Problem 37 - Elementary Algebra

Question

Use proportions to change each common fraction to a percent. $$\frac{11}{20}$$

Step-by-Step Solution

Answer

\( \frac{11}{20} \) is 55%.

Step 1: Set the Fraction as a Proportion

When converting a fraction to a percent, you can set it equal to \( \frac{x}{100} \) because percent is always out of 100. So, we have \( \frac{11}{20} = \frac{x}{100} \).

Step 2: Solve the Proportion for x

To solve the proportion, cross-multiply to find the value of \( x \). This gives \( 11 \times 100 = 20 \times x \).

Step 3: Compute for x

Calculate \( 11 \times 100 \), which equals 1100. Then divide both sides by 20 to solve for \( x \): \( x = \frac{1100}{20} \).

Step 4: Simplify the Division

Simplify \( \frac{1100}{20} \) by performing the division: \( 1100 \div 20 = 55 \).

Step 5: Interpret the Result as a Percent

Since \( x = 55 \), this means \( \frac{11}{20} \) is equivalent to 55%. So, the fraction \( \frac{11}{20} \) equals 55%.

Key Concepts

Understanding Proportions

When dealing with fractions and percentages, proportions offer a helpful way to relate two ratios. A proportion is simply an equation that states two ratios are equal. For our exercise, we used the fraction \( \frac{11}{20} \). To convert this into a percentage, we set it equal to \( \frac{x}{100} \) because a percent is always based on a denominator of 100.

• Remember, the number 100 in the denominator makes it easier to interpret any fraction directly as a percentage.
• By setting two ratios as equal (as we did with \( \frac{11}{20} = \frac{x}{100} \)), you form a proportion that you can solve to find \( x \), the percent value.

This is a simple and efficient technique to convert any fraction into a percentage.

Cross-Multiplication in Proportions

Once you have set your fractions equal using a proportion, like \( \frac{11}{20} = \frac{x}{100} \), the next step involves cross-multiplication. This method simplifies solving for unknowns in proportions by eliminating the denominators and making it easier to calculate.

Here's how it works:

• Take the numerator of the first fraction (11) and multiply it by the denominator of the second fraction (100). This gives you \( 11 \times 100 \).
• Next, take the denominator of the first fraction (20) and multiply it by the numerator of the second fraction (\( x \)). This results in \( 20 \times x \).

You'll now have the equation \( 1100 = 20x \). Cross-multiplying helps you directly go to an expression that you can solve for \( x \) by isolating it on one side of the equation.

Simplifying Fractions for Easy Calculation

The final step in solving our equation \( 1100 = 20x \) is to simplify. After cross-multiplication, we were able to isolate \( x \) by dividing both sides by 20. This resulted in \( x = \frac{1100}{20} \). Simplifying fractions is a crucial step to find the easiest way to interpret the result.

• To simplify, you might perform actual division: \( 1100 \div 20 = 55 \).
• This operation gives you a straightforward number which represents the percentage equivalent of your initial fraction.

So, through simplifying the fraction, we find that \( x = 55 \), meaning \( \frac{11}{20} \) is equal to 55%, making it much easier to understand at a glance.