Converting Repeating Decimals to Fractions
To convert a repeating decimal to a fraction in 8th grade, set the decimal equal to x, multiply by 10 to the power of the repeating block length to create a new equation with aligned repeating tails, then subtract the original equation to cancel the infinite tail. For 0.5 repeating: let x = 0.555..., multiply by 10 to get 10x = 5.555..., subtract to get 9x = 5, so x = 5/9. For 0.8 repeating 3: multiply by 100 and 10, subtract to get 90x = 75, so x = 5/6. This algebraic technique from enVision Mathematics, Grade 8, Chapter 1 proves that all repeating decimals are rational numbers.
Key Concepts
Property To convert a repeating decimal to a fraction, set the decimal equal to a variable $x$. Multiply the equation by powers of 10 to create two new equations where the infinitely repeating decimal parts are perfectly aligned. Subtracting the smaller equation from the larger one will cancel out the infinite repeating tail, leaving a simple algebraic equation to solve for $x$.
Examples Example 1 (Pure Repeating): Convert $0.\overline{5}$ to a fraction. Let $x = 0.555...$ Multiply by 10 (since 1 digit repeats): $10x = 5.555...$ Subtract the original equation: $$10x x = 5.555... 0.555...$$ $$9x = 5$$ $$x = \frac{5}{9}$$.
Example 2 (Mixed Repeating): Convert $0.8\overline{3}$ to a fraction. Let $x = 0.8333...$ Multiply by 10 and 100 to create two equations with aligned repeating parts: $$100x = 83.333...$$ $$10x = 8.333...$$ Subtract them: $$100x 10x = 83.333... 8.333...$$ $$90x = 75$$ $$x = \frac{75}{90} = \frac{5}{6}$$.
Common Questions
How do I convert a repeating decimal to a fraction?
Let x equal the repeating decimal. Multiply by 10 to the power of the repeating block length to create a new equation with aligned tails. Subtract the original equation to cancel the repeating part and solve for x.
Convert 0.7 repeating to a fraction.
Let x = 0.777... Multiply by 10: 10x = 7.777... Subtract: 9x = 7. So x = 7/9.
Convert 0.45 repeating to a fraction.
Let x = 0.454545... Multiply by 100: 100x = 45.4545... Subtract: 99x = 45. So x = 45/99 = 5/11.
Why does subtracting the two equations work?
When both equations have the same infinitely repeating decimal tail, subtracting makes those tails cancel out, leaving whole numbers that can be solved with basic algebra.
Is 0.333... rational or irrational?
It is rational. 0.333... = 1/3, which is a ratio of two integers. All repeating decimals are rational numbers.
When do 8th graders learn to convert repeating decimals to fractions?
Chapter 1 of enVision Mathematics, Grade 8 covers this in the Real Numbers unit.