/*! 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 10 Suppose you borrow \(P\) dollars... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 10

Suppose you borrow \(P\) dollars at a monthly interest rate of \(r\) (as a decimal) and wish to pay off the loan in \(t\) months. Then your monthly payment can be calculated using $$ M=\frac{\operatorname{Pr}(1+r)^{t}}{(1+r)^{t}-1} \text { dollars. } $$ Remember that for monthly compounding, you get the monthly rate by dividing the APR by 12. Suppose you borrow \(\$ 3500\) at \(9 \%\) APR (meaning that you use \(r=0.09 / 12\) in the preceding formula) and pay it back in 2 years. a. What is your monthly payment? b. Let's look ahead to the time when the loan is paid off. i. What is the total amount you paid to the bank? ii. How much of that was interest? c. The amount \(B\) that you still owe the bank after making \(k\) monthly payments can be calculated using the variables \(r, P\), and \(t\). The relationship is given by $$ B=P \times\left(\frac{(1+r)^{t}-(1+r)^{k}}{(1+r)^{t}-1}\right) \text { dollars . } $$ i. How much do you still owe the bank after 1 year of payments? ii. An amortization table is a table that shows how much you still owe the bank after each payment. Make an amortization table for this loan.

Short Answer

Expert verified
The monthly payment is $159.89. Total payment to the bank is $3837.36, with $337.36 in interest. After one year, $1864.68 is still owed.

Step by step solution

01

Determine Monthly Interest Rate

First, calculate the monthly interest rate from the annual percentage rate (APR). Given that the APR is 9%, convert it to decimal form and divide by 12: \( r = \frac{0.09}{12} = 0.0075 \).
02

Understanding Time Period Conversion

The loan period is given in years (2 years). Convert it into months since the formula requires months: \( t = 2 \times 12 = 24 \) months.
03

Calculate Monthly Payment (M)

Use the monthly payment formula \( M = \frac{Pr(1+r)^t}{(1+r)^t-1} \) with \( P = 3500 \), \( r = 0.0075 \), and \( t = 24 \). Plug these into the formula:\[ M = \frac{3500 \times 0.0075 \times (1+0.0075)^{24}}{(1+0.0075)^{24} - 1} \]Calculate it to find \( M \approx 159.89 \), which means your monthly payment is approximately $159.89.
04

Calculate Total Paid Amount

The total amount paid after 2 years can be found by multiplying the monthly payment by the number of months: \( ext{Total Paid} = M \times t = 159.89 \times 24 = 3837.36 \).
05

Calculate Interest Paid

To find the total interest paid, subtract the principal from the total amount paid: \( ext{Interest} = 3837.36 - 3500 = 337.36 \).
06

Calculate Remaining Balance After 1 Year (12 Payments)

To find out how much is still owed after 12 payments (1 year), use the remaining balance formula:\[ B = 3500 \times \left(\frac{(1+0.0075)^{24} - (1+0.0075)^{12}}{(1+0.0075)^{24} - 1}\right) \]Calculate this to find that \( B \approx 1864.68 \), meaning you still owe $1864.68 after one year.
07

Create an Amortization Table

An amortization table would detail each month's starting balance, the monthly payment amount, interest paid, principal paid, and ending balance. Repeat this process for each month using the formula for remaining balance, adjusting the principal and interest components based on remaining balance and monthly payment.

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 out a loan, understanding the monthly interest rate is crucial. The monthly interest rate is derived from dividing the Annual Percentage Rate (APR) by 12, as it is an annual rate and needs to be converted to a monthly format. Let's break this down:
  • Suppose the APR is 9%.
  • Convert the percentage to a decimal to get 0.09.
  • Divide by 12 to reflect monthly compounding: \( r = \frac{0.09}{12} = 0.0075 \).
This monthly rate is what impacts your monthly payments directly, as it is the rate used in the loan payment formula. Understanding this rate helps in anticipating how much interest will accrue monthly, influencing your total payment amount over time.
APR Calculation
The Annual Percentage Rate (APR) represents the annual cost of borrowing, expressed as a percentage. It includes the nominal interest rate and any additional costs or fees associated with the loan. However, for consistency and precise calculation of loan payments, the APR needs conversion into a monthly interest rate.Here's why the APR conversion is vital:
  • Many loans have monthly payments, so we need to know how much interest is affecting each payment cycle.
  • For accurate calculation of the monthly payment, use the formula: \( M = \frac{Pr(1+r)^t}{(1+r)^t-1} \), where \( r \) is the monthly interest rate.
Understanding APR conversion ensures you're not overpaying in interest and helps in comparing loan offers effectively.
Amortization Table
An amortization table is an excellent tool that shows how loan payments break down into interest and principal over time. This table provides a clear picture of how much you still owe, each month's interest costs, and how your payments reduce the overall loan balance. Creating an amortization table involves:
  • Starting with the total loan amount.
  • Calculating each monthly payment's interest and principal portions.
  • Subtracting the principal paid from the outstanding balance.
  • Updating these numbers consistently over the entire loan period.
The amortization table is crucial for loan tracking as it helps borrowers see the progress of their payments and aids in managing financial planning effectively.
Remaining Loan Balance
Calculating the remaining loan balance after several payments helps you understand how much is still owed and how close you are to paying off the debt. The formula to calculate the remaining balance \( B \) after \( k \) payments is:\[ B = P \times \left(\frac{(1+r)^t - (1+r)^k}{(1+r)^t - 1}\right) \]This formula allows you to see the remaining amount to pay at any point during the loan term.Key points:
  • \( t \) refers to the total months, while \( k \) is the number of payments already made.
  • Understanding the remaining balance enables better budgeting and potential early repayment strategies.
  • It provides a snapshot of the loan status, helping in decision-making for refinancing or additional payments.
Tracking the remaining balance gives you confidence in managing your loan efficiently, avoiding surprises.

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

The resale value \(V\), in thousands of dollars, of a boat is a function of the number of years \(t\) since the start of 2001 , and the formula is $$ V=12.5-1.1 t $$ a. Calculate \(V(3)\) and explain in practical terms what your answer means. b. In what year will the resale value be 7 thousand dollars? c. Solve for \(t\) in the formula above to obtain a formula expressing \(t\) as a function of \(V\). d. In what year will the resale value be \(4.8\) thousand dollars?

When a skydiver jumps from an airplane, her downward velocity, in feet per second, before she opens her parachute is given by \(v=176\left(1-0.834^{t}\right)\), where \(t\) is the number of seconds that have elapsed since she jumped from the airplane. We found earlier that the terminal velocity for the skydiver is 176 feet per second. How long does it take to reach \(90 \%\) of terminal velocity?

One model for the spread of epidemics gives the number of newly infected individuals \(t\) days after the outbreak of the epidemic as $$ \text { New cases }=\frac{\beta n(n+1)^{2} e^{(n+1) \beta t}}{\left(n+e^{(n+1) \beta t}\right)^{2}}. $$ Here \(n\) is the total number of people we expect to be infected over the course of the epidemic, and \(\beta\) depends on the nature of the infection as well as on other environmental factors. For a certain epidemic, the number of new cases is $$ \text { New cases }=\frac{75,150 e^{0.3 t}}{\left(500+e^{0.3 t}\right)^{2}} \text {. } $$ a. Make a graph of the number of new cases versus days since the outbreak. Include times up to 30 days. b. What is the greatest number of new cases we expect to see in 1 day, and when does that occur? c. The local medical facilities can handle no more than 25 new cases per day. During what time period will it be necessary to recruit help from outside sources?

A mole of a chemical compound is a fixed number, \({ }^{16}\) like a dozen, of molecules (or atoms in the case of an element) of that compound. A mole of water, for example, is about 18 grams, or just over a half an ounce in your kitchen. Chemists often use the mole as the measure of the amount of a chemical compound. A mole of carbon dioxide has a fixed mass, but the volume \(V\) that it occupies depends on pressure \(p\) and temperature \(T\); greater pressure tends to compress the gas into a smaller volume, whereas increasing temperature tends to make the gas expand into a larger volume. If we measure the pressure in atmospheres ( 1 atm is the pressure exerted by the atmosphere at sea level), the temperature in kelvins, and the volume in liters, then the relationship is given by the ideal gas law: $$ p V=0.082 T \text {. } $$ a. Solve the ideal gas law for the volume \(V\). b. What is the volume of 1 mole of carbon dioxide under \(3 \mathrm{~atm}\) of pressure at a temperature of 300 kelvins? c. Solve the ideal gas law for pressure. d. What is the pressure on 1 mole of carbon dioxide if it occupies a volume of \(0.4\) liter at a temperature of 350 kelvins? e. Solve the ideal gas law for temperature. f. At what temperature will 1 mole of carbon dioxide occupy a volume of 2 liters under a pressure of \(0.3 \mathrm{~atm}\) ?

An explosion produces a spherical shock wave whose radius \(R\) expands rapidly. The rate of expansion depends on the energy \(E\) of the explosion and the elapsed time \(t\) since the explosion. For many explosions, the relation is approximated closely by $$ R=4.16 E^{0.2} t^{0.4} . $$ Here \(R\) is the radius in centimeters, \(E\) is the energy in ergs, and \(t\) is the elapsed time in seconds. The relation is valid only for very brief periods of time, perhaps a second or so in duration. a. An explosion of 50 pounds of TNT produces an energy of about \(10^{15}\) ergs. See Figure \(2.85\). How long is required for the shock wave to reach a point 40 meters ( 4000 centimeters) away? b. A nuclear explosion releases much more energy than conventional explosions. A small nuclear device of yield 1 kiloton releases approximately \(9 \times 10^{20}\) ergs. How long would it take for the shock wave from such an explosion to reach a point 40 meters away? c. The shock wave from a certain explosion reaches a point 50 meters away in \(1.2\) seconds. How much energy was released by the explosion? The values of \(E\) in parts a and b may help you set an appropriate window. Note: In 1947 the government released film of the first nuclear explosion in 1945 , but the yield of the explosion remained classified. Sir Geoffrey Taylor used the film to determine the rate of expansion of the shock wave and so was able to publish a scientific paper \({ }^{26}\) concluding correctly that the yield was in the 20 -kiloton range.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

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.