Deandre is taking out an amortized loan for $$ \$ 84,000 $$ to open a small business and is deciding between the offers from two lenders. He wants to know which one would be the better deal over the life of the small business loan, and by how much. Answer each part. Do not round intermediate computations, and round your answers to the nearest cent. If necessary, refer to the list of financial formulas. (a) His credit union has offered him a 8 -year small business loan at an annual interest rate of $$ 10.4 \% $$. Find the monthly payment. s $$ \square 7 $$ (b) A savings and loan association has offered him a 9 -year small business loan at an annual interest rate of $$ 10.4 \% $$. Find the monthly payment. $$ s \square $$ (c) Suppose Deandre pays the monthly payment each month for the full term. Which lender's small business loan would have the lowest total amount to pay off, and by how much? Credit union The total amount paid would be $$ \$ \square $$ less than to the savings and loan association. Savings and loan association The total amount paid would be $$ s \square $$ less than to the credit union.

Question

UpStudy AI · Accepted Answer

To solve this problem, we will use the formula for the monthly payment $$ M $$ on an amortized loan, which is given by:

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

where:
- $$ P $$ is the loan amount (principal),
- $$ r $$ is the monthly interest rate (annual interest rate divided by 12),
- $$ n $$ is the total number of payments (loan term in months).

### Part (a): Monthly Payment for the Credit Union Loan

1. **Extract known values:**
   - $$ P = 84000 $$
   - Annual interest rate = $$ 10.4\% $$ → Monthly interest rate $$ r = \frac{10.4\%}{12} = \frac{0.104}{12} $$
   - Loan term = 8 years → Total payments $$ n = 8 \times 12 $$

2. **Calculate $$ r $$ and $$ n $$:**
   $$
   r = \frac{0.104}{12} = 0.00866667
   $$
   $$
   n = 8 \times 12 = 96
   $$

3. **Substitute into the formula:**
   $$
   M = 84000 \frac{0.00866667(1 + 0.00866667)^{96}}{(1 + 0.00866667)^{96} - 1}
   $$

Now, let's calculate the monthly payment $$ M $$ for the credit union loan.

### Part (b): Monthly Payment for the Savings and Loan Association Loan

1. **Extract known values:**
   - Loan term = 9 years → Total payments $$ n = 9 \times 12 $$

2. **Calculate $$ n $$:**
   $$
   n = 9 \times 12 = 108
   $$

3. **Substitute into the formula:**
   $$
   M = 84000 \frac{0.00866667(1 + 0.00866667)^{108}}{(1 + 0.00866667)^{108} - 1}
   $$

Now, let's calculate both monthly payments and the total amounts paid for each loan.

### Part (c): Total Amount Paid Comparison

1. **Total amount paid for each loan:**
   - Total amount for credit union = $$ M_{CU} \times 96 $$
   - Total amount for savings and loan = $$ M_{SLA} \times 108 $$

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{84000\left(0.00866667\left(1+0.00866667\right)^{108}\right)}{\left(\left(1+0.00866667\right)^{108}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{84000\times 0.00866667\left(1+0.00866667\right)^{108}}{\left(1+0.00866667\right)^{108}-1}$$
- step2: Add the numbers:
  $$\frac{84000\times 0.00866667\times 1.00866667^{108}}{\left(1+0.00866667\right)^{108}-1}$$
- step3: Add the numbers:
  $$\frac{84000\times 0.00866667\times 1.00866667^{108}}{1.00866667^{108}-1}$$
- step4: Convert the expressions:
  $$\frac{84000\times 0.00866667\left(\frac{100866667}{100000000}\right)^{108}}{1.00866667^{108}-1}$$
- step5: Convert the expressions:
  $$\frac{84000\times 0.00866667\left(\frac{100866667}{100000000}\right)^{108}}{\left(\frac{100866667}{100000000}\right)^{108}-1}$$
- step6: Multiply:
  $$\frac{\frac{18200007\times 100866667^{108}}{25000\times 100000000^{108}}}{\left(\frac{100866667}{100000000}\right)^{108}-1}$$
- step7: Subtract the numbers:
  $$\frac{\frac{18200007\times 100866667^{108}}{25000\times 100000000^{108}}}{\frac{100866667^{108}-100000000^{108}}{100000000^{108}}}$$
- step8: Multiply by the reciprocal:
  $$\frac{18200007\times 100866667^{108}}{25000\times 100000000^{108}}\times \frac{100000000^{108}}{100866667^{108}-100000000^{108}}$$
- step9: Rewrite the expression:
  $$\frac{18200007\times 100866667^{108}}{25000\times 100000000^{108}}\times \frac{25000^{108}\times 4000^{108}}{100866667^{108}-100000000^{108}}$$
- step10: Reduce the numbers:
  $$\frac{18200007\times 100866667^{108}}{100000000^{108}}\times \frac{25000^{107}\times 4000^{108}}{100866667^{108}-100000000^{108}}$$
- step11: Rewrite the expression:
  $$\frac{18200007\times 100866667^{108}}{25000^{108}\times 4000^{108}}\times \frac{25000^{107}\times 4000^{108}}{100866667^{108}-100000000^{108}}$$
- step12: Reduce the numbers:
  $$\frac{18200007\times 100866667^{108}}{25000\times 4000^{108}}\times \frac{4000^{108}}{100866667^{108}-100000000^{108}}$$
- step13: Rewrite the expression:
  $$\frac{18200007\times 100866667^{108}}{25000\times 4000^{108}}\times \frac{1000^{108}\times 4^{108}}{100866667^{108}-100000000^{108}}$$
- step14: Rewrite the expression:
  $$\frac{18200007\times 100866667^{108}}{1000\times 25\times 4000^{108}}\times \frac{1000^{108}\times 4^{108}}{100866667^{108}-100000000^{108}}$$
- step15: Reduce the numbers:
  $$\frac{18200007\times 100866667^{108}}{25\times 4000^{108}}\times \frac{1000^{107}\times 4^{108}}{100866667^{108}-100000000^{108}}$$
- step16: Rewrite the expression:
  $$\frac{18200007\times 100866667^{108}}{25\times 4000^{108}}\times \frac{125^{107}\times 8^{107}\times 4^{108}}{100866667^{108}-100000000^{108}}$$
- step17: Rewrite the expression:
  $$\frac{18200007\times 100866667^{108}}{5^{2}\times 4000^{108}}\times \frac{5^{321}\times 8^{107}\times 4^{108}}{100866667^{108}-100000000^{108}}$$
- step18: Reduce the numbers:
  $$\frac{18200007\times 100866667^{108}}{4000^{108}}\times \frac{5^{319}\times 8^{107}\times 4^{108}}{100866667^{108}-100000000^{108}}$$
- step19: Rewrite the expression:
  $$\frac{18200007\times 100866667^{108}}{125^{108}\times 32^{108}}\times \frac{5^{319}\times 8^{107}\times 4^{108}}{100866667^{108}-100000000^{108}}$$
- step20: Rewrite the expression:
  $$\frac{18200007\times 100866667^{108}}{5^{324}\times 32^{108}}\times \frac{5^{319}\times 8^{107}\times 4^{108}}{100866667^{108}-100000000^{108}}$$
- step21: Reduce the numbers:
  $$\frac{18200007\times 100866667^{108}}{5^{5}\times 32^{108}}\times \frac{8^{107}\times 4^{108}}{100866667^{108}-100000000^{108}}$$
- step22: Rewrite the expression:
  $$\frac{18200007\times 100866667^{108}}{5^{5}\times 2^{540}}\times \frac{2^{321}\times 4^{108}}{100866667^{108}-100000000^{108}}$$
- step23: Reduce the numbers:
  $$\frac{18200007\times 100866667^{108}}{5^{5}\times 2^{219}}\times \frac{4^{108}}{100866667^{108}-100000000^{108}}$$
- step24: Rewrite the expression:
  $$\frac{18200007\times 100866667^{108}}{5^{5}\times 2^{219}}\times \frac{2^{216}}{100866667^{108}-100000000^{108}}$$
- step25: Reduce the numbers:
  $$\frac{18200007\times 100866667^{108}}{5^{5}\times 2^{3}}\times \frac{1}{100866667^{108}-100000000^{108}}$$
- step26: Multiply the fractions:
  $$\frac{18200007\times 100866667^{108}}{25000\left(100866667^{108}-100000000^{108}\right)}$$
- step27: Multiply:
  $$\frac{18200007\times 100866667^{108}}{25000\times 100866667^{108}-25000\times 100000000^{108}}$$
Calculate or simplify the expression $$ 84000 * (0.00866667 * (1 + 0.00866667)^96) / ((1 + 0.00866667)^96 - 1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{84000\left(0.00866667\left(1+0.00866667\right)^{96}\right)}{\left(\left(1+0.00866667\right)^{96}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{84000\times 0.00866667\left(1+0.00866667\right)^{96}}{\left(1+0.00866667\right)^{96}-1}$$
- step2: Add the numbers:
  $$\frac{84000\times 0.00866667\times 1.00866667^{96}}{\left(1+0.00866667\right)^{96}-1}$$
- step3: Add the numbers:
  $$\frac{84000\times 0.00866667\times 1.00866667^{96}}{1.00866667^{96}-1}$$
- step4: Convert the expressions:
  $$\frac{84000\times 0.00866667\left(\frac{100866667}{100000000}\right)^{96}}{1.00866667^{96}-1}$$
- step5: Convert the expressions:
  $$\frac{84000\times 0.00866667\left(\frac{100866667}{100000000}\right)^{96}}{\left(\frac{100866667}{100000000}\right)^{96}-1}$$
- step6: Multiply:
  $$\frac{\frac{18200007\times 100866667^{96}}{25000\times 100000000^{96}}}{\left(\frac{100866667}{100000000}\right)^{96}-1}$$
- step7: Subtract the numbers:
  $$\frac{\frac{18200007\times 100866667^{96}}{25000\times 100000000^{96}}}{\frac{100866667^{96}-100000000^{96}}{100000000^{96}}}$$
- step8: Multiply by the reciprocal:
  $$\frac{18200007\times 100866667^{96}}{25000\times 100000000^{96}}\times \frac{100000000^{96}}{100866667^{96}-100000000^{96}}$$
- step9: Rewrite the expression:
  $$\frac{18200007\times 100866667^{96}}{25000\times 100000000^{96}}\times \frac{25000^{96}\times 4000^{96}}{100866667^{96}-100000000^{96}}$$
- step10: Reduce the numbers:
  $$\frac{18200007\times 100866667^{96}}{100000000^{96}}\times \frac{25000^{95}\times 4000^{96}}{100866667^{96}-100000000^{96}}$$
- step11: Rewrite the expression:
  $$\frac{18200007\times 100866667^{96}}{25000^{96}\times 4000^{96}}\times \frac{25000^{95}\times 4000^{96}}{100866667^{96}-100000000^{96}}$$
- step12: Reduce the numbers:
  $$\frac{18200007\times 100866667^{96}}{25000\times 4000^{96}}\times \frac{4000^{96}}{100866667^{96}-100000000^{96}}$$
- step13: Rewrite the expression:
  $$\frac{18200007\times 100866667^{96}}{25000\times 4000^{96}}\times \frac{1000^{96}\times 4^{96}}{100866667^{96}-100000000^{96}}$$
- step14: Rewrite the expression:
  $$\frac{18200007\times 100866667^{96}}{1000\times 25\times 4000^{96}}\times \frac{1000^{96}\times 4^{96}}{100866667^{96}-100000000^{96}}$$
- step15: Reduce the numbers:
  $$\frac{18200007\times 100866667^{96}}{25\times 4000^{96}}\times \frac{1000^{95}\times 4^{96}}{100866667^{96}-100000000^{96}}$$
- step16: Rewrite the expression:
  $$\frac{18200007\times 100866667^{96}}{25\times 4000^{96}}\times \frac{125^{95}\times 8^{95}\times 4^{96}}{100866667^{96}-100000000^{96}}$$
- step17: Rewrite the expression:
  $$\frac{18200007\times 100866667^{96}}{5^{2}\times 4000^{96}}\times \frac{5^{285}\times 8^{95}\times 4^{96}}{100866667^{96}-100000000^{96}}$$
- step18: Reduce the numbers:
  $$\frac{18200007\times 100866667^{96}}{4000^{96}}\times \frac{5^{283}\times 8^{95}\times 4^{96}}{100866667^{96}-100000000^{96}}$$
- step19: Rewrite the expression:
  $$\frac{18200007\times 100866667^{96}}{125^{96}\times 32^{96}}\times \frac{5^{283}\times 8^{95}\times 4^{96}}{100866667^{96}-100000000^{96}}$$
- step20: Rewrite the expression:
  $$\frac{18200007\times 100866667^{96}}{5^{288}\times 32^{96}}\times \frac{5^{283}\times 8^{95}\times 4^{96}}{100866667^{96}-100000000^{96}}$$
- step21: Reduce the numbers:
  $$\frac{18200007\times 100866667^{96}}{5^{5}\times 32^{96}}\times \frac{8^{95}\times 4^{96}}{100866667^{96}-100000000^{96}}$$
- step22: Rewrite the expression:
  $$\frac{18200007\times 100866667^{96}}{5^{5}\times 2^{480}}\times \frac{2^{285}\times 4^{96}}{100866667^{96}-100000000^{96}}$$
- step23: Reduce the numbers:
  $$\frac{18200007\times 100866667^{96}}{5^{5}\times 2^{195}}\times \frac{4^{96}}{100866667^{96}-100000000^{96}}$$
- step24: Rewrite the expression:
  $$\frac{18200007\times 100866667^{96}}{5^{5}\times 2^{195}}\times \frac{2^{192}}{100866667^{96}-100000000^{96}}$$
- step25: Reduce the numbers:
  $$\frac{18200007\times 100866667^{96}}{5^{5}\times 2^{3}}\times \frac{1}{100866667^{96}-100000000^{96}}$$
- step26: Multiply the fractions:
  $$\frac{18200007\times 100866667^{96}}{25000\left(100866667^{96}-100000000^{96}\right)}$$
- step27: Multiply:
  $$\frac{18200007\times 100866667^{96}}{25000\times 100866667^{96}-25000\times 100000000^{96}}$$
Calculate or simplify the expression $$ 1200.880933 * 108 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1200.880933\times 108$$
- step1: Multiply the numbers:
  $$129695.140764$$
Calculate or simplify the expression $$ 1292.47799 * 96 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1292.47799\times 96$$
- step1: Multiply the numbers:
  $$124077.88704$$
### Part (a): Monthly Payment for the Credit Union Loan

The monthly payment for the credit union loan is approximately:

$$
M_{CU} \approx 1292.48
$$

### Part (b): Monthly Payment for the Savings and Loan Association Loan

The monthly payment for the savings and loan association loan is approximately:

$$
M_{SLA} \approx 1200.88
$$

### Part (c): Total Amount Paid Comparison

1. **Total amount paid for the credit union loan:**
   $$
   \text{Total}_{CU} = M_{CU} \times 96 \approx 1292.48 \times 96 \approx 124077.89
   $$

2. **Total amount paid for the savings and loan association loan:**
   $$
   \text{Total}_{SLA} = M_{SLA} \times 108 \approx 1200.88 \times 108 \approx 129695.14
   $$

### Conclusion

- The credit union's total amount paid would be approximately $$ \$ 124077.89 $$.
- The savings and loan association's total amount paid would be approximately $$ \$ 129695.14 $$.

Thus, the credit union's small business loan would have the lowest total amount to pay off, and the difference is:

$$
\text{Difference} = 129695.14 - 124077.89 \approx 4617.25
$$

### Final Answers

- (a) Monthly payment for the credit union: $$ \$ 1292.48 $$
- (b) Monthly payment for the savings and loan association: $$ \$ 1200.88 $$
- (c) The credit union's total amount paid would be $$ \$ 4617.25 $$ less than to the savings and loan association.
(a) Monthly payment for the credit union: \$1,292.48 (b) Monthly payment for the savings and loan association: \$1,200.88 (c) The credit union's loan has a lower total amount to pay off by \$4,617.25 compared to the savings and loan association.

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