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Chapter 2: Problem 5

Use the formula to solve for the given variable. Solve \(i=\) Prt for \(r\), given that P= 600 dollars, t=2 \(\frac{1}{2}\) years, and i= 90 dollars. Express \(r\) as a percent.

Short Answer

Expert verified
The interest rate is 6%.

Step by step solution

01

Understand the Formula

We are given the formula for simple interest: \( i = Prt \), where \( i \) is the interest, \( P \) is the principal amount, \( r \) is the rate of interest as a decimal, and \( t \) is the time in years.
02

Substitute Known Values

Substitute the known values into the formula: \( i = 90 \) dollars, \( P = 600 \) dollars, and \( t = 2.5 \) years. The equation becomes \( 90 = 600 \times r \times 2.5 \).
03

Solve for \( r \)

To isolate \( r \), rearrange the equation: \( r = \frac{90}{600 \times 2.5} \). Calculate the expression by first finding \( 600 \times 2.5 = 1500 \), then \( r = \frac{90}{1500} \).
04

Simplify the Expression

Simplify \( \frac{90}{1500} \). The greatest common divisor of 90 and 1500 is 30, so \( r = \frac{90 \div 30}{1500 \div 30} = \frac{3}{50} = 0.06 \).
05

Convert the Result to a Percentage

Convert \( r \) from a decimal to a percentage by multiplying by 100. Therefore, \( r = 0.06 \times 100 = 6\% \).

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Key Concepts

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

Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In this context, we're using algebra to manipulate and solve a formula for a specific variable. The formula we are working with is for simple interest:
  • \(i = Prt\)
Where:
  • \(i\) is the interest earned,
  • \(P\) is the principal amount,
  • \(r\) is the rate of interest,
  • \(t\) is the time period in years.
Algebra helps us to rearrange this formula to solve for any one of the variables when the others are known. The principles of algebra are critical as they allow us to isolate and solve for the variable of interest, in this case, the interest rate \(r\). Once identified, these symbols represent the quantities involved in the problem and algebraic rules and operations help us to manipulate the formula to suit our needs.
Solving Equations
Solving equations is a fundamental skill in mathematics, particularly in algebra. When faced with a formula that you need to rearrange, like \(i = Prt\), the goal is to isolate the variable you want to solve. Here's how it's done:
  • Start by substituting the known values into the equation. So, in our example, \(i = 90\), \(P = 600\), and \(t = 2.5\), which gives us the equation:\[ 90 = 600 \times r \times 2.5 \]
  • Next, you want to solve for \(r\). Divide both sides of the equation by \(600 \times 2.5\) to isolate \(r\):\[ r = \frac{90}{600 \times 2.5} \]
  • By calculating \(600 \times 2.5\), you get \(1500\). Therefore, \[ r = \frac{90}{1500} \]
  • Lastly, simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 30:\[ r = \frac{3}{50} = 0.06 \]
Ultimately, solving equations is about manipulating them so that you can easily identify the value of the variable you're interested in. Always verify by back-substituting to ensure your solution is correct.
Interest Rate Conversion
The conversion of the rate from a decimal to a percentage is the final step in solving for the simple interest rate. Initially, the calculated interest rate was expressed as a decimal number 0.06. It鈥檚 crucial to convert this into a percentage for clear communication:
  • Multiply the decimal number by 100.
  • So, for the equation:\[ r = 0.06 \]multiplied by 100 gives:
  • \( r = 6\% \).
This step is important because interest rates are generally given as percentages rather than decimals, making it easier to interpret in everyday financial contexts. This conversion clarifies how much interest you would earn on an annual basis in percentage terms from the principal amount invested. Given this, always remember to express your final rate as a percentage to ensure your solutions are easily understood.

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