You wish to finance the purchase of a boat home for $$ \$ 38,600 $$. A finance company offers an APR of $$ 10 \% $$ on a 24 -month installment loan. After first using Table 13 -1 to find the finance charge, calculate the monthly payment. Select one: a. $$ \$ 1,781.23 $$ b. $$ \$ 1,619.08 $$ c. $$ \$ 1,769.17 $$ d. $$ \$ 1,608.33 $$

Question

UpStudy AI · Accepted Answer

To solve this problem, we need to follow these steps:

1. **Identify the known values:**
   - Loan amount (principal) $$ P = 38600 $$
   - Annual Percentage Rate (APR) $$ r = 10\% = 0.10 $$
   - Loan term $$ n = 24 $$ months

2. **Calculate the monthly interest rate:**
   $$
   r_m = \frac{r}{12} = \frac{0.10}{12}
   $$

3. **Calculate the finance charge using the formula for the total payment on an installment loan:**
   The formula for the monthly payment $$ M $$ on an installment loan is given by:
   $$
   M = P \cdot \frac{r_m(1 + r_m)^n}{(1 + r_m)^n - 1}
   $$

4. **Substitute the values into the formula and calculate $$ M $$.**

Let's perform the calculations step by step.

### Step 1: Calculate the monthly interest rate
$$
r_m = \frac{0.10}{12} = 0.0083333
$$

### Step 2: Substitute into the monthly payment formula
Now we can substitute $$ P $$, $$ r_m $$, and $$ n $$ into the formula for $$ M $$:
$$
M = 38600 \cdot \frac{0.0083333(1 + 0.0083333)^{24}}{(1 + 0.0083333)^{24} - 1}
$$

### Step 3: Calculate $$ M $$
Now, let's calculate $$ M $$ using the above formula.

I'll perform the calculation now.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{38600\left(0.0083333\left(1+0.0083333\right)^{24}\right)}{\left(\left(1+0.0083333\right)^{24}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{38600\times 0.0083333\left(1+0.0083333\right)^{24}}{\left(1+0.0083333\right)^{24}-1}$$
- step2: Add the numbers:
  $$\frac{38600\times 0.0083333\times 1.0083333^{24}}{\left(1+0.0083333\right)^{24}-1}$$
- step3: Add the numbers:
  $$\frac{38600\times 0.0083333\times 1.0083333^{24}}{1.0083333^{24}-1}$$
- step4: Convert the expressions:
  $$\frac{38600\times 0.0083333\left(\frac{10083333}{10000000}\right)^{24}}{1.0083333^{24}-1}$$
- step5: Convert the expressions:
  $$\frac{38600\times 0.0083333\left(\frac{10083333}{10000000}\right)^{24}}{\left(\frac{10083333}{10000000}\right)^{24}-1}$$
- step6: Multiply:
  $$\frac{\frac{16083269\times 10083333^{24}}{50000\times 10000000^{24}}}{\left(\frac{10083333}{10000000}\right)^{24}-1}$$
- step7: Subtract the numbers:
  $$\frac{\frac{16083269\times 10083333^{24}}{50000\times 10000000^{24}}}{\frac{10083333^{24}-10000000^{24}}{10000000^{24}}}$$
- step8: Multiply by the reciprocal:
  $$\frac{16083269\times 10083333^{24}}{50000\times 10000000^{24}}\times \frac{10000000^{24}}{10083333^{24}-10000000^{24}}$$
- step9: Rewrite the expression:
  $$\frac{16083269\times 10083333^{24}}{50000\times 10000000^{24}}\times \frac{50000^{24}\times 200^{24}}{10083333^{24}-10000000^{24}}$$
- step10: Reduce the numbers:
  $$\frac{16083269\times 10083333^{24}}{10000000^{24}}\times \frac{50000^{23}\times 200^{24}}{10083333^{24}-10000000^{24}}$$
- step11: Rewrite the expression:
  $$\frac{16083269\times 10083333^{24}}{50000^{24}\times 200^{24}}\times \frac{50000^{23}\times 200^{24}}{10083333^{24}-10000000^{24}}$$
- step12: Reduce the numbers:
  $$\frac{16083269\times 10083333^{24}}{50000\times 200^{24}}\times \frac{200^{24}}{10083333^{24}-10000000^{24}}$$
- step13: Rewrite the expression:
  $$\frac{16083269\times 10083333^{24}}{200\times 250\times 200^{24}}\times \frac{200^{24}}{10083333^{24}-10000000^{24}}$$
- step14: Reduce the numbers:
  $$\frac{16083269\times 10083333^{24}}{250\times 200^{24}}\times \frac{200^{23}}{10083333^{24}-10000000^{24}}$$
- step15: Rewrite the expression:
  $$\frac{16083269\times 10083333^{24}}{250\times 200^{24}}\times \frac{50^{23}\times 4^{23}}{10083333^{24}-10000000^{24}}$$
- step16: Rewrite the expression:
  $$\frac{16083269\times 10083333^{24}}{50\times 5\times 200^{24}}\times \frac{50^{23}\times 4^{23}}{10083333^{24}-10000000^{24}}$$
- step17: Reduce the numbers:
  $$\frac{16083269\times 10083333^{24}}{5\times 200^{24}}\times \frac{50^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}$$
- step18: Rewrite the expression:
  $$\frac{16083269\times 10083333^{24}}{5\times 200^{24}}\times \frac{25^{22}\times 2^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}$$
- step19: Rewrite the expression:
  $$\frac{16083269\times 10083333^{24}}{5\times 200^{24}}\times \frac{5^{44}\times 2^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}$$
- step20: Reduce the numbers:
  $$\frac{16083269\times 10083333^{24}}{200^{24}}\times \frac{5^{43}\times 2^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}$$
- step21: Rewrite the expression:
  $$\frac{16083269\times 10083333^{24}}{25^{24}\times 8^{24}}\times \frac{5^{43}\times 2^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}$$
- step22: Rewrite the expression:
  $$\frac{16083269\times 10083333^{24}}{5^{48}\times 8^{24}}\times \frac{5^{43}\times 2^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}$$
- step23: Reduce the numbers:
  $$\frac{16083269\times 10083333^{24}}{5^{5}\times 8^{24}}\times \frac{2^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}$$
- step24: Rewrite the expression:
  $$\frac{16083269\times 10083333^{24}}{5^{5}\times 2^{72}}\times \frac{2^{22}\times 4^{23}}{10083333^{24}-10000000^{24}}$$
- step25: Reduce the numbers:
  $$\frac{16083269\times 10083333^{24}}{5^{5}\times 2^{50}}\times \frac{4^{23}}{10083333^{24}-10000000^{24}}$$
- step26: Rewrite the expression:
  $$\frac{16083269\times 10083333^{24}}{5^{5}\times 2^{50}}\times \frac{2^{46}}{10083333^{24}-10000000^{24}}$$
- step27: Reduce the numbers:
  $$\frac{16083269\times 10083333^{24}}{5^{5}\times 2^{4}}\times \frac{1}{10083333^{24}-10000000^{24}}$$
- step28: Multiply the fractions:
  $$\frac{16083269\times 10083333^{24}}{50000\left(10083333^{24}-10000000^{24}\right)}$$
- step29: Multiply:
  $$\frac{16083269\times 10083333^{24}}{50000\times 10083333^{24}-50000\times 10000000^{24}}$$
The calculated monthly payment $$ M $$ is approximately $$ 1781.19 $$.

### Conclusion
The closest option to this calculated value is:
- a. $$ \$ 1,781.23 $$

Thus, the correct answer is **a. $$ \$ 1,781.23 $$**.
The monthly payment is approximately \$1,781.23.

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