How Do You Calculate and Convert a Percentage?

How Do You Calculate and Convert a Percentage?

A percentage is a proportion that shows a number as a part of a whole, with the “whole” always being \(100\). It can also be thought of as the numerator of a fraction with the denominator always equal to \(100\). The term “percentage” is derived from the Latin word “per centum”, which translates to “per hundred”. The symbol for percentage is \(\bf{\%}\). For example, if it rained \(34\) days in the last \(100\) days, we say that it rained \(34\%\) of the time.

Calculating Percentage

You can use the following steps to calculate a percentage:

1. Determine the whole amount.

2. Divide the number you want to express as a percentage by the total.

3. Multiply the result by \(100\).

Example:

Suppose Joseph has three yellow pencils out of a total of \(12\) pencils. What percentage of his pencils are yellow?

1. The whole (total) amount is \(12\).

2. Divide the number of yellow pencils by the total number of pencils Joseph has:

\[3 \div 12 = 0.25\]

3. Multiply the result by \(100\):

\[0.25 \times 100 = 25\]

Adding a \(\%\) symbol, the answer is \(25\%\). So, \(25\%\) of Joseph’s pencils are yellow.

You can convert percentages to decimals or fractions and vice versa. Let’s look at all the conversion methods.

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Conversions Using Percentages

Sometimes it is necessary to convert percentages to other numeric representations and vice versa. Here are the basics for doing these conversions.

Converting a Percentage to a Decimal

If we are given a percentage, it is very easy to convert it to a decimal:

1. Drop off the percentage sign (\(\%\)).

2. Move the decimal point two places left (this is the same as dividing by \(100\)).

Example:

What is \(82\%\) expressed as a decimal?

Solution:

1. First, we drop the percentage symbol:

\[82\]

2. Then, we simply move the decimal point two places left (i.e., divide by \(100\)):

\[82 \div 100 = 0.82\]

Note: If a decimal point is not shown explicitly, always assume a number has a decimal at the end of it (on the right side).

Converting a Decimal to a Percent

This is exactly the reverse process of converting a percent to a decimal:

1. Move the decimal point two places to the right (multiply by \(100\)).

2. Place a percentage sign (\(\%\)) at the end of the number.

Example:

What is \(0.07\) expressed as a percentage?

Solution:

1. First, we move the decimal point two places to the right (i.e., multiply the decimal by \(100\)):

\[0.05 \times 100 = 5\]

2. Then, we simply place a percentage symbol at the end:

\[5\%\]

So, \(0.07\) expressed as a percentage is \(7\%\).

Converting a Percent to a Fraction

We will look at two examples below to clarify how a percent is converted to a fraction. First, the steps:

1. Drop the percentage symbol.

2. Divide the number by \(100\). If the number is a decimal, move the decimal point until it becomes a whole number. Move the decimal point of the denominator the same number of places as you move the decimal point of the numerator.

3. Reduce, if possible.

Example 1:

What is \(24%\) expressed as a fraction and reduced to its lowest terms?

Solution:

1. First, we drop the percent sign:

\[24\]

2. and 3. Then, divide by \(100\) and reduce the fraction to its lowest terms:

\[\frac{24}{100} = \frac{6}{25}\]

Example 2:

What is \(37.5\%\) expressed as a fraction and reduced to its lowest terms?

Solution:

1. First, we drop the percent sign:

\[37.5\]

2. Dividing it by \(100\), we have:

\[\frac{37.5}{100}\]

Since the numerator is a decimal, we multiply \(37.5\) by \(10\) to make it into a whole number. We multiply the denominator by \(10\) as well:

\[\frac{37.5 \times 10}{100 \times 10} = \frac{375}{1\text{,}000}\]

3. Now, we just need to reduce this fraction to get the final answer:

\[\frac{375}{1\text{,}000} = \frac{15}{40} = \frac{3}{8}\]

Converting a Fraction to a Percent

The best way to convert a fraction to a percentage is to convert the fraction to a decimal first, then follow the steps of decimal-to-percentage conversion. An example will shed some light on this process.

Example:

What is \(\frac{4}{5}\) expressed as a percentage?

Solution:

Dividing \(4\) by \(5\), we have:

\[4 \div 5 = 0.8\]

We have changed the fraction to a decimal and now we follow the steps learned before for converting a decimal to a percentage:

\[0.8 \times 100 = 80\]

That is \(80\%\).

More Practice

Now that you have all the conversions ingrained in your brain, let’s look at a couple of examples: finding a certain percentage of a number and finding a number is what percent of another number.

Example 1:

Calculate \(30\%\) of \(120\).

Solution:

1. First, let’s convert the percentage to a decimal:

\[30 \div 100 = 0.3\]

2. Then, we multiply \(120\) by \(0.3\):

\[120 \times 0.3 = 36\]

Thus, \(30\%\) of \(120\) is \(36\).

Note: As the first step, we could have converted the percentage to a fraction instead of a decimal.

Example 2:

\(36\) is what percent of \(90\)?

Solution:

If we are given a problem in the form “\(n\) is what percent of \(m\)”, we always divide \(n\) by \(m\) and convert the result to a percentage, like so:

\[36 \div 90 = 0.4\] \[0.4 \times 100 = 40\] \[40\%\]

Note: When dividing \(36\) by \(90\), we ended up with a decimal. Reducing the fraction \(\frac{36}{90}\) to its lowest terms and then converting it to a percentage is also another correct process.

For additional help with percentages and other math concepts, or to test your knowledge, check out our math practice tests, study guides, and flashcards.

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