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Sagot :
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.
[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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