Lesson Objectives

- Demonstrate an understanding of the addition property of equality
- Demonstrate an understanding of the multiplication property of equality
- Learn how to convert a repeating decimal into a fraction
- Learn the shortcut to convert a repeating decimal into a fraction

## How to Convert a Repeating Decimal into a Fraction

In our pre-algebra course, we learned how to convert from a
decimal to a fraction and from a
fraction to a decimal.
Up to this point, we have not discussed how to convert
a repeating decimal into a fraction. A repeating decimal is a decimal number that repeats the same ending digit or pattern of ending digits forever.
We generally place a bar over the digit or pattern of digits that repeat. Let's take a look at a few examples.

0. » The digit "7" repeats forever

2.5 » The pattern "32" repeats forever

0. » The pattern "9138" repeats forever

We may also see an ellipsis used instead of an overbar:

0. = 0.777...

2.5 = 2.5323232...

0. » = 0.913891389138...

When we use the ellipsis, we must ensure that the digit that repeats or pattern of repeating digits is crystal clear.

To convert a repeating decimal to a fraction, we rely on our knowledge of the addition property of equality, along with the multiplication property of equality. Let's begin by looking at an example.

Suppose we want to convert 0. into a fraction. Let's begin by setting our repeating decimal equal to a variable like x:

x = 0.

It is normally easier to think about this problem using the ellipsis format. Let's convert our equation into:

x = 0.888...

Next, we will multiply both sides of the equation by 10

x = 0.888...

10 • x = 10 • 0.888...

10x = 8.888...

Now we will subtract the original equation away from the new equation:

10x - x = 8.888... - 0.888...

Before we go any further, think about why this is legal? We know the addition property of equality allows us to add or subtract the same value to or from both sides of the equation. Since x is equal to 0.888..., this means they are the same value. So when we subtract away x on the left side and subtract away 0.888... on the right side, we are subtracting away the same value.

Left Side:

10x - x = 9x

Right Side: $$\hspace{.75em}8.888...$$ $$\underline{-0.888...}$$ The numbers after the decimal point will cancel: $$\require{cancel}\hspace{.75em}8.\cancel{888...}$$ $$\underline{-0.\cancel{888...}}$$ $$\hspace{.75em}8$$ Our equation becomes:

9x = 8

Now we can solve for x by dividing each side by 9, the coefficient of x: $$9x = 8$$ $$x = \frac{8}{9}$$ Remember x was set equal to (0.888...) when we started. We can replace the x and show: $$0.888... = \frac{8}{9}$$

Example 1: Convert each repeating decimal into a fraction

0.5

We will write this number using an ellipsis:

0.5111...

Step 1) Set the repeating decimal equal to a variable

x = 0.5111...

Step 2) Multiply both sides of the equation by 10

n is the number of digits in the repeating string. Here we have one digit (1) that repeats forever. This means n is 1:

10 • x = 10 • 0.5111...

10x = 5.111...

Step 3) Subtract away the first equation from the second equation

Equation 1) x = 0.5111...

Equation 2) 10x = 5.111...

10x - x = 5.111... - 0.5111...

Left side:

10x - x = 9x

Right side: $$\hspace{.75em}5.1111...$$ $$\underline{-0.5111...}$$ Cancel: $$\hspace{.75em}5.1\cancel{111...}$$ $$\underline{-0.5\cancel{111...}}$$ $$\hspace{.75em}4.6$$ Our equation becomes:

9x = 4.6

Step 4) Solve the equation $$9x = 4.6$$ Clear the decimal on the right, multiply each side by 10: $$90x = 46$$ Divide each side of the equation by 90: $$x = \frac{46}{90}$$ Simplify: $$x = \frac{23}{45}$$ $$0.5111... = \frac{23}{45}$$ Example 2: Convert each repeating decimal into a fraction

0.6

We will write this number using an ellipsis:

0.6259259259...

Step 1) Set the repeating decimal equal to a variable

x = 0.6259259259...

Step 2) Multiply both sides of the equation by 10

n is the number of digits in the repeating string. Here we have three digits (259) that repeat forever. This means n is 3:

10

1000 • x = 1000 • 0.6259259259...

1000x = 625.9259259...

Step 3) Subtract away the first equation from the second equation

Equation 1) x = 0.6259259259...

Equation 2)

1000x = 625.9259259...

1000x - x = 625.9259259... - 0.6259259...

Left side:

1000x - x = 999x

Right side:

$$\hspace{1.2em}625.9259259...$$ $$\underline{-\hspace{1.5em}0.6259259...}$$ Cancel: $$\hspace{1.2em}625.9\cancel{259259...}$$ $$\underline{-\hspace{1.5em}0.6\cancel{259259...}}$$ $$\hspace{1.2em}625.3$$ Our equation becomes:

999x = 625.3

Clear the decimal on the right, multiply each side by 10:

9990x = 6253

Step 4) Solve the equation $$9990x = 6253$$ Divide each side of the equation by 9990: $$x = \frac{6253}{9990}$$ Simplify: $$x = \frac{169}{270}$$ $$0.6259259259... = \frac{169}{270}$$

Example 3: Convert each repeating decimal into a fraction

0.

Since we have "72" as the repeating part, we will place this in the numerator of our fraction. The denominator will be two 9's or 99 since there are two digits in the number 72: $$\frac{72}{99}$$ Simplify: $$\frac{72}{99} = \frac{8}{11}$$ $$0.\overline{72} = \frac{8}{11}$$

0. » The digit "7" repeats forever

2.5 » The pattern "32" repeats forever

0. » The pattern "9138" repeats forever

We may also see an ellipsis used instead of an overbar:

0. = 0.777...

2.5 = 2.5323232...

0. » = 0.913891389138...

When we use the ellipsis, we must ensure that the digit that repeats or pattern of repeating digits is crystal clear.

To convert a repeating decimal to a fraction, we rely on our knowledge of the addition property of equality, along with the multiplication property of equality. Let's begin by looking at an example.

Suppose we want to convert 0. into a fraction. Let's begin by setting our repeating decimal equal to a variable like x:

x = 0.

It is normally easier to think about this problem using the ellipsis format. Let's convert our equation into:

x = 0.888...

Next, we will multiply both sides of the equation by 10

^{n}, where n is the number of digits in the repeating pattern. In our case, we have one digit 8, that repeats forever. This means n is 1 and 10 to the power of 1 is just 10. We will multiply both sides of our equation by 10:x = 0.888...

10 • x = 10 • 0.888...

10x = 8.888...

Now we will subtract the original equation away from the new equation:

10x - x = 8.888... - 0.888...

Before we go any further, think about why this is legal? We know the addition property of equality allows us to add or subtract the same value to or from both sides of the equation. Since x is equal to 0.888..., this means they are the same value. So when we subtract away x on the left side and subtract away 0.888... on the right side, we are subtracting away the same value.

Left Side:

10x - x = 9x

Right Side: $$\hspace{.75em}8.888...$$ $$\underline{-0.888...}$$ The numbers after the decimal point will cancel: $$\require{cancel}\hspace{.75em}8.\cancel{888...}$$ $$\underline{-0.\cancel{888...}}$$ $$\hspace{.75em}8$$ Our equation becomes:

9x = 8

Now we can solve for x by dividing each side by 9, the coefficient of x: $$9x = 8$$ $$x = \frac{8}{9}$$ Remember x was set equal to (0.888...) when we started. We can replace the x and show: $$0.888... = \frac{8}{9}$$

### Converting a Repeating Decimal into a Fraction

- Set the repeating decimal equal to a variable
- We will refer to this as equation #1

- Multiply both sides of the equation by 10
^{n}where n is the number of digits in the repeating string- We will refer to this as equation #2

- Subtract away equation #1 from equation #2
- We do this by subtracting the left side of equation #1 away from the left side of equation #2 and the right side of equation #1 from the right side of equation #2

- Solve the equation, this will give us the fractional equivalent for our repeating decimal

Example 1: Convert each repeating decimal into a fraction

0.5

We will write this number using an ellipsis:

0.5111...

Step 1) Set the repeating decimal equal to a variable

x = 0.5111...

Step 2) Multiply both sides of the equation by 10

^{n}n is the number of digits in the repeating string. Here we have one digit (1) that repeats forever. This means n is 1:

10 • x = 10 • 0.5111...

10x = 5.111...

Step 3) Subtract away the first equation from the second equation

Equation 1) x = 0.5111...

Equation 2) 10x = 5.111...

10x - x = 5.111... - 0.5111...

Left side:

10x - x = 9x

Right side: $$\hspace{.75em}5.1111...$$ $$\underline{-0.5111...}$$ Cancel: $$\hspace{.75em}5.1\cancel{111...}$$ $$\underline{-0.5\cancel{111...}}$$ $$\hspace{.75em}4.6$$ Our equation becomes:

9x = 4.6

Step 4) Solve the equation $$9x = 4.6$$ Clear the decimal on the right, multiply each side by 10: $$90x = 46$$ Divide each side of the equation by 90: $$x = \frac{46}{90}$$ Simplify: $$x = \frac{23}{45}$$ $$0.5111... = \frac{23}{45}$$ Example 2: Convert each repeating decimal into a fraction

0.6

We will write this number using an ellipsis:

0.6259259259...

Step 1) Set the repeating decimal equal to a variable

x = 0.6259259259...

Step 2) Multiply both sides of the equation by 10

^{n}n is the number of digits in the repeating string. Here we have three digits (259) that repeat forever. This means n is 3:

10

^{3}= 10001000 • x = 1000 • 0.6259259259...

1000x = 625.9259259...

Step 3) Subtract away the first equation from the second equation

Equation 1) x = 0.6259259259...

Equation 2)

1000x = 625.9259259...

1000x - x = 625.9259259... - 0.6259259...

Left side:

1000x - x = 999x

Right side:

$$\hspace{1.2em}625.9259259...$$ $$\underline{-\hspace{1.5em}0.6259259...}$$ Cancel: $$\hspace{1.2em}625.9\cancel{259259...}$$ $$\underline{-\hspace{1.5em}0.6\cancel{259259...}}$$ $$\hspace{1.2em}625.3$$ Our equation becomes:

999x = 625.3

Clear the decimal on the right, multiply each side by 10:

9990x = 6253

Step 4) Solve the equation $$9990x = 6253$$ Divide each side of the equation by 9990: $$x = \frac{6253}{9990}$$ Simplify: $$x = \frac{169}{270}$$ $$0.6259259259... = \frac{169}{270}$$

### Shortcut - Converting Repeating Decimals to Fractions

If the series of repeating digits begins immediately after the decimal point, we can use a shortcut to perform our conversion.- Write the repeating part of the number in the numerator of a fraction
- Write the denominator of the fraction as the same number of 9's as we have digits in the numerator
- Simplify the fraction

Example 3: Convert each repeating decimal into a fraction

0.

Since we have "72" as the repeating part, we will place this in the numerator of our fraction. The denominator will be two 9's or 99 since there are two digits in the number 72: $$\frac{72}{99}$$ Simplify: $$\frac{72}{99} = \frac{8}{11}$$ $$0.\overline{72} = \frac{8}{11}$$

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