Problem 47 - Elementary Algebra

Question

Use proportions to change each common fraction to a percent. $$\frac{12}{5}$$

Step-by-Step Solution

Answer

The fraction \(\frac{12}{5}\) as a percent is 240%.

Step 1: Set up the Proportion

A proportion is an equation that states that two ratios are equal. To convert a fraction into a percent, we can set up the proportion where the given fraction is equal to another fraction with 100 as the denominator. For \(\frac{12}{5}\), we set it equal to \(\frac{x}{100}\) where \(x\) represents the percent value.\[\frac{12}{5} = \frac{x}{100}\]

Step 2: Cross-Multiply to Solve for x

Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction. Multiply across the established equation:\[12 \times 100 = 5 \times x\]This simplifies to:\[1200 = 5x\]

Step 3: Solve for x

To find the value of \(x\), you need to isolate \(x\) on one side of the equation. Do this by dividing both sides of the equation by 5:\[\frac{1200}{5} = x\]This simplifies to:\[x = 240\]

Step 4: Conclusion

Since \(x\) represents the percent, we conclude that the fraction \(\frac{12}{5}\) is equivalent to 240%. Therefore, \(\frac{12}{5}\) converted to a percent is 240%.

Key Concepts

Understanding Proportions

A proportion is a helpful mathematical concept that states that two ratios are equal. It is instrumental when comparing fractions and determining equivalent values like percentages. To convert a fraction into a percent, you can set up a simple proportion. Here's how:

• Take the given fraction, like \( \frac{12}{5} \).
• Set it equal to another fraction with 100 as its denominator. The formula looks like this: \( \frac{12}{5} = \frac{x}{100} \).

This setup means that \(12\) parts out of \(5\) will correspond to \(x\) parts out of \(100\), where \(x\) is unknown but represents the percentage. By understanding this setup, you'll be able to manipulate fractions effectively for conversion.

Learning Cross-Multiplication

Cross-multiplication is a handy technique used to solve equations involving fractions by eliminating them. At first glance, fractions can seem tricky, but cross-multiplication makes it simple.To cross-multiply:

• Multiply the numerator (top number) of one fraction by the denominator (bottom number) of the other fraction.
• Repeat this with the other numerator and denominator. The products you get from the multiplications are set equal to each other.

For the fraction \( \frac{12}{5} = \frac{x}{100} \), you cross-multiply like so:

• \(12 \times 100 = 5 \times x\)

Which gives you:

• \(1200 = 5x\)

This step eliminates the fractions, simplifying solving the proportion.

Solving for x

Once you've used cross-multiplication to set up the equation, you'll often find yourself needing to solve for \( x \). This process involves simple algebra to isolate \( x \) on one side of the equation.Take \( 1200 = 5x \) as derived from cross-multiplication:

• To solve for \( x \), divide both sides by 5.
• Perform the division: \( \frac{1200}{5} = x \).
• Calculate the result: \( x = 240 \).

You've now found the percent value that corresponds to the original fraction. In this case, \( \frac{12}{5} \) equals \( 240\% \). By mastering these steps, you can confidently tackle any fractions-to-percent conversion problem.