Math / Algebra

QuestionPREVIOUS ANSWERS ASK YOUR TEACHER PRACTICE ANOTHER (Round your answers to the nearest cent.) (a) What monthly payment will she be required to make if the car is financed over a period of 60 months? Over a period of 72 months?
60 \$ 487.22 review the concepts you need. review the concepts you need. (b) What will the interest charges be if she elects the 60 -month plan? The 72 -month plan?
Need Help? Read It Submit Answer Viewing Saved Work Revert to Last Response

Studdy Solution

STEP 1

What is this asking? We need to figure out the monthly payments and total interest paid for a car loan with two different loan lengths. Watch out! Don't forget to correctly calculate the loan amount after the down payment and be careful with the interest rate, it's given yearly, but we need it monthly!

STEP 2

1. Calculate Loan Amount
2. Calculate Monthly Payment (60 months)
3. Calculate Total Interest (60 months)
4. Calculate Monthly Payment (72 months)
5. Calculate Total Interest (72 months)

STEP 3

Alright, first things first!
We've got a car that costs $\$32,000$ and a down payment of $25\%$ .
Let's calculate the down payment amount: $0.25 \cdot \$32,000 = \$8,000$ .

STEP 4

Now, subtract that down payment from the original price to find the loan amount: $\$32,000 - \$8,000 = \$24,000$ .
This is how much we're actually borrowing!

STEP 5

Time for the monthly payment formula!
It's a bit of a mouthful, but stick with me: $M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}$ .
Don't worry, we'll break it down!

STEP 6

$M$ is our monthly payment, $P$ is the principal (our loan amount, $\$24,000$ ), $r$ is the monthly interest rate, and $n$ is the number of payments.

STEP 7

The annual interest rate is $8\%$ , so the monthly rate is $0.08 / 12 \approx 0.00666667$ .
For 60 months, $n = 60$ .

STEP 8

Plugging everything in: $M = \$24,000 \cdot \frac{0.00666667(1+0.00666667)^{60}}{(1+0.00666667)^{60} - 1}$ .

STEP 9

Calculating carefully, we get $M \approx \$487.22$ .
That's our monthly payment for the 60-month loan!

STEP 10

The total amount paid over 60 months is $60 \cdot \$487.22 = \$29,233.20$ .

STEP 11

Subtract the initial loan amount to find the total interest: $\$29,233.20 - \$24,000 = \$5,233.20$ .

STEP 12

We use the same formula as before, but now $n = 72$ : $M = \$24,000 \cdot \frac{0.00666667(1+0.00666667)^{72}}{(1+0.00666667)^{72} - 1}$ .

STEP 13

Crunching the numbers, we get $M \approx \$429.87$ .
Lower monthly payments, but let's see what happens to the interest!

STEP 14

Total paid over 72 months: $72 \cdot \$429.87 = \$30,950.64$ .

STEP 15

Total interest: $\$30,950.64 - \$24,000 = \$6,950.64$ .
See how that extra time adds up in interest?

STEP 16

(a) 60-month payment: $\$487.22$ ; 72-month payment: $\$429.87$ . (b) 60-month interest: $\$5,233.20$ ; 72-month interest: $\$6,950.64$ .

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Calculate Loan Amount
2. Calculate Monthly Payment (60 months)
3. Calculate Total Interest (60 months)
4. Calculate Monthly Payment (72 months)
5. Calculate Total Interest (72 months)

STEP 3

Alright, first things first!
We've got a car that costs $\$32,000$ and a down payment of $25\%$ .
Let's calculate the down payment amount: $0.25 \cdot \$32,000 = \$8,000$ .

STEP 4

Now, subtract that down payment from the original price to find the loan amount: $\$32,000 - \$8,000 = \$24,000$ .
This is how much we're actually borrowing!

STEP 5

Time for the monthly payment formula!
It's a bit of a mouthful, but stick with me: $M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}$ .
Don't worry, we'll break it down!

STEP 6

$M$ is our monthly payment, $P$ is the principal (our loan amount, $\$24,000$ ), $r$ is the monthly interest rate, and $n$ is the number of payments.

STEP 7

The annual interest rate is $8\%$ , so the monthly rate is $0.08 / 12 \approx 0.00666667$ .
For 60 months, $n = 60$ .

STEP 8

Plugging everything in: $M = \$24,000 \cdot \frac{0.00666667(1+0.00666667)^{60}}{(1+0.00666667)^{60} - 1}$ .

STEP 9

Calculating carefully, we get $M \approx \$487.22$ .
That's our monthly payment for the 60-month loan!

STEP 10

The total amount paid over 60 months is $60 \cdot \$487.22 = \$29,233.20$ .

STEP 11

Subtract the initial loan amount to find the total interest: $\$29,233.20 - \$24,000 = \$5,233.20$ .

STEP 12

We use the same formula as before, but now $n = 72$ : $M = \$24,000 \cdot \frac{0.00666667(1+0.00666667)^{72}}{(1+0.00666667)^{72} - 1}$ .

STEP 13

Crunching the numbers, we get $M \approx \$429.87$ .
Lower monthly payments, but let's see what happens to the interest!

STEP 14

Total paid over 72 months: $72 \cdot \$429.87 = \$30,950.64$ .

STEP 15

Total interest: $\$30,950.64 - \$24,000 = \$6,950.64$ .
See how that extra time adds up in interest?

STEP 16

(a) 60-month payment: $\$487.22$ ; 72-month payment: $\$429.87$ . (b) 60-month interest: $\$5,233.20$ ; 72-month interest: $\$6,950.64$ .

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

1. Calculate Loan Amount2. Calculate Monthly Payment (60 months)3. Calculate Total Interest (60 months)4. Calculate Monthly Payment (72 months)5. Calculate Total Interest (72 months)

STEP 3

Alright, first things first!We've got a car that costs $32,000\$32,000$32,000 and a **down payment** of 25%25\%25%.Let's calculate the down payment amount: 0.25⋅$32,000=$8,0000.25 \cdot \$32,000 = \$8,0000.25⋅$32,000=$8,000.

STEP 4

Now, subtract that **down payment** from the **original price** to find the **loan amount**: $32,000−$8,000=$24,000\$32,000 - \$8,000 = \$24,000$32,000−$8,000=$24,000.This is how much we're actually borrowing!

STEP 5

Time for the **monthly payment** formula!It's a bit of a mouthful, but stick with me: M=P⋅r(1+r)n(1+r)n−1M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}M=P⋅(1+r)n−1r(1+r)n​.Don't worry, we'll break it down!

STEP 6

MMM is our **monthly payment**, PPP is the **principal** (our loan amount, $24,000\$24,000$24,000), rrr is the **monthly interest rate**, and nnn is the **number of payments**.

STEP 7

The **annual interest rate** is 8%8\%8%, so the **monthly rate** is 0.08/12≈0.006666670.08 / 12 \approx 0.006666670.08/12≈0.00666667.For 60 months, n=60n = 60n=60.

STEP 8

Plugging everything in: M=$24,000⋅0.00666667(1+0.00666667)60(1+0.00666667)60−1M = \$24,000 \cdot \frac{0.00666667(1+0.00666667)^{60}}{(1+0.00666667)^{60} - 1}M=$24,000⋅(1+0.00666667)60−10.00666667(1+0.00666667)60​.

STEP 9

Calculating carefully, we get M≈$487.22M \approx \$487.22M≈$487.22.That's our **monthly payment** for the 60-month loan!

STEP 10

The **total amount paid** over 60 months is 60⋅$487.22=$29,233.2060 \cdot \$487.22 = \$29,233.2060⋅$487.22=$29,233.20.

STEP 11

Subtract the **initial loan amount** to find the **total interest**: $29,233.20−$24,000=$5,233.20\$29,233.20 - \$24,000 = \$5,233.20$29,233.20−$24,000=$5,233.20.

STEP 12

We use the same formula as before, but now n=72n = 72n=72: M=$24,000⋅0.00666667(1+0.00666667)72(1+0.00666667)72−1M = \$24,000 \cdot \frac{0.00666667(1+0.00666667)^{72}}{(1+0.00666667)^{72} - 1}M=$24,000⋅(1+0.00666667)72−10.00666667(1+0.00666667)72​.

STEP 13

Crunching the numbers, we get M≈$429.87M \approx \$429.87M≈$429.87.Lower monthly payments, but let's see what happens to the interest!

STEP 14

**Total paid** over 72 months: 72⋅$429.87=$30,950.6472 \cdot \$429.87 = \$30,950.6472⋅$429.87=$30,950.64.

STEP 15

**Total interest**: $30,950.64−$24,000=$6,950.64\$30,950.64 - \$24,000 = \$6,950.64$30,950.64−$24,000=$6,950.64.See how that extra time adds up in interest?

STEP 16

(a) 60-month payment: $487.22\$487.22$487.22; 72-month payment: $429.87\$429.87$429.87. (b) 60-month interest: $5,233.20\$5,233.20$5,233.20; 72-month interest: $6,950.64\$6,950.64$6,950.64.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Calculate Loan Amount
2. Calculate Monthly Payment (60 months)
3. Calculate Total Interest (60 months)
4. Calculate Monthly Payment (72 months)
5. Calculate Total Interest (72 months)

Alright, first things first!
We've got a car that costs $\$32,000$ and a down payment of $25\%$ .
Let's calculate the down payment amount: $0.25 \cdot \$32,000 = \$8,000$ .

Now, subtract that down payment from the original price to find the loan amount: $\$32,000 - \$8,000 = \$24,000$ .
This is how much we're actually borrowing!

Time for the monthly payment formula!
It's a bit of a mouthful, but stick with me: $M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}$ .
Don't worry, we'll break it down!

$M$ is our monthly payment, $P$ is the principal (our loan amount, $\$24,000$ ), $r$ is the monthly interest rate, and $n$ is the number of payments.

The annual interest rate is $8\%$ , so the monthly rate is $0.08 / 12 \approx 0.00666667$ .
For 60 months, $n = 60$ .

Plugging everything in: $M = \$24,000 \cdot \frac{0.00666667(1+0.00666667)^{60}}{(1+0.00666667)^{60} - 1}$ .

Calculating carefully, we get $M \approx \$487.22$ .
That's our monthly payment for the 60-month loan!

The total amount paid over 60 months is $60 \cdot \$487.22 = \$29,233.20$ .

Subtract the initial loan amount to find the total interest: $\$29,233.20 - \$24,000 = \$5,233.20$ .

We use the same formula as before, but now $n = 72$ : $M = \$24,000 \cdot \frac{0.00666667(1+0.00666667)^{72}}{(1+0.00666667)^{72} - 1}$ .

Crunching the numbers, we get $M \approx \$429.87$ .
Lower monthly payments, but let's see what happens to the interest!

Total paid over 72 months: $72 \cdot \$429.87 = \$30,950.64$ .

Total interest: $\$30,950.64 - \$24,000 = \$6,950.64$ .
See how that extra time adds up in interest?

(a) 60-month payment: $\$487.22$ ; 72-month payment: $\$429.87$ . (b) 60-month interest: $\$5,233.20$ ; 72-month interest: $\$6,950.64$ .