Converting Mixed Repeating Decimals to Fractions
Converting a decimal with both non-repeating and repeating parts to a fraction requires multiplying the original decimal by two different powers of 10 to align the repeating tails, then subtracting. For 0.1 repeating 2: multiply by 100 to get 12.222... and by 10 to get 1.222..., subtract to get 90x = 11, so x = 11/90. For 0.8 repeating 3: 100x = 83.333... and 10x = 8.333..., subtract to get 90x = 75, so x = 5/6. This algebraic technique from enVision Mathematics, Grade 8, Chapter 1 handles the more complex case of mixed repeating decimals.
Key Concepts
To convert a decimal with a non repeating part, create two equations by multiplying the original decimal, $x$, by two different powers of 10. The goal is to create two new equations where the repeating decimal parts are perfectly aligned. Subtracting the smaller equation from the larger one will then eliminate the repeating tail, allowing you to solve for $x$.
Common Questions
How do I convert a mixed repeating decimal like 0.18 repeating to a fraction?
Let x = 0.1888... Multiply by 100 and by 10 to align the tails: 100x = 18.888... and 10x = 1.888... Subtract: 90x = 17, so x = 17/90.
Convert 0.1 repeating 2 to a fraction.
Let x = 0.1222... Multiply by 100: 100x = 12.222... Multiply by 10: 10x = 1.222... Subtract: 90x = 11. x = 11/90.
Convert 0.8 repeating 3 to a fraction.
Let x = 0.8333... Multiply by 100: 100x = 83.333... Multiply by 10: 10x = 8.333... Subtract: 90x = 75. x = 75/90 = 5/6.
Why multiply by two different powers of 10?
You need two equations where the repeating tails align exactly so they cancel when subtracted. The exponents of 10 are chosen based on how many non-repeating digits and how many repeating digits there are.
How do I choose which two powers of 10 to use?
Multiply by 10 to the power of (non-repeating digits + repeating digits) and by 10 to the power of (non-repeating digits only). Subtracting eliminates the repeating part.
When do 8th graders learn to convert mixed repeating decimals?
Chapter 1 of enVision Mathematics, Grade 8 covers this in the Real Numbers unit.