Math / Algebra

QuestionGENERAL EDUCATION COMMON ASSIGNMENT FOR MATH 1324 Scott wants to purchase a Toyota Rave XLE. The model he wants is available for \ $26,789. Scott has saved \$4,000 for a down payment, and the dealer has offered$ \ $3,500$ for his trade-in. He is considening three financing options Toyota's financial department is offering O\% interest for three years Scott's credit union is offering $1.98 \%$ compounded monthly for four years His bank is offering $2.97 \%$ compounded monthly for five vears Your Assignment: (a) What would Scott's monthly loan payment be for each option? (b) What total amount of interest would Scot pay for each option? (c) Describe the advantages and disadvantages of cach financing opt on (d) it you were Scott, which of these three fimancink copt ams would ta- Use and why?

Studdy Solution

STEP 1

What is this asking? We need to figure out which loan is best for Scott when buying a car, by comparing monthly payments and total interest paid for three different loan options. Watch out! Don't forget to subtract the down payment and trade-in value from the car price to get the loan amount!
Also, be careful with the interest rates and compounding periods.

STEP 2

1. Calculate Loan Amount
2. 0% Interest Loan Calculations
3. Credit Union Loan Calculations
4. Bank Loan Calculations
5. Compare and Choose

STEP 3

Alright, let's start by figuring out how much Scott actually needs to borrow.
The car costs $\$26,789$ , and he's got a $\$4,000$ down payment and a $\$3,500$ trade-in.

STEP 4

So, we subtract those from the car price: $\$26,789 - \$4,000 - \$3,500 = \$19,289$ .
This means Scott needs to borrow $\$19,289$ .
Let's call this the principal, $P = \$19,289$ .

STEP 5

This one's easy peasy!
With 0% interest, Scott only needs to pay back the principal over three years.
Since there are 12 months in a year, that's $3 \cdot 12 = 36$ months.

STEP 6

So, the monthly payment is just the principal divided by the number of months: $\$19,289 / 36 \approx \$535.81$ .
And the total interest paid is... well, $\$0$ !
Zero interest, zero extra cost.
Amazing!

STEP 7

Now things get a little more interesting!
The credit union offers $1.98\%$ interest compounded monthly for four years.
First, let's convert the annual interest rate to a monthly interest rate: $0.0198 / 12 = 0.00165$ .

STEP 8

Next, we calculate the number of months in four years: $4 \cdot 12 = 48$ months.

STEP 9

We can use the monthly loan payment formula: $M = \frac{P \cdot r \cdot (1+r)^n}{(1+r)^n - 1}$ , where $M$ is the monthly payment, $P$ is the principal ( $\$19,289$ ), $r$ is the monthly interest rate ( $0.00165$ ), and $n$ is the number of months ( $48$ ).

STEP 10

Plugging in the values: $M = \frac{\$19,289 \cdot 0.00165 \cdot (1+0.00165)^{48}}{(1+0.00165)^{48} - 1} \approx \$427.94$ .

STEP 11

To find the total interest paid, we multiply the monthly payment by the number of months and subtract the principal: $\$427.94 \cdot 48 - \$19,289 \approx \$1,052.12$ .

STEP 12

The bank offers $2.97\%$ interest compounded monthly for five years.
The monthly interest rate is $0.0297 / 12 = 0.002475$ .

STEP 13

The number of months is $5 \cdot 12 = 60$ months.

STEP 14

Using the same monthly loan payment formula as before, with $P = \$19,289$ , $r = 0.002475$ , and $n = 60$ : $M = \frac{\$19,289 \cdot 0.002475 \cdot (1+0.002475)^{60}}{(1+0.002475)^{60} - 1} \approx \$362.14$ .

STEP 15

The total interest paid is $\$362.14 \cdot 60 - \$19,289 \approx \$2,439.40$ .

STEP 16

Let's recap!
The 0% loan has a monthly payment of $\$535.81$ with $\$0$ interest.
The credit union loan has a monthly payment of $\$427.94$ with $\$1,052.12$ interest.
The bank loan has a monthly payment of $\$362.14$ with $\$2,439.40$ interest.

STEP 17

(a) Monthly payments: 0% - $\$535.81$ ; Credit Union - $\$427.94$ ; Bank - $\$362.14$ . (b) Total interest: 0% - $\$0$ ; Credit Union - $\$1,052.12$ ; Bank - $\$2,439.40$ . (c) 0%: highest payment, but no interest!
Credit Union: lower payment than 0%, but some interest.
Bank: lowest payment, but most interest overall. (d) Many valid answers, depending on Scott's financial situation.
If he can afford the higher payment, the 0% loan is best.
If he needs a lower payment, the credit union is a good compromise.

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Calculate Loan Amount
2. 0% Interest Loan Calculations
3. Credit Union Loan Calculations
4. Bank Loan Calculations
5. Compare and Choose

STEP 3

Alright, let's start by figuring out how much Scott actually needs to borrow.
The car costs $\$26,789$ , and he's got a $\$4,000$ down payment and a $\$3,500$ trade-in.

STEP 4

So, we subtract those from the car price: $\$26,789 - \$4,000 - \$3,500 = \$19,289$ .
This means Scott needs to borrow $\$19,289$ .
Let's call this the principal, $P = \$19,289$ .

STEP 5

This one's easy peasy!
With 0% interest, Scott only needs to pay back the principal over three years.
Since there are 12 months in a year, that's $3 \cdot 12 = 36$ months.

STEP 6

So, the monthly payment is just the principal divided by the number of months: $\$19,289 / 36 \approx \$535.81$ .
And the total interest paid is... well, $\$0$ !
Zero interest, zero extra cost.
Amazing!

STEP 7

Now things get a little more interesting!
The credit union offers $1.98\%$ interest compounded monthly for four years.
First, let's convert the annual interest rate to a monthly interest rate: $0.0198 / 12 = 0.00165$ .

STEP 8

Next, we calculate the number of months in four years: $4 \cdot 12 = 48$ months.

STEP 9

We can use the monthly loan payment formula: $M = \frac{P \cdot r \cdot (1+r)^n}{(1+r)^n - 1}$ , where $M$ is the monthly payment, $P$ is the principal ( $\$19,289$ ), $r$ is the monthly interest rate ( $0.00165$ ), and $n$ is the number of months ( $48$ ).

STEP 10

Plugging in the values: $M = \frac{\$19,289 \cdot 0.00165 \cdot (1+0.00165)^{48}}{(1+0.00165)^{48} - 1} \approx \$427.94$ .

STEP 11

To find the total interest paid, we multiply the monthly payment by the number of months and subtract the principal: $\$427.94 \cdot 48 - \$19,289 \approx \$1,052.12$ .

STEP 12

The bank offers $2.97\%$ interest compounded monthly for five years.
The monthly interest rate is $0.0297 / 12 = 0.002475$ .

STEP 13

The number of months is $5 \cdot 12 = 60$ months.

STEP 14

Using the same monthly loan payment formula as before, with $P = \$19,289$ , $r = 0.002475$ , and $n = 60$ : $M = \frac{\$19,289 \cdot 0.002475 \cdot (1+0.002475)^{60}}{(1+0.002475)^{60} - 1} \approx \$362.14$ .

STEP 15

The total interest paid is $\$362.14 \cdot 60 - \$19,289 \approx \$2,439.40$ .

STEP 16

Let's recap!
The 0% loan has a monthly payment of $\$535.81$ with $\$0$ interest.
The credit union loan has a monthly payment of $\$427.94$ with $\$1,052.12$ interest.
The bank loan has a monthly payment of $\$362.14$ with $\$2,439.40$ interest.

STEP 17

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. 0% Interest Loan Calculations3. Credit Union Loan Calculations4. Bank Loan Calculations5. Compare and Choose

STEP 3

Alright, let's **start** by figuring out how much Scott actually needs to borrow.The car costs $26,789\$26,789$26,789, and he's got a $4,000\$4,000$4,000 down payment and a $3,500\$3,500$3,500 trade-in.

STEP 4

So, we **subtract** those from the car price: $26,789−$4,000−$3,500=$19,289 \$26,789 - \$4,000 - \$3,500 = \$19,289 $26,789−$4,000−$3,500=$19,289.This means Scott needs to borrow $19,289\$19,289$19,289.Let's call this the **principal**, P=$19,289P = \$19,289P=$19,289.

STEP 5

This one's easy peasy!With **0% interest**, Scott only needs to pay back the principal over three years.Since there are 12 months in a year, that's 3⋅12=363 \cdot 12 = 363⋅12=36 months.

STEP 6

So, the **monthly payment** is just the principal divided by the number of months: $19,289/36≈$535.81\$19,289 / 36 \approx \$535.81$19,289/36≈$535.81.And the **total interest paid** is... well, $0\$0$0!Zero interest, zero extra cost.Amazing!

STEP 7

Now things get a little more interesting!The credit union offers 1.98%1.98\%1.98% interest **compounded monthly** for four years.First, let's convert the **annual interest rate** to a **monthly interest rate**: 0.0198/12=0.00165 0.0198 / 12 = 0.00165 0.0198/12=0.00165.

STEP 8

Next, we calculate the number of months in four years: 4⋅12=484 \cdot 12 = 484⋅12=48 months.

STEP 9

STEP 10

Plugging in the values: M=$19,289⋅0.00165⋅(1+0.00165)48(1+0.00165)48−1≈$427.94M = \frac{\$19,289 \cdot 0.00165 \cdot (1+0.00165)^{48}}{(1+0.00165)^{48} - 1} \approx \$427.94M=(1+0.00165)48−1$19,289⋅0.00165⋅(1+0.00165)48​≈$427.94.

STEP 11

To find the **total interest paid**, we multiply the monthly payment by the number of months and subtract the principal: $427.94⋅48−$19,289≈$1,052.12 \$427.94 \cdot 48 - \$19,289 \approx \$1,052.12 $427.94⋅48−$19,289≈$1,052.12.

STEP 12

The bank offers 2.97%2.97\%2.97% interest compounded monthly for five years.The **monthly interest rate** is 0.0297/12=0.0024750.0297 / 12 = 0.0024750.0297/12=0.002475.

STEP 13

The number of months is 5⋅12=605 \cdot 12 = 605⋅12=60 months.

STEP 14

STEP 15

The **total interest paid** is $362.14⋅60−$19,289≈$2,439.40 \$362.14 \cdot 60 - \$19,289 \approx \$2,439.40 $362.14⋅60−$19,289≈$2,439.40.

STEP 16

STEP 17

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. 0% Interest Loan Calculations
3. Credit Union Loan Calculations
4. Bank Loan Calculations
5. Compare and Choose

Alright, let's start by figuring out how much Scott actually needs to borrow.
The car costs $\$26,789$ , and he's got a $\$4,000$ down payment and a $\$3,500$ trade-in.

So, we subtract those from the car price: $\$26,789 - \$4,000 - \$3,500 = \$19,289$ .
This means Scott needs to borrow $\$19,289$ .
Let's call this the principal, $P = \$19,289$ .

This one's easy peasy!
With 0% interest, Scott only needs to pay back the principal over three years.
Since there are 12 months in a year, that's $3 \cdot 12 = 36$ months.

So, the monthly payment is just the principal divided by the number of months: $\$19,289 / 36 \approx \$535.81$ .
And the total interest paid is... well, $\$0$ !
Zero interest, zero extra cost.
Amazing!

Now things get a little more interesting!
The credit union offers $1.98\%$ interest compounded monthly for four years.
First, let's convert the annual interest rate to a monthly interest rate: $0.0198 / 12 = 0.00165$ .

Next, we calculate the number of months in four years: $4 \cdot 12 = 48$ months.

Plugging in the values: $M = \frac{\$19,289 \cdot 0.00165 \cdot (1+0.00165)^{48}}{(1+0.00165)^{48} - 1} \approx \$427.94$ .

To find the total interest paid, we multiply the monthly payment by the number of months and subtract the principal: $\$427.94 \cdot 48 - \$19,289 \approx \$1,052.12$ .

The bank offers $2.97\%$ interest compounded monthly for five years.
The monthly interest rate is $0.0297 / 12 = 0.002475$ .

The number of months is $5 \cdot 12 = 60$ months.

The total interest paid is $\$362.14 \cdot 60 - \$19,289 \approx \$2,439.40$ .