Problem 41 - Elementary Algebra

Question

Use proportions to change each common fraction to a percent. $$\frac{1}{6}$$

Step-by-Step Solution

Answer

\( \frac{1}{6} \approx 16.67\% \).

Step 1: Set up the proportion

We want to convert the fraction \( \frac{1}{6} \) into a percent. To do this, we'll use the proportion method. Set up the proportion by writing \( \frac{1}{6} = \frac{x}{100} \), where \( x \) represents the percent.

Step 2: Cross-multiply

Solve the proportion by cross-multiplying: \( 1 \times 100 = 6 \times x \). This simplifies to \( 100 = 6x \).

Step 3: Solve for x

Divide both sides of the equation by 6 to isolate \( x \): \( x = \frac{100}{6} \).

Step 4: Simplify the division

Calculate \( \frac{100}{6} \). Doing the division gives \( x \approx 16.67 \).

Step 5: Interpret the solution

The value of \( x \) that we calculated represents the percentage. Therefore, \( \frac{1}{6} \) is approximately \( 16.67\% \).

Key Concepts

Understanding Common Fractions

Common fractions are mathematical expressions that represent a part of a whole. They are made up of two numbers: the numerator and the denominator. The numerator, positioned on top, indicates how many parts we are considering, while the denominator, located below, signifies the total number of equal parts the whole is divided into.
For example, the fraction \( \frac{1}{6} \) indicates that we are considering 1 part out of a whole divided into 6 equal parts. This is a basic concept in mathematics that helps us in understanding proportions, percentages, and ratios.

• Numerator: The number on top of the fraction.
• Denominator: The number below the fraction.
• Example: In \( \frac{1}{6} \), 1 is the numerator and 6 is the denominator.

Common fractions are essential in many mathematical calculations and everyday situations like cooking, budgeting, and measuring. Recognizing how to relate these to percentages is a fundamental skill.

Converting Fractions to Percentages

Percent conversion is the process of changing a fraction into a percentage. A percentage is another way to express a fraction, typically out of 100. In many cases, we want to understand a number in terms of its equivalence out of 100, which is why converting fractions to percentages can be useful.
To convert a fraction like \( \frac{1}{6} \) to a percent, follow these steps:

• Set up a proportion: \( \frac{1}{6} = \frac{x}{100} \).
• This means you're finding what number \( x \) represents 100 when \( 1 \) represents 6.

By understanding that a percentage is another form of a fraction with 100 as the denominator, you leverage simple arithmetic to find this equivalence, making data and facts easier to grasp in everyday life.

Mastering Cross-Multiplication

Cross-multiplication is a technique used to solve proportions. It is especially useful in converting fractions to percentages. When you have a proportion such as \( \frac{1}{6} = \frac{x}{100} \), cross-multiplication helps you find the unknown value, \( x \).
Here’s how it works:

• Multiply the numerator of the first fraction by the denominator of the second fraction: \( 1 \times 100 = 100 \).
• Multiply the denominator of the first fraction by the numerator of the second fraction: \( 6 \times x = 6x \).
• Set these two products equal to solve for \( x \): \( 100 = 6x \).
• Solve for \( x \) by dividing both sides by 6: \( x = \frac{100}{6} \).

This method is straightforward and powerful for solving equations involving proportions. Mastering cross-multiplication is vital for tasks involving rate, ratio, and scaling in varied academic and real-world applications.