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Chapter 8: Problem 4

In Exercises \(1-8\), the principal \(P\) is borrowed at simple interest rater for a period of time t. Find the simple interest owed for the use of the money. Assume 360 days in a year. \(P=\$ 260, r=4 \%, t=3\) years

Short Answer

Expert verified
The total simple interest owed for the use of the money is \$31.2.

Step by step solution

01

Convert Percentage Rate into Decimal

In this problem, the interest rate \(r\) is given in percentage. We must therefore first convert the rate from percentage to a decimal. The conversion is done by dividing the percentage rate by 100. Hence, \(r = \frac{4}{100} = 0.04 \).
02

Substitute the values into the Simple Interest Formula

Once the values have been correctly interpreted, they are substituted into the Simple Interest formula \(I = Prt\). Substituting the values, we get \(I = 260 * 0.04 * 3\)
03

Calculate the Simple Interest

After substituting the values into the formula, simplify the arithmetic to find the simple interest. Hence, the calculation becomes \(I = \$ 31.2\).

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Key Concepts

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

Percentage Rate Conversion
To start calculating simple interest, you first need to convert the given percentage rate into a more manageable decimal form. This is because percentages represent a portion of 100, and dealing with decimals allows us to perform calculations more easily.
  • Take the percentage rate given; for this example, it's 4%.
  • Divide the percentage by 100 to convert it into a decimal. So, \( r = \frac{4}{100} = 0.04 \), meaning 4% is equivalent to 0.04 when used in calculations.
This step is important to ensure that the units match and that you have an accurate basis for further calculations.
Simple Interest Formula
Simple interest is calculated using a straightforward formula, which is perfect for understanding how interest accumulates over time with a fixed rate. This formula is: \[ I = Prt \]where:
  • \(I\) is the simple interest.
  • \(P\) is the principal amount, or the initial amount of money borrowed or invested.
  • \(r\) is the annual interest rate in decimal form (not percentage).
  • \(t\) is the time the money is borrowed or invested, expressed in years.
Plugging the known values from the exercise - \(P = 260\), \(r = 0.04\), and \(t = 3\) years - into the formula gives us the equation: \[ I = 260 \times 0.04 \times 3 \].This formula gives us a way to determine how much interest will be owed at the end of the time period.
Arithmetic Simplification
Once you've set up the formula with the given values, you鈥檒l need to perform the arithmetic operations to simplify it and find the interest. This step is crucial as it helps you to focus on basic calculations to get the simple interest amount.
  • Multiply the principal \(260\) by the rate \(0.04\). This first multiplication gives you \(10.4\).
  • Next, multiply the result \(10.4\) by the time period \(3\) years, which results in \(31.2\).
Concluding this calculation provides the simple interest owed, which is \(\$ 31.2\). The arithmetic simplification highlights the logical steps needed to reach the final answer, each based on simple multiplication.

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Most popular questions from this chapter

In Exercises 1-10, use $$ P M T=\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} . $$ Round answers to the nearest dollar. Suppose that you are thinking about buying a car and have narrowed down your choices to two options: The new-car option: The new car costs \(\$ 68,000\) and can be financed with a four-year loan at \(7.14 \%\). The used-car option: A three-year old model of the same car costs \(\$ 28,000\) and can be financed with a four-year loan at \(7.92 \%\). What is the difference in monthly payments between financing the new car and financing the used car?

What is a mortgage?

a. Suppose that between the ages of 22 and 40 , you contribute \(\$ 3000\) per year to a \(401(\mathrm{k})\) and your employer contributes \(\$ 1500\) per year on your behalf. The interest rate is \(8.3 \%\) compounded annually. What is the value of the \(401(\mathrm{k})\), rounded to the nearest dollar, after 18 years? b. Suppose that after 18 years of working for this firm, you move on to a new job. However, you keep your accumulated retirement funds in the \(401(\mathrm{k})\). How much money, to the nearest dollar, will you have in the plan when you reach age \(65 ?\) c. What is the difference between the amount of money you will have accumulated in the \(401(\mathrm{k})\) and the amount you contributed to the plan?

What is a loan amortization schedule?

In Exercises 1-10, use $$ P M T=\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} $$ to determine the regular payment amount, rounded to the nearest dollar. The price of a home is \(\$ 220,000\). The bank requires a \(20 \%\) down payment and three points at the time of closing. The cost of the home is financed with a 30 -year fixed-rate mortgage at \(7 \%\). a. Find the required down payment. b. Find the amount of the mortgage. c. How much must be paid for the three points at closing? d. Find the monthly payment (excluding escrowed taxes and insurance). e. Find the total cost of interest over 30 years.
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