

Answer

Step 1: Recall the compound interest formula for total amount repaid: $A = P(1+\frac{r}{n})^{nt}$, where $A$ = total amount paid back, $P$ = principal ($2500), $r$ = annual interest rate (decimal), $n$ = number of times interest is compounded per year (12, since monthly), $t$ = time in years (duration in months / 12). Step 2: Calculate total amount for Loan A: $r=0.095$, $t=12/12=1$. $A = 2500(1+\frac{0.095}{12})^{12*1} ≈ 2500*(1.007917)^{12} ≈ 2500*1.0992 ≈ $2748.00. Total interest paid: $2748 - 2500 = $248. Step 3: Calculate total amount for Loan B: $r=0.0875$, $t=24/12=2$. $A = 2500(1+\frac{0.0875}{12})^{12*2} ≈ 2500*(1.007292)^{24} ≈ 2500*1.1881 ≈ $2970.25. Total interest paid: $2970.25 - 2500 = $470.25. Step 4: Calculate total amount for Loan C: $r=0.0775$, $t=36/12=3$. $A = 2500(1+\frac{0.0775}{12})^{12*3} ≈ 2500*(1.006458)^{36} ≈ 2500*1.2521 ≈ $3130.25. Total interest paid: $3130.25 - 2500 = $630.25. Step 5: Calculate total amount for Loan D: $r=0.0660$, $t=48/12=4$. $A = 2500(1+\frac{0.0660}{12})^{12*4} ≈ 2500*(1.0055)^{48} ≈ 2500*1.3003 ≈ $3250.75. Total interest paid: $3250.75 - 2500 = $750.75. Step 6: Compare total interest paid: Loan A has the lowest total interest ($248), so it is the cheapest in the long run.