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Chapter 7: Problem 19

Change the percent to a simplified fraction or mixed number. $$115 \%$$

Short Answer

Expert verified
115% = 1 \(\frac{3}{20}\)

Step by step solution

01

Convert Percent to Fraction

Express 115% as a fraction by writing it over 100. This turns 115% into \( \frac{115}{100} \).
02

Simplify the Fraction

Find the greatest common divisor (GCD) of 115 and 100. The GCD of 115 and 100 is 5. Divide both numerator and denominator by 5: \( \frac{115 \div 5}{100 \div 5} = \frac{23}{20} \).
03

Convert to Mixed Number (if needed)

Since \( \frac{23}{20} \) is an improper fraction, convert it to a mixed number by dividing 23 by 20, which gives 1 whole and a remainder of 3. Therefore, \( \frac{23}{20} = 1 \frac{3}{20} \).

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Key Concepts

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

percent to fraction conversion
Converting a percentage to a fraction is a vital skill in mathematics. The process is straightforward. You start by expressing the percent value as a fraction with a denominator of 100. To convert 115% to a fraction, write it as: \( \frac{115}{100}\). This step ensures that you move from a percentage directly into a fraction form, making it easier to simplify and manipulate later. The key takeaway here is that the denominator in this initial fraction is always 100, reflecting the 'per hundred' nature of percentages.
simplifying fractions
Once the percentage is successfully converted into a fraction, the next step is to simplify it. Simplifying fractions means reducing them to their simplest form, where numerator and denominator share no common factors other than 1.
To simplify \( \frac{115}{100}\), we first identify the greatest common divisor (GCD) of the numerator and denominator.
The GCD of 115 and 100 is 5. We then divide both by this GCD: \( \frac{115 \, \text{div} \, 5}{100 \, \text{div} \, 5} = \frac{23}{20} \). The result is \( \frac{23}{20} \), a simplified fraction. Simplifying fractions enhances clarity and often makes further mathematical operations easier.
improper fractions
An improper fraction is one where the numerator is larger than the denominator. This means the fraction has a value greater than or equal to 1. In our example, the fraction \( \frac{23}{20} \) is an improper fraction because 23 (numerator) is greater than 20 (denominator).
Understanding improper fractions is fundamental because they often need further simplification or conversion into mixed numbers, making calculations and interpretations more intuitive.
mixed numbers
Converting improper fractions to mixed numbers can make them easier to understand and use. A mixed number consists of a whole number and a proper fraction. To convert \( \frac{23}{20} \) to a mixed number, you divide the numerator by the denominator. Here, 23 divided by 20 equals 1 with a remainder of 3. This gives us the mixed number: 1 \( \frac{3}{20} \). Mixed numbers provide a clear representation of quantities, combining whole numbers with fractional parts.
Mastering the conversion between fractions, improper fractions, and mixed numbers is essential for tackling a broad range of mathematical problems.

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