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Chapter 8: Problem 40

Convert to fraction notation. $$ 87 \frac{1}{2} \% $$

Short Answer

Expert verified
\frac{7}{8}

Step by step solution

01

Understand the Problem

The goal is to convert the mixed number percentage into a fraction notation.
02

Convert Mixed Number to Improper Fraction

First, convert the mixed number \( 87 \frac{1}{2} \) into an improper fraction. Multiply the whole number part by the denominator of the fractional part, then add the numerator of the fractional part. \begin{align*}87 \frac{1}{2} &= 87 + \frac{1}{2} = \frac{174}{2} + \frac{1}{2} = \frac{174 + 1}{2} = \frac{175}{2}. Thus, \(87 \frac{1}{2} \)%= \frac{175}{2}.
03

Convert Percentage to a Fraction

Divide the resulting fraction by 100. Remember that dividing by 100 is equivalent to multiplying by \frac{1}{100}. Therefore:\begin{align*}\frac{175}{2} \times \frac{1}{100} &= \frac{175}{200} = \frac{175}{200}.Simplify the fraction by finding the greatest common divisor (GCD) of 175 and 200 which is 25 . Therefore:$$ \frac{175 梅25}{200 梅 25} = \frac{7}{8}$$

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Key Concepts

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

Mixed Number to Improper Fraction
A mixed number is a combination of a whole number and a fraction. To convert it into an improper fraction, follow a systematic process. First, multiply the whole number part by the denominator of the fractional part. Then, add the numerator of the fractional part to this product. This sum will be your new numerator, while the denominator remains the same. For example, converting the mixed number 87 \(\frac{1}{2}\) involves multiplying 87 by 2 (the denominator) to get 174. Then, add 1 (the numerator) to get a total numerator of 175. The improper fraction becomes \(\frac{175}{2}\).

This step is crucial because it ensures you can work with the entire quantity as one fraction.
Percentage Conversion
A percentage represents a part out of 100. To convert a percentage into a fraction, simply rewrite the percentage as a fraction over 100. This means you divide the percentage value by 100. For example, converting \(87 \frac{1}{2} \%\) involves first changing the mixed number to an improper fraction (\frac{175}{2}) and then dividing by 100. Mathematically, dividing by 100 is the same as multiplying by \(\frac{1}{100}\). Therefore, we get \(\frac{175}{2} \times \frac{1}{100} = \frac{175}{200}\).

This method highlights the direct relationship between percentages and their equivalent fractions.
Simplifying Fractions
Simplifying fractions makes them easier to work with and understand. Start by determining the Greatest Common Divisor (GCD) of the numerator and denominator. The GCD is the largest number that evenly divides both the numerator and the denominator. In our example, the GCD of 175 and 200 is 25. Divide both the numerator and the denominator by their GCD to simplify the fraction. Therefore, \(\frac{175}{200} \) simplifies to \( \frac{175 \div 25}{200 \div 25} = \frac{7}{8} \).

Remember, simplifying fractions helps in comparing and reducing complexity in calculations.

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