1. Core Focus on Linear Equations
Lesson 4.9
Core Focus on Linear Equations
Converting Repeating Decimals
to Fractions

2. Warm-Up 1. Convert the decimal 0.5 into a fraction.
3. Convert the number 4.12 into a mixed number.
4. Convert the number into a mixed number.

3. Converting Repeating Decimals to Fractions
Lesson 4.9
Converting Repeating Decimals to Fractions
Convert repeating decimals to fractions.

4. Vocabulary
Rational Number A number that can be expressed as a fraction of two integers. Good to know!  Every rational number can be written as a decimal number.  Some of these decimals will terminate, while others will repeat.
Terminating Decimals
Repeating Decimals

5. Converting a Repeating Decimal to a Fraction
Let x equal the repeating portion of the decimal number.
Multiply both sides of the equation by a power of ten to move the repeating digit(s) to the left side of the decimal point.
Subtract x from both sides of the equation.
Solve for x.

6. Remember that x is equal to 0.444…
Example 1
Convert to a fraction. Let x equal the repeating decimal. Multiply both sides of the equation by 10. This moves the repeating digit to the left side of the decimal point. Subtract x from both sides of the equation. Divide both sides of the equation by 9.
Remember that x is equal to 0.444…
The fraction form of is .

7. Example 2
David had ounces of silver. What is this amount as a fraction?
Let x equal the repeating decimal. Multiply both sides of the equation by 100. This moves the repeating digit to the left side of the decimal point. Subtract x from both sides of the equation. Divide both sides of the equation by 99. The fraction form of is .
Ignore the whole number while finding the fraction.
Remember: x = …
Now that the fraction is found, add back in the whole number.

8. Any other power of 10 will work.
Example 3
What is as a fraction?
When the repeating decimal digits fall after digits that do not repeat, you need to set up a system of equations to find the equivalent fractions.
Let x equal the repeating decimal. Multiply both sides of the equation by 100. This moves the repeating digit (3) to the left side of the decimal point. Create another equation by multiplying both sides of the original equation by a different power of 10.
Any other power of 10 will work.

9. Example 3 Continued… What is as a fraction?
Subtract the equations from one another. Divide both sides of the equation by 90. The fraction form of is .

10. Communication Prompts
What are some reasons people would choose to use repeating decimals rather than fractions?
What are some reasons people would choose to use fractions rather than using repeating decimals?

11. Exit Problems Convert each repeating decimal to a rational number. 1.2. 3.