21.1.1: Comparing 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
\(\frac{2}{3}<\:\frac{5}{7}\)
Example \(\PageIndex{2}\)
In statistics, we say that two events are independent if the probability of the second occurring is equal to the probability of the second occurring given that the first occurs. The probability of rolling two dice and having the sum equal to 7 is \(\frac{6}{36}\). If you know that the first die lands on a 4, then the probability that the sum of the two dice is a 7 is \(\frac{1}{6}\). Are these events independent?
Solution
We need to compare \(\frac{6}{36}\)and \(\frac{1}{6}\). The common denominator is 36. We convert the second fraction to
\[\frac{1}{6}\frac{6}{6}=\frac{6}{36}\nonumber \]
Now we can see that the two fractions are equal, so the events are independent.
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.
Example \(\PageIndex{3}\)
In computing probabilities for a uniform distribution, fractions come up. Given that the number of ounces in a medium sized drink is uniformly distributed between 15 and 26 ounces, the probability that a randomly selected medium sized drink is less than 22 ounces is \(\frac{7}{11}\). Given that the weight of in a medium sized American is uniformly distributed between 155 and 212 pounds, the probability that a randomly selected medium sized American is less than 195 pounds is \(\frac{40}{57}\). Is it more likely to select a medium sized drink that is less than 22 ounces or to select a medium sized American who is less than 195 pounds?
Solution
We could get a common denominator and build the fractions, but it is much easier to just turn both fractions into decimal numbers and then compare. We have:
\[\frac{7}{11}\approx0.6364\nonumber \]
and
\[\frac{40}{57}\approx0.7018\nonumber \]
Notice that
\[0.6364\:<\:0.7018 \nonumber \]
Hence, we can conclude that it is less likely to pick the medium sized 22 ounce or less drink than to pick the 195 pound or lighter medium sized person.
Exercise
If you guess on 10 true or false questions, the probability of getting at least 9 correct is \(\frac{11}{1024}\). If you guess on six multiple choice questions with three choices each, then the probability of getting at least five of the six correct is \(\frac{7}{729}\). Which of these is more likely?
Comparing Fractions, Decimals and Percents
When you want to compare a fraction to a decimal or a percent, it is usually easiest to convert to a decimal number first, and then compare the decimal numbers.
Example \(\PageIndex{4}\)
Compare 0.52 and \(\frac{7}{13}\).
Solution
We first convert \(\frac{7}{13}\) to a decimal by dividing to get 0.5385. Now notice that
\[0.52 < 0.5385\nonumber \]
Thus
\[0.52\:<\frac{\:7}{13}\nonumber \]
Example \(\PageIndex{5}\)
When we preform a hypothesis test in statistics, We have to compare a number called the p-value to another number called the level of significance. Suppose that the p-value is calculated as 0.0641 and the level of significance is 5%. Compare these two numbers.
Solution
We first convert the level of significance, 5%, to a decimal number. Recall that to convert a percent to a decimal, we move the decimal over two places to the right. This gives us 0.05. Now we can compare the two decimals:
\[0.0641 > 0.05\nonumber \]
Therefore, the p-value is greater than the level of significance.
This is an application of comparing fractions to probability.