/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 Marie can afford a \(\$ 250\) pe... [FREE SOLUTION] | 91Ó°ÊÓ

91Ó°ÊÓ

Marie can afford a \(\$ 250\) per month car payment. She's found a 5 year loan at \(7 \%\) interest. a. How expensive of a car can she afford? b. How much total money will she pay the loan company? c. How much of that money is interest?

Short Answer

Expert verified
a. $13,195.15 b. $15,000 c. $1,804.85

Step by step solution

01

Understand the Loan Structure

Marie wants to buy a car using a loan with a fixed interest rate of 7% over 5 years. The loan requires monthly payments, which she can afford up to $250 per month.
02

Use the Loan Amortization Formula

To find out how much car Marie can afford, calculate the present value of the annuity (the car loan) using the formula for a loan: \[ P = \frac{C \times (1 - (1 + r)^{-n})}{r} \]where:- \(P\) is the loan principal (amount Marie can borrow for the car),- \(C\) is the monthly payment \($250\),- \(r\) is the monthly interest rate (7% per year or \( \frac{0.07}{12} \)),- \(n\) is the total number of payments (5 years or 60 months).
03

Calculate the Monthly Interest Rate

Convert the annual interest rate to a monthly interest rate:\[ r = \frac{0.07}{12} \approx 0.0058333 \]
04

Calculate Total Number of Payments

Calculate the total number of monthly payments over the span of the loan:\[ n = 5 \times 12 = 60 \]
05

Plug Values into the Amortization Formula

Substitute the values into the formula:\[ P = \frac{250 \times (1 - (1 + 0.0058333)^{-60})}{0.0058333} \]
06

Compute the Loan Principal

Calculate the result from the amortization formula:\[ P \approx \frac{250 \times (1 - (1.0058333)^{-60})}{0.0058333} \approx \frac{250 \times (1 - 0.694008)}{0.0058333} \approx \frac{76.49795}{0.0058333} \approx 13195.15 \]Marie can afford a car worth approximately \( \$ 13,195.15 \).
07

Calculate Total Money Paid

To find the total amount paid to the loan company:\[ \text{Total payments} = \text{Monthly Payment} \times \text{Total Number of Payments} \]\[ \text{Total payments} = 250 \times 60 = 15000 \]
08

Calculate Total Interest Paid

To find the total interest paid, subtract the principal from the total payments:\[ \text{Interest paid} = \text{Total payments} - P \]\[ \text{Interest paid} = 15000 - 13195.15 = 1804.85 \]
09

Review Answers

To summarize:- The most expensive car Marie can afford is \( \\( 13,195.15 \).- She will pay a total of \( \\) 15,000 \) to the loan company.- The total interest paid will be \( \$ 1,804.85 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interest Rate Calculation
Interest rate calculation is crucial when taking out a loan, as it affects how much you'll end up paying over time. The interest rate on a loan, like Marie's car loan at 7%, represents the cost of borrowing money. This rate is often given as an annual percentage but needs to be converted for monthly payments.

To convert an annual interest rate to a monthly rate, divide by 12 (the number of months in a year). In Marie's case, the monthly interest rate is \[ r = \frac{0.07}{12} \approx 0.0058333 \]. Understanding this conversion is pivotal when calculating loan principles and monthly payments, as seen in her car loan scenario. By using the appropriate formula, individuals can better understand their financial obligations and manage their budgets accordingly.
  • Annual interest rates need to be converted for monthly payment calculations.
  • Interest rate calculation helps predict the total cost of a loan.
Financial Literacy
Financial literacy refers to understanding and effectively using various financial skills, including personal financial management, budgeting, and investing. It's the cornerstone of making informed decisions about money, like whether to take out a loan.
Marie faced questions about how much car she can afford and the total cost of her loan. By applying financial literacy principles, she understood that the maximum car price she can handle considers her budget limit of $250 monthly payments.
Being financially literate enables you to evaluate loan terms and understand the long-term financial impact. It helps in assessing how much of a purchase will be repaid as interest, ultimately shedding light on additional costs that may not be apparent at first glance. Becoming financially literate involves gaining knowledge about interest, loans, savings, and more, empowering you to make wise financial decisions.
  • Helps assess personal budget limits and affordability.
  • Empowers better decision making with loans and interest calculations.
Monthly Payments
Monthly payments are regular payments made to repay a loan over a term, in Marie's case, each month for 60 months. These payments typically comprise both principal and interest, meaning you're not only paying off the car's cost but also the cost of borrowing the money.
The amount of the monthly payment is influenced by several factors: - Loan principal: The amount borrowed, which was approximately $13,195 for Marie's car. - Interest rate: The cost of borrowing, converted to a monthly figure. - Loan term: The duration over which the loan is to be repaid, impacting the size of each payment. Understanding monthly payments helps in planning and maintaining financial stability. They ensure that the borrowed amount is repaid systematically. Marie's commitment to a $250 monthly payment over five years will result in her paying a total of $15,000; a substantial part of that is the interest, illustrating why comprehension of the loan structure is vital.
  • Important for budgeting and financial planning.
  • Includes both the loan principal and interest.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

You deposit \(\$ 10,000\) in an account earning \(4 \%\) interest compounded monthly. a. How much will you have in the account in 25 years? b. How much interest will you earn?

Suppose that 10 years ago you bought a home for \(\$ 110,000\), paying \(10 \%\) as a down payment, and financing the rest at \(9 \%\) interest for 30 years. a. Let's consider your existing mortgage: i. How much money did you pay as your down payment? ii. How much money was your mortgage (loan) for? iii. What is your current monthly payment? iv. How much total interest will you pay over the life of the loan? b. This year, you check your loan balance. Only part of your payments have been going to pay down the loan; the rest has been going towards interest. You see that you still have \(\$ 88,536\) left to pay on your loan. Your house is now valued at \(\$ 150,000\). i. How much of the loan have you paid off? (i.e., how much have you reduced the loan balance by? Keep in mind that interest is charged each month - it's not part of the loan balance.) ii. How much money have you paid to the loan company so far? iii. How much interest have you paid so far? iv. How much equity do you have in your home (equity is value minus remaining debt) c. Since interest rates have dropped, you consider refinancing your mortgage at a lower \(6 \%\) rate. i. If you took out a new 30 year mortgage at \(6 \%\) for your remaining loan balance, what would your new monthly payments be? ii. How much interest will you pay over the life of the new loan? d. Notice that if you refinance, you are going to be making payments on your home for another 30 years. In addition to the 10 years you've already been paying, that's 40 years total. i. How much will you save each month because of the lower monthly payment? ii. How much total interest will you be paying (you need to consider the amount from \(2 \mathrm{c}\) and \(3 \mathrm{~b}\) ) iii. Does it make sense to refinance? (there isn't a correct answer to this question. Just give your opinion and your reason)

You deposit \$200 each month into an account earning \(3 \%\) interest compounded monthly. a. How much will you have in the account in 30 years? b. How much total money will you put into the account? c. How much total interest will you earn?

Suppose you are looking to buy a \(\$ 5000\) face value 26 -week T-bill. If you want to earn at least \(1 \%\) annual interest, what is the most you should pay for the T-bill?

You deposit \(\$ 300\) in an account earning \(5 \%\) interest compounded annually. How much will you have in the account in 10 years?

See all solutions

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.