How to Convert a Decimal to a Fraction
Knowing how to convert a decimal to a fraction is a handy skill. It’s not only useful on math tests, but also in real-life situations where you want to do a part to whole comparison. Multiplying fractions is also arguably simpler than multiplying decimals. In this post, we’ll break this skill down into steps so that you can quickly do these conversions whenever you encounter them, whether it’s real life or on a test.
How to Convert a Decimal to a Fraction
Though the concept might seem difficult at first, converting a decimal to a fraction can be done in just a few simple steps.
First, you need to confirm that the decimal you’re working with is a terminating one. This means it has a finite number of digits rather than a sequence of repeating digits that continue infinitely. Some examples of terminating decimals are: \(.25,\:.782,\:.9672154,\) etc. Most decimals are terminating, and this is a good news since these are the easiest decimals to convert to a fraction.
Once you know that a decimal is terminating, follow the steps below to convert it to a fraction.
How to Convert a Terminating Decimal to a Fraction
Step 1: Create a fraction by putting the decimal over 1.
Write the decimal as a fraction, with the decimal as the numerator and \(1\) as the denominator. So, for example, if you’re given the decimal \(.38\), you will write it as:
\(\frac{.38}{1}\)
Step 2: Multiply by the right power of 10 to get rid of the fraction.
Next you’ll need to get rid of the decimal in the fraction you just created. You can’t simply erase it, though, You’ll first need to account for place value. If you know your place value, you know that the decimal \(.38\) is read 38 hundredths, which gives you an important clue about what the denominator is about to become.
If you don’t know your place values, though, no need to worry; there’s a simple trick to help you out. Simply count the number of digits to the right of the decimal. In the example above there are two. So, you’ll need to multiply both sides of the fraction by \(10\), two times:
\(\frac{.38\:\times\:10\:\times\:10}{1\:\times\:10\:\times\:10}=\frac{38}{100}\)
As you can see in the example above, the decimal \(.38\) then converts to the fraction \(\frac{38}{100}\), but your job isn’t done yet.
Step 3: Reduce the fraction using common factors.
Your last step is reducing the fraction. To do this you need to find common factors of both the numerator and the denominator, and divide each by them again and again until you cannot reduce the numbers any further.
In the example above, both \(38\) and \(100\) can be divided by \(2\), and then divided by \(2\) again. See how it works below:
\(\frac{38\:\div\:2}{100\:\div\:2}=\frac{19}{50}\)
Because \(19\) and \(50\) do not have any more common factors, this fraction cannot be reduced any further. Therefore we now know that:
\(.38=\frac{19}{50}\)
Let’s look at a simple example. To convert the common decimal \(.2\) to a fraction you’d follow the same steps.
Example 1:
Step 1:
\(\frac{.2}{1}\)
Step 2:
\(\frac{.2\:\times\:10}{1\:\times\:10}=\frac{2}{10}\)
Step 3:
\(\frac{2\:\div\:2}{10\:\div\:2}=\frac{1}{5}\)
So, we now know that:
\(.2=\frac{1}{5}\)
Let’s take a look at another, more complicated example. Let’s convert \(.8535\) to a fraction.
Example 2:
Step 1:
\(\frac{.8535}{1}\)
Step 2:
\(\frac{.8535\:\times\:10\:\times\:10\:\times\:10\:\times\:10}{1\:\times\:10\:\times\:10\:\times\:10\:\times\:10}=\frac{8535}{10000}\)
Step 3:
\(\frac{8535\:\div\:5}{10000\:\div\:5}=\frac{1707}{2000}\)
Because \(1707\) and \(2000\) have no more factors in common, the fraction cannot be reduced further. Therefore:
\(.8535=\frac{1707}{2000}\)
How to Convert a Non-Terminating, Repeating Decimal to a Fraction
Of course, these examples are only of terminating decimals. Things get a little more complicated when you look at recurring decimals like \(.666666666. . . \) that repeat infinitely. These sometimes appear as rounded decimals like \(.666666667\) or with a bar above the repeating digit(s). These might look like different numbers, but they are really all a way of describing the same digit or series of digits that repeat themselves forever.
In this care, you’ll need to create two algebraic equations to calculate the fraction.
Step 1: Set x equal to the repeating part of the decimal.
For the first equation, let \(x\) be the repeating decimals you’re solving for. In this case \(x=.6\) because the single digit \(6\) is repeating itself.
Step 2: Multiply the equation by the right power of 10 to get one set of the repeating digits to the left of the decimal.
For the second equation, multiply both sides of the equation above by \(10\) enough times to get the repeating digit or digits to the left of the decimal. For the example above, because the repeating decimal is a single digit, you will only need to multiply the equation by \(10\) once to get the equation \(10x = 6.6\).
(Note that if the repeating decimal had more digits, like \(.581581581581\), you’d need to multiply your original equation by ten multiple times to get a complete set of all repeating digits to the left of the decimal. In that instance, you’d end up with \(1000x=581.581581581\).)
Step 3: Solve for x using the two equations.
Use your two equations to solve for \(x\).
\(x = .6\)
\(10x = 6.6\)
\(10x-x = 6.6-.6\)
\(9x = 6\)
\(x = \frac{6}{9} = {2}{3}\)
Though converting a decimal to a fraction can sometimes be a complex process, when you break it down into a step by step approach, you’ll keep your work organized and always know what comes next.
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