Convert Repeating Decimals To Fractions: A Step-by-Step Guide
Converting repeating decimals to fractions can seem tricky, but it's a valuable skill in mathematics. In this guide, we'll break down the process step-by-step with clear explanations and examples. Let's dive in!
Understanding Repeating Decimals
Before we get started, let's make sure we all know what a repeating decimal actually is. A repeating decimal, also known as a recurring decimal, is a decimal number that has a digit or a block of digits that repeat indefinitely. These repeating digits are often denoted with a bar over them or enclosed in parentheses.
For example, 0.333... (or ) is a repeating decimal where the digit '3' repeats forever. Similarly, 0.142857142857... (or ) is a repeating decimal with the block of digits '142857' repeating endlessly.
Why is understanding this important? Well, recognizing that a decimal is repeating is the first step to converting it into a fraction. Not all decimals can be neatly turned into fractions, but repeating decimals always can, and that's what we're going to learn how to do today.
When we talk about converting repeating decimals to fractions, we're essentially finding a ratio of two integers that equals the decimal. This is super useful in various areas of math, like algebra and calculus, where fractions are often easier to work with than decimals. Plus, it's a neat trick to have up your sleeve for impressing your friends (or at least understanding your math homework better!).
So, keep in mind that the key to this whole process is spotting the repeating pattern and using a little algebraic manipulation to get rid of the infinitely repeating part. Once you get the hang of it, you'll be converting repeating decimals to fractions like a pro. Now, let's get to those examples!
Example 1: Convert 2.(5) to a Fraction
The first example we're going to tackle is converting the repeating decimal 2.(5) to a fraction. When you first look at a repeating decimal like this, it might seem a bit daunting, but don't worry, we'll take it one step at a time.
Step 1: Set up an equation.
Let's start by assigning the repeating decimal to a variable. We'll call it x. So, we write:
x = 2.5555...
This is our foundation. The variable x now represents the value we want to convert into a fraction. The repeating decimal is expressed in its expanded form to highlight the repeating digit, which in this case is '5'.
Step 2: Multiply by a power of 10.
Next, we need to multiply both sides of the equation by a power of 10 that will shift the repeating part to the left of the decimal point. Since only one digit repeats (the '5'), we'll multiply by 10:
10x = 25.5555...
Multiplying by 10 moves the decimal point one place to the right. This is a crucial step because it sets us up to eliminate the repeating part in the next step. Notice how the decimal part is still '.5555...', which is the same as in the original equation.
Step 3: Subtract the original equation.
Now, we're going to subtract the original equation (x = 2.5555...) from the new equation (10x = 25.5555...). This will eliminate the repeating decimal part:
10x - x = 25.5555... - 2.5555...
This simplifies to:
9x = 23
See how the repeating decimals neatly cancel each other out? That's the magic of this method! We're left with a simple algebraic equation that we can easily solve.
Step 4: Solve for x.
To find the value of x, we need to isolate x by dividing both sides of the equation by 9:
x = 23 / 9
So, the repeating decimal 2.(5) is equal to the fraction 23/9. You can check this by dividing 23 by 9 on a calculator, and you'll see that it gives you 2.5555... Hooray, we did it!
Example 2: Convert 8.(16) to a Fraction
Now let's convert the repeating decimal 8.(16) into a fraction. This one is a little different because we have a repeating block of two digits, '16'. But don't worry, the process is very similar.
Step 1: Set up an equation.
As before, we assign the repeating decimal to a variable, x:
x = 8.161616...
Here, x represents the value we want to convert, and we write the repeating decimal with the repeating block '16' clearly shown.
Step 2: Multiply by a power of 10.
Since two digits are repeating, we need to multiply by 100 (which is ) to shift the repeating block to the left of the decimal point:
100x = 816.161616...
Multiplying by 100 moves the decimal point two places to the right, maintaining the repeating decimal part '.161616...'.
Step 3: Subtract the original equation.
Subtract the original equation (x = 8.161616...) from the new equation (100x = 816.161616...):
100x - x = 816.161616... - 8.161616...
This simplifies to:
99x = 808
The repeating decimals cancel each other out, leaving us with a straightforward equation.
Step 4: Solve for x.
To isolate x, divide both sides of the equation by 99:
x = 808 / 99
Thus, the repeating decimal 8.(16) is equal to the fraction 808/99. Give it a try on your calculator to confirm!
Example 3: Convert 4.(2) to a Fraction
Let's convert the repeating decimal 4.(2) to a fraction. This example reinforces the technique with a slightly different number.
Step 1: Set up an equation.
Assign the repeating decimal to a variable, x:
x = 4.2222...
Step 2: Multiply by a power of 10.
Since one digit repeats, multiply by 10:
10x = 42.2222...
Step 3: Subtract the original equation.
Subtract the original equation (x = 4.2222...) from the new equation (10x = 42.2222...):
10x - x = 42.2222... - 4.2222...
This simplifies to:
9x = 38
Step 4: Solve for x.
To isolate x, divide both sides of the equation by 9:
x = 38 / 9
So, the repeating decimal 4.(2) is equal to the fraction 38/9.
Example 4: Convert 7.(13) to a Fraction
Now let's convert the repeating decimal 7.(13) into a fraction. This example will further clarify the process when dealing with repeating blocks of digits.
Step 1: Set up an equation.
Assign the repeating decimal to a variable, x:
x = 7.131313...
Step 2: Multiply by a power of 10.
Since two digits are repeating, we multiply by 100:
100x = 713.131313...
Step 3: Subtract the original equation.
Subtract the original equation (x = 7.131313...) from the new equation (100x = 713.131313...):
100x - x = 713.131313... - 7.131313...
This simplifies to:
99x = 706
Step 4: Solve for x.
To isolate x, divide both sides of the equation by 99:
x = 706 / 99
Therefore, the repeating decimal 7.(13) is equal to the fraction 706/99.
Conclusion
Converting repeating decimals to fractions involves setting up equations, multiplying by powers of 10, subtracting to eliminate the repeating part, and solving for the unknown variable. By following these steps, you can confidently convert any repeating decimal into a fraction. This skill is not only useful for mathematical problems but also enhances your understanding of number systems. Practice these examples and you'll master the art of converting repeating decimals to fractions in no time!