# Converting between Fractions, Decimals, and Percentages

A **rational number** is any number than can be written as a ratio of two integers (a *fraction*), such as \(\frac{4}{5}\) or \(\frac{8}{25}\). Percentages (such as 32%) and decimals (such as .32) are other forms in which we can write rational numbers, and it is often helpful to re-express them in different ways. For example, we might note that \(\frac{8}{25} = .32 = 32\%\) are all different ways of writing the same value.

## Converting Fractions to Decimals

To convert a fraction to a decimal, divide the numerator by the denominator. For example, to convert \(\frac{1}{4}\) to a decimal, we divide 1 by 4 to get the quotient 0.25.

\[\frac{1}{4}=0.25\]

## What is the decimal representation of \(\frac{1}{6}?\)

Performing long division when \(1\) is divided by \(6\), the quotient doesn't have a terminating point in that the 6 goes on forever. So, to represent a repeating number, we use a hyphen on the top: \[\frac{1}{6}=0.1\overline{6}. \]

## What is the decimal representation of \(\frac{1}{5}?\)

Performing long division when 1 is divided by 5, we get the quotient to be 0.2. Thus,\[\frac{1}{5}=0.2.\]

## Converting Decimals to Fractions

To write an **integer** as a fraction, make the denominator one. For example, the number \(9\) can be written in fractional form as \(\frac{9}{1}\).

To write a **terminating decimal** as a fraction, write the decimal number into an equivalent fraction by making the numerator the number itself and the denominator 1. For example, if the number is \(0.6\), it becomes \(\frac{0.6}{1}\). After that, multiply both the numerator and the denominator with a number that takes the decimal place of the numerator into the last place. So multiply by 10 \(\left(\frac{0.6\times 10}{1 \times 10}\right)\) to get \(\frac{6}{10}\). Simplify to get \(\frac{3}{5}\).

## What is the fractional representation of \(0.623?\)

We have

\[0.623=\frac{0.623}{1}.\]

Multiply both the numerator and the denominator by 1000, to move the decimal point of the numerator to the zero's place:

\[\frac{0.623\times1000}{1\times 1000}=\frac{623}{1000},\]

which can't be simplified any further. \(_\square\)

To write a **repeating decimal,** such as 0.343434..., as a fraction, we can begin by setting it equal to a variable:
\[0.\overline{34} = x.\]

Our decimal repeats after the hundredths place, so we can write another equation, multiplying both sides of our original equation by 100: \[34.\overline{34}... = 100x.\]

Subtracting the first equation from the second, we get \(34 = 99x, \) or \(x = \frac{34}{99}.\)

## What is the fractional representation of \(0.\overline{8}?\)

We have

\[0.\overline{8}=x.\]

Multiply both sides of the equation by 10:

\[8\overline{8}=10x.\]

Subtracting the first equation from the second, we have \(8 = 9x\) and \(x=\frac{8}{9}.\)

You can read about other methods for converting repeating decimals to fractions here.

## Converting Percentages

In everyday life, we often use percentages to express the amount of something relative to the whole. A percent is a ratio expressed as a fraction of 100, so the whole is equal to "100%". In math and science, however, decimals are more frequently used to express portions. When using decimals, we consider "1" as the whole. Therefore we have 1=100%, and we come to a formula that gives the relation between decimals and percentages:

\[\begin{align} a&=100\times a\text{ %},\\ b\text{ %}&=b\times0.01, \end{align}\]

where \(0\leq a\leq1\) and \(0\leq b\leq100\). Thus, the percentage is expressed as a number 100 times larger than the decimal. Now let's take a look at a few examples.

## Convert 75% into decimals.

Using the formula above, we have

\[75\text{ %}=75\times0.01=0.75.\ _\square\]

## How can we express 'half' using decimals? What is this in percentage?

When we use decimals, we consider "1" as whole. Therefore "0.5" would be equal to half.

When using percentages, we consider "100%" as whole. Hence "50%" would be equal to half.

Using the conversion formula above, we can verify this as shown below:

\[0.5=100\times0.5\text{ %}=50\text{ %}.\ _\square\]

**Cite as:**Converting between Fractions, Decimals, and Percentages.

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