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Chapter 5: Problem 13

Compute the specified quantity. Round all answers to the nearest month, the nearest cent, or the nearest \(0.001 \%\), as appropriate. Simple Loans You take out a 6 -month, \(\$ 5,000\) loan at \(8 \%\) simple interest. How much would you owe at the end of the 6 months?

Short Answer

Expert verified
The total amount owed at the end of the 6 months would be $5,400.

Step by step solution

01

Write down the given information

We are given the following information: - Principal amount (P): $5,000 - Simple Interest rate (r): 8% (or 0.08 in decimal form) - Time period (t): 6 months Note that we need to convert the time period in years.
02

Convert the time period to years

To convert the time period from months to years, we simply divide the given period (6 months) by the total number of months in a year (12 months). t (in years) = \(\frac{6}{12}\) t = 0.5 years
03

Calculate the simple interest using the formula

To calculate the simple interest, we will use the formula: Simple Interest (SI) = P * r * t where P, r, and t are defined as mentioned in step 1. SI = \(5,000 * 0.08 * 0.5\) SI = \(400\) So, the simple interest after 6 months is $400.
04

Calculate the total amount owed

To find out how much would be owed at the end of the 6 months, we will add the simple interest calculated in step 3 to the principal amount. Total Amount = Principal Amount + Simple Interest Total Amount = \(5,000 + 400\) Total Amount = $5,400 After 6 months, a total of $5,400 would be owed.

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Key Concepts

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

Principal Amount
The principal amount is the initial sum of money borrowed or invested, before any interest is applied. It's the base for all interest calculations in simple interest scenarios. In loan contexts, it's the amount that the borrower has taken from the lender, intending to pay back with additional interest over time.
  • For example, in our scenario, the principal amount is $5,000. This is the amount initially received by the borrower.
  • No matter the length of the loan or the interest rate, this principal amount remains unchanged in simple interest calculations until the debt is fully paid off or the investment matures.
Understanding the principal amount is crucial because it helps determine how much interest will be paid or received. The larger the principal, the higher the amount of interest accrued.
Interest Rate
The interest rate is a percentage charged on the principal by the lender or earned on an investment. It signifies the cost of borrowing money or the reward of investing it. Usually expressed per annum, it reflects the cost or yield for each year, making it easier to compare different financial products.
  • In our example, the interest rate is 8%. This means that annually, 8% of the principal will be added as simple interest. To express this in decimal form for calculations, divide by 100, yielding 0.08.
  • An 8% simple interest rate means for every $100 borrowed, $8 is charged as interest each year, assuming the period is exactly one year.
The interest rate influences the total cost of borrowing and guides how much extra will be paid over the lifecycle of the loan. A clear understanding can assist in better financial planning and budgeting.
Loan Period
The loan period is the duration over which the loan is active, and interest is charged or earned. In simple interest calculations, understanding the time frame is essential for accurate computation.
  • In the problem presented, the loan period is 6 months. However, for simplicity, calculations often require converting this period into years. This is done by dividing the number of months by 12 (the number of months in a year).
  • Thus, for our calculations, 6 months converts to 0.5 years (since \(\frac{6}{12} = 0.5\)).
The length of the loan period directly impacts the total interest accrued; the longer the time, the more interest will be accumulated. This period aids borrowers and investors in understanding their commitment or the maturity timeline of their investments.
Total Amount Owed
The total amount owed at the end of a loan period includes both the principal and the interest accrued over that period. Calculating this amount helps to determine the exact payment required to settle the debt entirely.
  • To determine the total amount owed, add the calculated simple interest to the principal. In our case, this is the $5,000 principal plus the $400 simple interest.
  • This gives us a total of $5,400 due at the end of the 6-month period.
Understanding the total amount owed is crucial for financial planning as it informs the borrower of their total debt obligation by the end of the loan term. This insight is essential for avoiding surprises and ensuring that funds are allocated appropriately to meet repayment schedules.

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Most popular questions from this chapter

At the start of 1985, the incidence of AIDS was doubling every 6 months and 40,000 cases had been reported in the United States. Assuming this trend would have continued, determine when, to the nearest tenth of a year, the number of cases would have reached 1 million.

Find the periodic payments necessary to accumulate the amounts given in Exercises \(7-12\) in a sinking fund. (Assume end-of-period deposits and compounding at the same intervals as deposits.) \(\$ 20,000\) in a fund paying \(5 \%\) per year, with monthly payments for 5 years, if the fund contains \(\$ 10,000\) at the start

You take out an adjustable rate mortgage for \(\$ 100,000\) for 20 years. For the first 5 years, the rate is \(4 \%\). It then rises to \(7 \%\) for the next 10 years, and then \(9 \%\) for the last 5 years. What are your monthly payments in the first 5 years, the next 10 years, and the last 5 years? (Assume that each time the rate changes, the payments are recalculated to amortize the remaining debt if the interest rate were to remain constant for the remaining life of the mortgage.)

Are based on the following table, which shows the 2008 annual inflation rates in several Latin American countries. \({ }^{13}\) Assume that the rates shown continue indefinitely. $$\begin{array}{|l|c|c|c|c|c|c|c|} \hline \text { Country } & \text { Argentina } & \text { Brazil } & \text { Bolivia } & \text { Nicaragua } & \text { Venezuela } & \text { Mexico } & \text { Uruguay } \\ \hline \text { Currency } & \text { Peso } & \text { Real } & \text { Boliviano } & \begin{array}{c} \text { Gold } \\ \text { cordoba } \end{array} & \text { Bolivar } & \text { Peso } & \text { Peso } \\ \hline \begin{array}{l} \text { Inflation } \\ \text { Rate (\%) } \end{array} & 9.2 & 6.3 & 15.1 & 13.8 & 25.7 & 5.0 & 8.5 \\ \hline \end{array}$$ If an item in Argentina now costs 1,000 pesos, what do you expect it to cost 5 years from now? (Answer to the nearest peso.)

Determine the periodic payments on the loans given: \(\$ 100,000\) borrowed at \(5 \%\) for 20 years, with quarterly payments
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