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Mimi has a business loan of [tex]\[tex]$12{,}000[/tex] from a bank with an annual interest rate of [tex]5\%[/tex]. The loan is to be repaid over 2 years.

What is Mimi's monthly repayment?

A. [tex]\$[/tex]12{,}000 \cdot \frac{0.05(1.05)^{24}}{(1.05)^{24}-1} = \[tex]$12{,}000 \cdot \frac{0.1613}{2.2251} = \$[/tex]869.89[/tex]

B. [tex]\[tex]$12{,}000 \cdot \frac{\frac{0.05}{12}\left(1+\frac{0.05}{12}\right)^{24}}{\left(1+\frac{0.05}{12}\right)^{24}-1} = \$[/tex]12{,}000 \cdot \frac{0.0042(1.0042)^{24}}{(1.0042)^{24}-1} = \frac{\[tex]$55.733}{0.1058} = \$[/tex]526.78[/tex]

C. [tex]\[tex]$12{,}000 \cdot 0.05 \cdot \frac{1}{24} = \$[/tex]25.00[/tex]

D. [tex]\[tex]$12{,}000 \cdot 0.05 \cdot 2 = \$[/tex]1{,}200.00[/tex]



Answer :

GinnyAnswer
To determine Mimi's monthly repayment for the business loan, we will use the formula for calculating annuity payments, specifically the formula for monthly repayments on an installment loan. The formula is:

[tex]\[ M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1} \][/tex]

Where:
- [tex]\( M \)[/tex] is the monthly repayment.
- [tex]\( P \)[/tex] is the loan amount.
- [tex]\( r \)[/tex] is the monthly interest rate.
- [tex]\( n \)[/tex] is the total number of monthly payments.

Given:
- [tex]\( P = \$12,000 \)[/tex]
- The annual interest rate is [tex]\( 5\% \)[/tex], or [tex]\( 0.05 \)[/tex].
- The repayment period is [tex]\( 2 \)[/tex] years.

First, we need to convert the annual interest rate to a monthly interest rate:
[tex]\[ r = \frac{0.05}{12} = 0.004167 \][/tex]

Next, we calculate the total number of monthly payments over the 2 years:
[tex]\[ n = 2 \times 12 = 24 \][/tex]

Now we can plug these values into the formula:

1. Calculate [tex]\( (1 + r)^n \)[/tex]:
[tex]\[ (1 + 0.004167)^{24} = 1.10494 \][/tex]

2. Calculate the numerator [tex]\( r \cdot (1 + r)^n \)[/tex]:
[tex]\[ 0.004167 \cdot 1.10494 = 0.004604 \][/tex]

3. Calculate the denominator [tex]\( (1 + r)^n - 1 \)[/tex]:
[tex]\[ 1.10494 - 1 = 0.104941 \][/tex]

4. Finally, calculate the monthly repayment [tex]\( M \)[/tex]:
[tex]\[ M = 12000 \cdot \frac{0.004604}{0.104941} = 12000 \cdot 0.043871 \approx 526.4567 \][/tex]

Therefore, Mimi's monthly repayment is approximately \$526.46.

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