Repeating Decimals to Fractions Practice Problems & Conceptual Mastery
Repeating decimals are rational numbers because they can always be expressed as a ratio of two integers using algebraic subtraction.
📝 Worked Step-by-Step Example
1. Let x = 0.2727... 2. 100x = 27.2727... 3. 100x − x = 27 ➔ 99x = 27. 4. x = 27/99 = 3/11.
🧠 Core Problem-Solving Strategies
1. Set x equal to the decimal
Let x = 0.aaaa... to define an algebraic equation.
2. Multiply by powers of 10
Multiply by 10, 100, etc. so the repeating decimal block aligns perfectly after the decimal point.
3. Subtract to eliminate repeating part
Subtracting the equations cancels out the infinite repeating tail completely.
📖 Key Mathematical Terms & Vocabulary
Repeating Decimal
A decimal with a digit or sequence of digits that repeats infinitely.
Period / Repetend
The repeating group of digits in a recurring decimal.
❓ Frequently Asked Questions
Why is the denominator often 9 or 99?
Because multiplying by 10 or 100 and subtracting 1x leaves 9x or 99x.
Is every repeating decimal rational?
Yes! Every single repeating decimal can be written as a fraction a/b.
What notation represents repeating digits?
A bar placed directly over the repeating digits (e.g. 0.3̄).
🗺️ Pedagogical Learning Roadmap
Comparing Fractions→
Master core Comparing Fractions concepts to build computational fluency.
PrerequisiteFractions on a Number Line→
Master core Fractions on a Number Line concepts to build computational fluency.
Next StepOrdering Fractions (Least to Greatest)→
Advance to Ordering Fractions (Least to Greatest) practice drills and timed challenges.
Next StepShape Fractions & Unit Fractions→
Advance to Shape Fractions & Unit Fractions practice drills and timed challenges.
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