What is the total amount of monthly payments for a $$ \$ 6,100 $$, two-year loan with an APR of $$ 5 \% $$ $$ \$ 406.94 $$ $$ \$ 322.88 $$ $$ \$ 6,422.88 $$ $$ \$ 267.62 $$

Question

UpStudy AI · Accepted Answer

To find the total amount of monthly payments for a loan, we can use the formula for the monthly payment on an installment loan, which is given by:

$$
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
$$

Where:
- $$ M $$ is the monthly payment
- $$ P $$ is the loan principal (amount borrowed)
- $$ r $$ is the monthly interest rate (annual interest rate divided by 12)
- $$ n $$ is the number of payments (loan term in months)

### Step 1: Extract known conditions
- Loan amount $$ P = 6100 $$
- Annual interest rate $$ APR = 5\% $$
- Loan term $$ = 2 $$ years $$ = 24 $$ months

### Step 2: Calculate the monthly interest rate
The monthly interest rate $$ r $$ is calculated as follows:

$$
r = \frac{APR}{12} = \frac{5\%}{12} = \frac{0.05}{12}
$$

### Step 3: Calculate the number of payments
The total number of payments $$ n $$ is:

$$
n = 2 \times 12 = 24
$$

### Step 4: Calculate the monthly payment $$ M $$
Now we can substitute the values into the formula to find $$ M $$:

$$
M = 6100 \cdot \frac{\frac{0.05}{12}(1 + \frac{0.05}{12})^{24}}{(1 + \frac{0.05}{12})^{24} - 1}
$$

Let's calculate $$ M $$ using the above formula.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{6100\left(\frac{0.05}{12}\left(1+\frac{0.05}{12}\right)^{24}\right)}{\left(\left(1+\frac{0.05}{12}\right)^{24}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{6100\times \frac{0.05}{12}\left(1+\frac{0.05}{12}\right)^{24}}{\left(1+\frac{0.05}{12}\right)^{24}-1}$$
- step2: Divide the terms:
  $$\frac{6100\times \frac{0.05}{12}\left(1+\frac{1}{240}\right)^{24}}{\left(1+\frac{0.05}{12}\right)^{24}-1}$$
- step3: Add the numbers:
  $$\frac{6100\times \frac{0.05}{12}\left(\frac{241}{240}\right)^{24}}{\left(1+\frac{0.05}{12}\right)^{24}-1}$$
- step4: Divide the terms:
  $$\frac{6100\times \frac{0.05}{12}\left(\frac{241}{240}\right)^{24}}{\left(1+\frac{1}{240}\right)^{24}-1}$$
- step5: Add the numbers:
  $$\frac{6100\times \frac{0.05}{12}\left(\frac{241}{240}\right)^{24}}{\left(\frac{241}{240}\right)^{24}-1}$$
- step6: Divide the terms:
  $$\frac{6100\times \frac{1}{240}\left(\frac{241}{240}\right)^{24}}{\left(\frac{241}{240}\right)^{24}-1}$$
- step7: Multiply:
  $$\frac{\frac{61\times 241^{24}}{12\times 5^{23}\times 48^{24}}}{\left(\frac{241}{240}\right)^{24}-1}$$
- step8: Subtract the numbers:
  $$\frac{\frac{61\times 241^{24}}{12\times 5^{23}\times 48^{24}}}{\frac{241^{24}-240^{24}}{240^{24}}}$$
- step9: Multiply by the reciprocal:
  $$\frac{61\times 241^{24}}{12\times 5^{23}\times 48^{24}}\times \frac{240^{24}}{241^{24}-240^{24}}$$
- step10: Rewrite the expression:
  $$\frac{61\times 241^{24}}{12\times 5^{23}\times 48^{24}}\times \frac{12^{24}\times 20^{24}}{241^{24}-240^{24}}$$
- step11: Reduce the numbers:
  $$\frac{61\times 241^{24}}{5^{23}\times 48^{24}}\times \frac{12^{23}\times 20^{24}}{241^{24}-240^{24}}$$
- step12: Rewrite the expression:
  $$\frac{61\times 241^{24}}{5^{23}\times 12^{24}\times 4^{24}}\times \frac{12^{23}\times 20^{24}}{241^{24}-240^{24}}$$
- step13: Reduce the numbers:
  $$\frac{61\times 241^{24}}{5^{23}\times 12\times 4^{24}}\times \frac{20^{24}}{241^{24}-240^{24}}$$
- step14: Rewrite the expression:
  $$\frac{61\times 241^{24}}{5^{23}\times 12\times 4^{24}}\times \frac{5^{24}\times 4^{24}}{241^{24}-240^{24}}$$
- step15: Reduce the numbers:
  $$\frac{61\times 241^{24}}{12\times 4^{24}}\times \frac{5\times 4^{24}}{241^{24}-240^{24}}$$
- step16: Rewrite the expression:
  $$\frac{61\times 241^{24}}{4\times 3\times 4^{24}}\times \frac{5\times 4^{24}}{241^{24}-240^{24}}$$
- step17: Reduce the numbers:
  $$\frac{61\times 241^{24}}{3\times 4}\times \frac{5}{241^{24}-240^{24}}$$
- step18: Multiply the fractions:
  $$\frac{61\times 241^{24}\times 5}{12\left(241^{24}-240^{24}\right)}$$
- step19: Multiply:
  $$\frac{305\times 241^{24}}{12\times 241^{24}-12\times 240^{24}}$$
The calculated monthly payment $$ M $$ is approximately $$ 267.62 $$.

### Step 5: Calculate the total amount of monthly payments
To find the total amount of monthly payments over the loan term, we multiply the monthly payment by the number of payments:

$$
\text{Total Payments} = M \times n = 267.62 \times 24
$$

Now, let's calculate the total amount of monthly payments.
Calculate the value by following steps:
- step0: Calculate:
  $$267.62\times 24$$
- step1: Multiply the numbers:
  $$6422.88$$
The total amount of monthly payments over the two-year loan is $$ 6422.88 $$.

### Summary of Results
- Monthly Payment: $$ 267.62 $$
- Total Amount of Monthly Payments: $$ 6422.88 $$

Thus, the correct answer is $$ \$ 6,422.88 $$.
The total amount of monthly payments is \$6,422.88.

What is the total amount of monthly payments for a \( \$ 6,100 \), two-year loan with an APR of \( 5 \% \) \( \$ 406.94 \) \( \$ 322.88 \) \( \$ 6,422.88 \) \( \$ 267.62 \)

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