Problem 39 - Elementary Algebra

Question

Use proportions to change each common fraction to a percent. $$\frac{3}{5}$$

Step-by-Step Solution

Answer

The fraction \( \frac{3}{5} \) is 60%.

Step 1: Understand the Problem

We need to convert the fraction \( \frac{3}{5} \) into a percent. A percent is a ratio with a denominator of 100.

Step 2: Set Up the Proportion

To convert \( \frac{3}{5} \) to a percentage, set up a proportion where \( \frac{3}{5} = \frac{x}{100} \). Here, \( x \) represents the percentage of the fraction.

Step 3: Solve the Proportion

Cross-multiply the proportion to solve for \( x \). This gives: \( 3 \times 100 = 5 \times x \). Simplify to get \( 300 = 5x \).

Step 4: Solve for x

Divide both sides of the equation by 5 to isolate \( x \): \( x = \frac{300}{5} \). Simplify to find \( x = 60 \).

Step 5: Interpret the Result

The value of \( x \) is 60, which means \( \frac{3}{5} \) is equivalent to 60%.

Key Concepts

Fractions to Percentages

When converting fractions to percentages, it's helpful to remember that a percentage is simply a fraction with a denominator of 100. This means that to convert any fraction to a percentage, you need to find an equivalent fraction that has 100 in the denominator. Let's break this down:

• Identify the fraction you want to convert. In this case, \( \frac{3}{5} \) is our starting fraction.
• To convert this fraction to a percentage, we want to find an equivalent fraction with a denominator of 100.
• Once we find this fraction, the numerator will be the percentage.

This method makes conversions more visual and tangible. By equating the fraction to a percentage, you're essentially asking, "What would the numerator have to be for a denominator of 100?" This forms the basis for our next method: the proportion method.

Proportion Method

The proportion method is a handy tool to convert fractions to percentages. It uses the relationship of ratios to establish an equivalent percentage.

• Set up a proportion by equating your initial fraction to a fraction with a denominator of 100. For \( \frac{3}{5} \), you use the proportion \( \frac{3}{5} = \frac{x}{100} \).
• Here, \( x \) is the unknown and represents the percentage we are solving for.
• This setup allows us to maintain the relative size of the comparison while converting to a format that makes calculating the final percentage straightforward.

With this proportion, we can solve for \( x \), which brings us to the next concept: cross multiplication.

Cross Multiplication

Cross multiplication is an efficient and clear method to solve proportions, especially when you're working with percentages. By cross-multiplying, you eliminate the fractions, making it easier to solve for the unknown.

• For the proportion \( \frac{3}{5} = \frac{x}{100} \), perform cross multiplication by multiplying diagonally: 3 times 100 and 5 times \( x \).
• This gives the equation \( 3 \times 100 = 5 \times x \).
• Perform the multiplication to get \( 300 = 5x \).
• Finally, solve for \( x \) by dividing both sides by 5, yielding \( x = 60 \).

With cross multiplication, you can neatly solve for the percentage, providing a straightforward and systematic approach to converting fractions. It ensures accuracy and reinforces the relationship between the numbers in your fraction and the desired percentage.