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Chapter 4: Problem 37

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

Short Answer

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

Step by step solution

01

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 \( \frac{11}{20} = \frac{x}{100} \).
02

Solve the Proportion for x

To solve the proportion, cross-multiply to find the value of \( x \). This gives \( 11 \times 100 = 20 \times x \).
03

Compute for x

Calculate \( 11 \times 100 \), which equals 1100. Then divide both sides by 20 to solve for \( x \): \( x = \frac{1100}{20} \).
04

Simplify the Division

Simplify \( \frac{1100}{20} \) by performing the division: \( 1100 \div 20 = 55 \).
05

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%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Thirty ounces of a punch that contains \(10 \%\) grapefruit juice is added to 50 ounces of a punch that contains \(20 \%\) grapefruit juice. Find the percent of grapefruit juice in the resulting mixture.

Solve each of the following problems. Keep in mind the suggestions we offered in this section. How long will it take $$\$ 4000$$ to double itself if it is invested at \(8 \%\) simple interest?

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ y=-7 x+10 \text { for } x $$

For Problems \(1-12\), solve each of the equations. These equations are the types you will be using in Problems 13-40. $$ 16 t+8\left(\frac{9}{2}-t\right)=60 $$

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ y=m x+b \quad \text { for } x $$
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