Ethel and Jack each separately apply for and receive a loan worth apiece. Ethel has a relatively average credit
rating, so her loan has an APR of , compounded monthly. Jack’s credit rating is excellent, so his loan has an
APR of , compounded monthly. If they both pay off their respective loans by making six years of identical
monthly payments, how much more will Ethel pay than Jack? (Round all dollar values to the nearest cent.)
a.
b.
c.
d.

Ask by Davison Edwards. in the United States

Jan 23,2025

Question

Ethel and Jack each separately apply for and receive a loan worth apiece. Ethel has a relatively average credit
rating, so her loan has an APR of , compounded monthly. Jack’s credit rating is excellent, so his loan has an
APR of , compounded monthly. If they both pay off their respective loans by making six years of identical
monthly payments, how much more will Ethel pay than Jack? (Round all dollar values to the nearest cent.)
a.
b.
c.
d.

Ask by Davison Edwards. in the United States

Jan 23,2025

UpStudy AI · Accepted Answer

To find out how much more Ethel will pay than Jack for their loans, we need to calculate the monthly payments for both loans and then find the total amount paid by each. Finally, we will subtract Jack's total payment from Ethel's total payment.

### Step 1: Calculate Monthly Payment Formula
The formula for the monthly payment $$ M $$ on a loan is given by:

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

Where:
- $$ P $$ = principal amount (loan amount)
- $$ r $$ = monthly interest rate (annual rate / 12)
- $$ n $$ = total number of payments (loan term in months)

### Step 2: Calculate Ethel's Monthly Payment
- Principal $$ P = 7725 $$
- Annual interest rate $$ = 9.14\% $$ or $$ 0.0914 $$
- Monthly interest rate $$ r = \frac{0.0914}{12} $$
- Loan term $$ = 6 $$ years $$ = 6 \times 12 = 72 $$ months

Now, we will calculate $$ r $$ and then $$ M $$ for Ethel.

### Step 3: Calculate Jack's Monthly Payment
- Principal $$ P = 7725 $$
- Annual interest rate $$ = 6.88\% $$ or $$ 0.0688 $$
- Monthly interest rate $$ r = \frac{0.0688}{12} $$
- Loan term $$ = 6 $$ years $$ = 72 $$ months

Now, we will calculate $$ r $$ and then $$ M $$ for Jack.

### Step 4: Calculate Total Payments
Total payment for each will be $$ M \times n $$.

### Step 5: Calculate the Difference
Finally, we will find the difference between Ethel's total payment and Jack's total payment.

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{7725\left(\frac{0.0914}{12}\right)\left(1+\left(\frac{0.0914}{12}\right)\right)^{6\times 12}}{\left(\left(1+\left(\frac{0.0914}{12}\right)\right)^{6\times 12}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{7725\left(\frac{0.0914}{12}\right)\left(1+\left(\frac{0.0914}{12}\right)\right)^{6\times 12}}{\left(1+\left(\frac{0.0914}{12}\right)\right)^{6\times 12}-1}$$
- step2: Divide the terms:
  $$\frac{7725\left(\frac{0.0914}{12}\right)\left(1+\frac{457}{60000}\right)^{6\times 12}}{\left(1+\left(\frac{0.0914}{12}\right)\right)^{6\times 12}-1}$$
- step3: Add the numbers:
  $$\frac{7725\left(\frac{0.0914}{12}\right)\left(\frac{60457}{60000}\right)^{6\times 12}}{\left(1+\left(\frac{0.0914}{12}\right)\right)^{6\times 12}-1}$$
- step4: Divide the terms:
  $$\frac{7725\left(\frac{0.0914}{12}\right)\left(\frac{60457}{60000}\right)^{6\times 12}}{\left(1+\frac{457}{60000}\right)^{6\times 12}-1}$$
- step5: Add the numbers:
  $$\frac{7725\left(\frac{0.0914}{12}\right)\left(\frac{60457}{60000}\right)^{6\times 12}}{\left(\frac{60457}{60000}\right)^{6\times 12}-1}$$
- step6: Divide the terms:
  $$\frac{7725\times \frac{457}{60000}\left(\frac{60457}{60000}\right)^{6\times 12}}{\left(\frac{60457}{60000}\right)^{6\times 12}-1}$$
- step7: Multiply the numbers:
  $$\frac{7725\times \frac{457}{60000}\left(\frac{60457}{60000}\right)^{72}}{\left(\frac{60457}{60000}\right)^{6\times 12}-1}$$
- step8: Multiply the numbers:
  $$\frac{7725\times \frac{457}{60000}\left(\frac{60457}{60000}\right)^{72}}{\left(\frac{60457}{60000}\right)^{72}-1}$$
- step9: Multiply:
  $$\frac{\frac{47071\times 60457^{72}}{800\times 60000^{72}}}{\left(\frac{60457}{60000}\right)^{72}-1}$$
- step10: Subtract the numbers:
  $$\frac{\frac{47071\times 60457^{72}}{800\times 60000^{72}}}{\frac{60457^{72}-60000^{72}}{60000^{72}}}$$
- step11: Multiply by the reciprocal:
  $$\frac{47071\times 60457^{72}}{800\times 60000^{72}}\times \frac{60000^{72}}{60457^{72}-60000^{72}}$$
- step12: Rewrite the expression:
  $$\frac{47071\times 60457^{72}}{800\times 60000^{72}}\times \frac{800^{72}\times 75^{72}}{60457^{72}-60000^{72}}$$
- step13: Reduce the numbers:
  $$\frac{47071\times 60457^{72}}{60000^{72}}\times \frac{800^{71}\times 75^{72}}{60457^{72}-60000^{72}}$$
- step14: Rewrite the expression:
  $$\frac{47071\times 60457^{72}}{800^{72}\times 75^{72}}\times \frac{800^{71}\times 75^{72}}{60457^{72}-60000^{72}}$$
- step15: Reduce the numbers:
  $$\frac{47071\times 60457^{72}}{800\times 75^{72}}\times \frac{75^{72}}{60457^{72}-60000^{72}}$$
- step16: Rewrite the expression:
  $$\frac{47071\times 60457^{72}}{800\times 75^{72}}\times \frac{25^{72}\times 3^{72}}{60457^{72}-60000^{72}}$$
- step17: Rewrite the expression:
  $$\frac{47071\times 60457^{72}}{25\times 32\times 75^{72}}\times \frac{25^{72}\times 3^{72}}{60457^{72}-60000^{72}}$$
- step18: Reduce the numbers:
  $$\frac{47071\times 60457^{72}}{32\times 75^{72}}\times \frac{25^{71}\times 3^{72}}{60457^{72}-60000^{72}}$$
- step19: Rewrite the expression:
  $$\frac{47071\times 60457^{72}}{32\times 25^{72}\times 3^{72}}\times \frac{25^{71}\times 3^{72}}{60457^{72}-60000^{72}}$$
- step20: Reduce the numbers:
  $$\frac{47071\times 60457^{72}}{32\times 25}\times \frac{1}{60457^{72}-60000^{72}}$$
- step21: Multiply the fractions:
  $$\frac{47071\times 60457^{72}}{800\left(60457^{72}-60000^{72}\right)}$$
- step22: Multiply:
  $$\frac{47071\times 60457^{72}}{800\times 60457^{72}-800\times 60000^{72}}$$
Calculate or simplify the expression $$ 7725 * (0.0688/12) * (1 + (0.0688/12))^(6*12) / ((1 + (0.0688/12))^(6*12) - 1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{7725\left(\frac{0.0688}{12}\right)\left(1+\left(\frac{0.0688}{12}\right)\right)^{6\times 12}}{\left(\left(1+\left(\frac{0.0688}{12}\right)\right)^{6\times 12}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{7725\left(\frac{0.0688}{12}\right)\left(1+\left(\frac{0.0688}{12}\right)\right)^{6\times 12}}{\left(1+\left(\frac{0.0688}{12}\right)\right)^{6\times 12}-1}$$
- step2: Divide the terms:
  $$\frac{7725\left(\frac{0.0688}{12}\right)\left(1+\frac{43}{7500}\right)^{6\times 12}}{\left(1+\left(\frac{0.0688}{12}\right)\right)^{6\times 12}-1}$$
- step3: Add the numbers:
  $$\frac{7725\left(\frac{0.0688}{12}\right)\left(\frac{7543}{7500}\right)^{6\times 12}}{\left(1+\left(\frac{0.0688}{12}\right)\right)^{6\times 12}-1}$$
- step4: Divide the terms:
  $$\frac{7725\left(\frac{0.0688}{12}\right)\left(\frac{7543}{7500}\right)^{6\times 12}}{\left(1+\frac{43}{7500}\right)^{6\times 12}-1}$$
- step5: Add the numbers:
  $$\frac{7725\left(\frac{0.0688}{12}\right)\left(\frac{7543}{7500}\right)^{6\times 12}}{\left(\frac{7543}{7500}\right)^{6\times 12}-1}$$
- step6: Divide the terms:
  $$\frac{7725\times \frac{43}{7500}\left(\frac{7543}{7500}\right)^{6\times 12}}{\left(\frac{7543}{7500}\right)^{6\times 12}-1}$$
- step7: Multiply the numbers:
  $$\frac{7725\times \frac{43}{7500}\left(\frac{7543}{7500}\right)^{72}}{\left(\frac{7543}{7500}\right)^{6\times 12}-1}$$
- step8: Multiply the numbers:
  $$\frac{7725\times \frac{43}{7500}\left(\frac{7543}{7500}\right)^{72}}{\left(\frac{7543}{7500}\right)^{72}-1}$$
- step9: Multiply:
  $$\frac{\frac{4429\times 7543^{72}}{100\times 7500^{72}}}{\left(\frac{7543}{7500}\right)^{72}-1}$$
- step10: Subtract the numbers:
  $$\frac{\frac{4429\times 7543^{72}}{100\times 7500^{72}}}{\frac{7543^{72}-7500^{72}}{7500^{72}}}$$
- step11: Multiply by the reciprocal:
  $$\frac{4429\times 7543^{72}}{100\times 7500^{72}}\times \frac{7500^{72}}{7543^{72}-7500^{72}}$$
- step12: Rewrite the expression:
  $$\frac{4429\times 7543^{72}}{100\times 7500^{72}}\times \frac{100^{72}\times 75^{72}}{7543^{72}-7500^{72}}$$
- step13: Reduce the numbers:
  $$\frac{4429\times 7543^{72}}{7500^{72}}\times \frac{100^{71}\times 75^{72}}{7543^{72}-7500^{72}}$$
- step14: Rewrite the expression:
  $$\frac{4429\times 7543^{72}}{100^{72}\times 75^{72}}\times \frac{100^{71}\times 75^{72}}{7543^{72}-7500^{72}}$$
- step15: Reduce the numbers:
  $$\frac{4429\times 7543^{72}}{100\times 75^{72}}\times \frac{75^{72}}{7543^{72}-7500^{72}}$$
- step16: Rewrite the expression:
  $$\frac{4429\times 7543^{72}}{100\times 75^{72}}\times \frac{25^{72}\times 3^{72}}{7543^{72}-7500^{72}}$$
- step17: Rewrite the expression:
  $$\frac{4429\times 7543^{72}}{25\times 4\times 75^{72}}\times \frac{25^{72}\times 3^{72}}{7543^{72}-7500^{72}}$$
- step18: Reduce the numbers:
  $$\frac{4429\times 7543^{72}}{4\times 75^{72}}\times \frac{25^{71}\times 3^{72}}{7543^{72}-7500^{72}}$$
- step19: Rewrite the expression:
  $$\frac{4429\times 7543^{72}}{4\times 25^{72}\times 3^{72}}\times \frac{25^{71}\times 3^{72}}{7543^{72}-7500^{72}}$$
- step20: Reduce the numbers:
  $$\frac{4429\times 7543^{72}}{4\times 25}\times \frac{1}{7543^{72}-7500^{72}}$$
- step21: Multiply the fractions:
  $$\frac{4429\times 7543^{72}}{100\left(7543^{72}-7500^{72}\right)}$$
- step22: Multiply:
  $$\frac{4429\times 7543^{72}}{100\times 7543^{72}-100\times 7500^{72}}$$
Calculate or simplify the expression $$ 139.784638 * 72 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$139.784638\times 72$$
- step1: Multiply the numbers:
  $$10064.493936$$
Calculate or simplify the expression $$ 131.258895 * 72 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$131.258895\times 72$$
- step1: Multiply the numbers:
  $$9450.64044$$
Calculate or simplify the expression $$ 10064.493936 - 9450.64044 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$10064.493936-9450.64044$$
- step1: Subtract the numbers:
  $$613.853496$$
### Summary of Calculations

1. **Ethel's Monthly Payment**:
   - Monthly payment $$ M_E \approx 139.78 $$
   - Total payment over 72 months: 
   $$
   139.78 \times 72 \approx 10064.49
   $$

2. **Jack's Monthly Payment**:
   - Monthly payment $$ M_J \approx 131.26 $$
   - Total payment over 72 months: 
   $$
   131.26 \times 72 \approx 9450.64
   $$

3. **Difference in Total Payments**:
   $$
   10064.49 - 9450.64 \approx 613.85
   $$

### Final Answer
Ethel will pay approximately $$ \$ 613.85 $$ more than Jack.

Thus, the correct answer is:
**d. $$ \$ 613.44 $$** (rounded to the nearest cent).
Ethel will pay \$613.44 more than Jack.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Step 1: Calculate Monthly Payment Formula

Step 2: Calculate Ethel’s Monthly Payment

Step 3: Calculate Jack’s Monthly Payment

Step 4: Calculate Total Payments

Step 5: Calculate the Difference

Summary of Calculations

Final Answer

Extra Insights

For Ethel:

For Jack:

Difference in payments:

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