/*! 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 22 The Flemings secured a bank loan... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 22

The Flemings secured a bank loan of $$\$ 288,000$$ to help finance the purchase of a house. The bank charges interest at a rate of \(9 \% /\) year on the unpaid balance. and interest computations are made at the end of each month. The Flemings have agreed to repay the loan in equal monthly installments over \(25 \mathrm{yr}\). What should be the size of each repayment if the loan is to be amortized at the end of the term?

Short Answer

Expert verified
The size of each repayment should be approximately \(\$ 2,349.78\). This is the amount the Flemings have to pay each month to amortize the loan at the end of the term (25 years).

Step by step solution

01

Find the monthly interest rate

To find the monthly interest rate, we need to convert the annual interest rate to a decimal and then divide it by 12 (the number of months in a year): \[r = \frac{9\%}{12} = \frac{0.09}{12} = 0.0075\]
02

Find the total number of payments

The loan will be repaid in equal monthly installments for 25 years. So we need to find the total number of payments by multiplying the number of years (25) by the number of months per year (12): \[n = 25\times 12 = 300\]
03

Plug in the values in the loan formula and solve for PMT

Now that we have the monthly interest rate (r) and the total number of payments (n), we can plug these values into the loan formula along with the loan amount (PV) to find the monthly payment (PMT): \[ \$ 288,000 = PMT \frac{1 - (1 + 0.0075)^{-300}}{0.0075} \]
04

Solve for PMT to find the size of each repayment

Rearrange the formula and solve for PMT: \[ PMT = \frac{\$ 288,000 \times 0.0075}{1 - (1 + 0.0075)^{-300}} \] Calculate the monthly payment: \[PMT \approx \$ 2,349.78\] The size of each repayment should be approximately \(\$ 2,349.78\). This is the amount the Flemings have to pay each month to amortize the loan at the end of the term (25 years).

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 91影视!

Key Concepts

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

Monthly Interest Rate
When taking a loan, interest is applied periodically. To understand costs accurately, it's vital to convert the annual rate into a monthly interest rate. This enables easy calculation of monthly payments. The Flemings have an annual interest rate of 9%.
To find the monthly interest rate:
  • Convert the annual rate to a decimal form: \(0.09\)
  • Divide by 12 (the number of months in a year): \(\frac{0.09}{12} = 0.0075\)
The monthly interest rate is thus 0.75%. With this rate, one can easily assess the monthly cost of borrowing for the Flemings' loan.
Loan Formula
The loan formula is a mathematical tool that helps in determining monthly payments. It incorporates various factors like the principal amount, interest rate, and loan tenure. For borrowers like the Flemings, understanding this formula is crucial to plan repayments.
The loan formula is expressed as:\[ PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \]Where:
  • \(PV\) is the principal (initial loan amount)
  • \(PMT\) is the monthly payment
  • \(r\) is the monthly interest rate
  • \(n\) is the total number of payments
By plugging the known values into this formula, one can solve for \(PMT\) to determine the monthly installment amount.
Monthly Installments
Monthly installments are crucial for budgeting loan repayments. They ensure systematic payment without letting debts accumulate. In the Flemings' case, each monthly installment is calculated using the loan formula.
To find the monthly installment:
  • Use the loan formula with the given loan information.
  • Substitute the known values: principal \(\\(288,000\), monthly rate \(0.0075\), and number of payments \(300\).
  • Calculate \(PMT\):\[PMT = \frac{\\)288,000 \times 0.0075}{1 - (1 + 0.0075)^{-300}}\]
  • Resulting in a monthly installment of approximately \(\$2,349.78\).
This amount is what the Flemings will need to pay monthly for 25 years to fully amortize their loan.
Amortization Schedule
An amortization schedule is a detailed, month-by-month accounting of the loan repayment process. It shows each payment over time, breaking down the amounts applied to interest and principal.
For the Flemings:
  • Their total payment period is 300 months (25 years).
  • Each monthly payment is approximately \(\$2,349.78\).
  • Initially, a larger portion of the payment goes towards interest.
  • Over time, more of the payment reduces the principal as the interest on the remaining loan decreases.
An amortization schedule helps borrowers understand how their payments are applied, ensuring effective budgeting over the life of the loan.

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

Since he was 22 years old, Ben has been depositing $$\$ 200$$ at the end of each month into a taxfree retirement account earning interest at the rate of \(6.5 \%\) /year compounded monthly. Larry, who is the same age as Ben, decided to open a tax-free retirement account 5 yr after Ben opened his. If Larry's account earns interest at the same rate as Ben's, determine how much Larry should deposit each month into his account so that both men will have the same amount of money in their accounts at age 65 .

If Jackson deposits $$\$ 100$$ at the end of each month in a savings account earning interest at the rate of \(8 \%\) /year compounded monthly, how much will he have on deposit in his savings account at the end of \(6 \mathrm{yr}\), assuming that he makes no withdrawals during that period?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The future value of an annuity can be found by adding together all the payments that are paid into the account.

Suppose an initial investment of $$\$ P$$ grows to an accumulated amount of $$\$ A$$ in \(t\) yr. Show that the effective rate (annual effective yield) is $$ r_{\text {eff }}=(A / P)^{1 / t}-1 $$ Use the formula given in Exercise 71 to solve Exercises \(72-76 .\)

The management of Gibraltar Brokerage Services anticipates a capital expenditure of $$\$ 20,000$$ in 3 yr for the purchase of new computers and has decided to set up a sinking fund to finance this purchase. If the fund earns interest at the rate of \(10 \% /\) year compounded quarterly, determine the size of each (equal) quarterly installment that should be deposited in the fund.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation
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.