Converting Repeating Decimals into Fractions

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  • 0:01 A Repeating Decimal
  • 1:54 Writing Two Equations
  • 4:11 Example
  • 6:10 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will be able to convert any repeating decimal that you come across into a fraction. Learn how to write the two equations that are needed for the conversion.

A Repeating Decimal

If you go shopping at all, then you are very familiar with decimal numbers. Decimal numbers are our numbers with a decimal point. When you are shopping, you see decimal numbers all around you. Your favorite game might have a price tag of $24.99. That's a decimal number!

If you do cooking or baking, then you are familiar with fractions such as 1/3 and 1/4. As you know, fractions and decimal numbers are related to each other. To convert a fraction into a decimal, all we need to do is divide the fraction's numerator by its denominator. 1/4 turns into 0.25. That's a nice decimal that we can easily write out.

But what about 1/3? What kind of a decimal number does that turn into? Going ahead with our division, we get 0.3333…. We get a decimal that keeps on repeating a series of numbers. We call this type of decimal number a repeating decimal.

In this case, the repeating series of numbers is 3. You recall that to convert a decimal into a fraction, you write the number after the decimal point in the numerator and you put a 1 in the denominator followed by zeroes. The number of zeroes you put in the denominator will equal the number of digits you have in the numerator. So, for the decimal 0.25, you would write it as 25/100. You can then simplify this fraction down to 1/4.

While you know how to convert a decimal number that ends into a fraction, you are unsure about repeating decimals. How do you go about converting these into fractions? The process is actually very different than what you are familiar with. But we will see that it's not all that difficult as long as you follow the same process each time. So, put your thinking cap on, for the rest of this lesson requires just a bit of brain muscle.

Writing Two Equations

Let's try and convert the repeating decimal 0.3333… to see what we get. The process of converting repeating decimals requires writing two equations and then subtracting them to find the fraction. First, we will say that x equals 0.333… We will call this equation our defining equation.

x = 0.333…

To find our first equation, we first note which numbers are being repeated. It is the number 3. So, we now need to figure out what to multiply our defining equation, x = 0.333…, by so that our repeating numbers are on the left side of the decimal point. If we multiply the 0.333… by 10, then we will have moved the decimal point one space to the right and we will have our repeating number to the left of the decimal point. Because we are using our defining equation, we also need to multiply the left side of our equation by 10. We get 10x = 3.333... for our first equation.

To find our second equation, we again use our defining equation. This time, we need to manipulate it so that the repeating numbers are to the right of the decimal point. Looking at our defining equation, we see that our repeating numbers are already to the right of the decimal point, so we don't need to change anything. Our second equation is x = 0.333….

Finding the Fraction

Now that we have our two equations, we now subtract them from each other. We get 10x - x = 3.333… - 0.333…. See how we simply subtracted the two sides from each other on both sides of the equation? We can now go ahead and solve this equation for x. This will give us our fraction.

10x - x = 3.333… - 0.333…

9x = 3

x = 3/9

x = 1/3

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