Problem 39 - Prealgebra

Question

Solve each problem using the percent equation. 17.6 is \(133 \frac{1}{3} \%\) of what number?

Step-by-Step Solution

Answer

17.6 is 133 1/3% of approximately 13.2.

Step 1: Understand the Percent Equation

The percent equation is represented as: \( = r \cdot b \), where \( n \) is the part, \( r \) is the rate (percent), and \( b \) is the base (the whole or original number). In this problem, \( n = 17.6 \) and \( r = 133 \frac{1}{3} \% \). We need to find \( b \).

Step 2: Convert Percentage to a Decimal

Convert \( 133 \frac{1}{3} \% \) to a decimal by first converting the fraction: \( \frac{1}{3} \approx 0.3333 \). So, \( 133 \frac{1}{3} \% = 133.3333 \% \). To convert it to a decimal, divide by 100: \( 133.3333 \div 100 = 1.333333 \).

Step 3: Set Up the Equation

Using the percent equation \( n = r \cdot b \), substitute the known values: \( 17.6 = 1.333333 \cdot b \).

Step 4: Solve for the Unknown Base

To find \( b \), divide both sides by 1.333333: \( b = \frac{17.6}{1.333333} \). Calculate this using a calculator for accuracy.

Step 5: Calculate the Base

Perform the division: \( b = \frac{17.6}{1.333333} \approx 13.2 \). Thus, the original number is approximately 13.2.

Key Concepts

Converting Percentages

Understanding how to convert percentages is an essential skill in solving problems involving percentages. A percentage is a way to express a number as a fraction of 100. For instance, the percentage 50% means 50 out of 100, or simply 0.5 when converted to a decimal form.
To convert a percentage to a decimal, divide the percentage by 100. This is because "percent" literally means "per hundred". So, for the exercise provided, you're dealing with 133 1/3%. Begin by converting the fraction to a decimal:

• \( \frac{1}{3} \approx 0.3333 \)

Then, you would add this to 133 to get a total of 133.3333%. To convert this percentage to a decimal:

• Divide 133.3333 by 100, resulting in approximately 1.333333.

This conversion is crucial because working with decimals is often easier when performing calculations like multiplication or division. It's one part of setting up the percent equation effectively.

Multiplication and Division

Multiplication and division are fundamental mathematical operations crucial in solving equations involving percentages. Once you convert a percentage to a decimal, it plays a significant role in the percent equation, which is structured as \( n = r \cdot b \). Here:

• \( n \) represents the part (17.6 in the provided exercise).
• \( r \) is the rate or percent as a decimal (1.333333).
• \( b \) is the base or the unknown whole you need to find.

In this problem, you're tasked with finding \( b \). To set up the equation, substitute the known values:

• \( 17.6 = 1.333333 \cdot b \)

To isolate \( b \), you perform division by dividing both sides of the equation by 1.333333. Think about it as undoing the multiplication to solve for \( b \). Calculations will reveal:

• \( b = \frac{17.6}{1.333333} \)

This step will solve for \( b \), providing the answer and demonstrating the practical use of division in percentage problems.

Fraction to Decimal Conversion

Converting fractions to decimals is a crucial preliminary step in percentage-related calculations. A fraction represents a part of a whole, similar to what a decimal does, just in a different form. When working with percentages, you often need to switch between these two representations for accuracy.
In the exercise, you are initially dealing with a mixed percentage, 133 1/3%. Here, the fraction part, 1/3, needs to be converted into a decimal to facilitate further calculations. To convert the fraction:

• Divide the numerator by the denominator: \( \frac{1}{3} = 0.3333 \).

You then add this decimal to the whole number in the percentage, turning it into a complete decimal form:

• Representing 133 1/3% as 133.3333%.

Converting this percentage further to a decimal (by dividing by 100) provides 1.333333. This conversion lays the groundwork for solving the problem with greater precision, emphasizing the importance of fraction to decimal conversions in real-world calculations involving percentages.