Problem 66 - Algebra for College Students

Question

Use your calculator to help solve each formula for the indicated variable. Solve \(i=\) Prt for \(i\), given that \(P=\$ 875, r=12 \frac{1}{2} \%\), and \(t=4\) years.

Step-by-Step Solution

Answer

The interest is \$437.50.

Step 1: Understand the Formula and Variables

The formula given is \( i = Prt \). This is the simple interest formula where \( i \) stands for interest, \( P \) is the principal amount, \( r \) is the rate of interest, and \( t \) is the time in years. We need to solve for \( i \) using the provided values.

Step 2: Convert the Interest Rate to Decimal

The interest rate given is \( 12 \frac{1}{2}\% \). First, we convert this percentage to a decimal for calculation.\ \( 12 \frac{1}{2}\% \) can be written as \( 12.5\% \).\ As a decimal, \( 12.5\% = \frac{12.5}{100} = 0.125 \).

Step 3: Substitute the Values into the Formula

Substitute the given values into the formula \( i = Prt \).\ \( P = 875 \), \( r = 0.125 \), \( t = 4 \).\ So the equation becomes \( i = 875 \times 0.125 \times 4 \).

Step 4: Calculate the Interest

Perform the multiplication to find \( i \).\ First, calculate \( 875 \times 0.125 = 109.375 \).\ Next, multiply that result by \( 4 \): \( 109.375 \times 4 = 437.5 \).\ Thus, the value of \( i \) is \( 437.5 \).

Step 5: Write the Solution

The interest \( i \) calculated for the given principal, rate, and time is \( 437.5 \). This means that the interest earned over 4 years is \$437.50.

Key Concepts

Interest Calculation

Interest calculation is a fundamental concept in financial mathematics, helping us determine the cost of borrowing money or the return on investment. In simple interest, the calculation is direct and straightforward.The formula for simple interest is:\[ i = Prt \]where:

• \( i \): interest to be calculated
• \( P \): principal amount, the initial sum borrowed or invested
• \( r \): annual interest rate (in decimal form)
• \( t \): time the money is invested or borrowed for, in years

To calculate the interest, you multiply the principal by the rate and the time. For example, if you have \( P = 875 \), \( r = 0.125 \), and \( t = 4 \), you substitute these values into the formula to find \( i \). The calculated interest is important as it tells you how much extra will need to be paid back on a loan or how much profit one can earn from an investment over time. Keep in mind that this formula assumes constant conditions without compounding.

Financial Mathematics

Financial mathematics is a field that uses mathematical formulas and principles to solve monetary problems. It covers a broad range of topics from calculating interest to analyzing investment returns, making it essential for personal and professional financial planning. One of the key aspects of financial mathematics is understanding how different financial instruments like loans, mortgages, and investments work.

• Interest calculations give insight into how much one will pay or earn with regard to loans or savings over time.
• By understanding these basics, individuals and businesses can make informed decisions when comparing financial products such as choosing between fixed and variable interest rates.

In our simple interest example, learning the calculation helps with visualizing how increments in time and variations in rates affect the total amount of interest. Therefore, mastering simple calculations builds a foundation for tackling more complex financial scenarios, making financial mathematics a powerful tool for effectively managing money.

Percentage Conversion

Percentage conversion is a necessary step in calculations involving percentages, especially when using formulas like simple interest. Percentages must be converted to decimals for them to be accurately incorporated into most mathematical expressions.Here's how you convert a percentage to a decimal:

• Take the percentage and divide it by 100 (as percentages are essentially per hundred).
• For example, \( 12 \frac{1}{2} \% \) becomes \( 12.5 \% \) first.
• Then transform \( 12.5 \% \) into a decimal by calculating \( \frac{12.5}{100} = 0.125 \).
• This decimal is then used in any subsequent calculations, such as for \( r \) in our simple interest formula \( i = Prt \).

Understanding how to convert percentages correctly ensures accuracy in calculations, which is vital. Whether it's applied in interest rate calculations or interpreting statistical data, percentage conversion is an essential skill in various mathematical and real-world applications.