Compare Fractions, Decimals, and Percents

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Learning Outcomes

Compare two fractions
Compare two numbers given in different forms

In this section, we will go over techniques to compare two numbers. These numbers could be presented as fractions, decimals or percents and may not be in the same form. For example, when we look at a histogram, we can compute the fraction of the group that occurs the most frequently. We might be interested in whether that fraction is greater than 25% of the population. By the end of this section we will know how to make this comparison.

Comparing Two Fractions

Whether you like fractions or not, they come up frequently in statistics. For example, a probability is defined as the number of ways a sought after event can occur over the total number of possible outcomes. It is commonly asked to compare two such probabilities to see if they are equal, and if not, which is larger. There are two main approaches to comparing fractions.

Approach 1: Change the fractions to equivalent fractions with a common denominator and then compare the numerators

The procedure of approach 1 is to first find the common denominator and then multiply the numerator and the denominator by the same whole number to make the denominators common.

Example \(\PageIndex{1}\)

Compare: \(\frac{2}{3}\) and \(\frac{5}{7}\)

Solution

A common denominator is the product of the two: \(3\:\times7\:=\:21\). We convert:

\[\frac{2}{3}\:\frac{7}{7}\:=\frac{14}{21}\nonumber \]

and

\[\frac{5}{7}\:\frac{3}{3}=\frac{15}{21}\nonumber \]

Next we compare the numerators and see that \(14\:<\:15\), hence

\(\frac{2}{3}<\:\frac{5}{7}\)

The diagram below is about Converting Between Fractions, Decimals and Percents. If you start with a fraction such as \(\frac{7}{8}\), you divide the numerator by the denominator. For this example, you get 0.875. You can turn that into a percent by multiplying by 100 or moving the decimal 2 places to the right to get 87.5%. You can also go in the other direction. If you start with a percent such as 87.5%, you can turn that into a decimal by dividing by 100 or moving the decimal 2 places to the right to get 0.875. Then you can write that as a fraction by writing the number over 100 to get \(\frac{87.5}{100} = \frac{875}{1000}\). Next divide top and bottom by 5 to get \(\frac{175}{200}\). Divide top and bottom again by 5 to get \(\frac{35}{40}\). Divide top and bottom by 5 one more time to get \(\frac{7}{8}\).

Approach 2: Use a calculator or computer to convert the fractions to decimals and then compare the decimals

If it is easy to build up the fractions so that we have a common denominator, then Approach 1 works well, but often the fractions are not simple, so it is easier to make use of the calculator or computer.