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FDP: Recurring Decimals
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Foundational Skills
Understand recurring decimal notation
\[ 0.272727… \rightarrow 0.\dot{2}\dot{7} \]
Write a recurring decimal using dot notation.
Read recurring decimal notation
\[ 0.\dot{1}\dot{8} = 0.1818… \]
Write out a recurring decimal from its dot notation.
Identify terminating vs recurring
\[ \frac{7}{12} \rightarrow \text{?} \]
Determine if a fraction gives terminating or recurring decimal.
Identify the recurring digits
\[ \frac{5}{12} \rightarrow \text{which recurs?} \]
Identify which digits form the recurring part.
Fraction to Recurring Decimal
Thirds to recurring decimal
\[ \frac{2}{3} = 0.\dot{6} \]
Convert a fraction with denominator 3 to recurring decimal.
Ninths to recurring decimal
\[ \frac{7}{9} = 0.\dot{7} \]
Convert a fraction with denominator 9 to recurring decimal.
Sixths to recurring decimal
\[ \frac{5}{6} = 0.8\dot{3} \]
Convert a fraction with denominator 6 to recurring decimal.
Elevenths to recurring decimal
\[ \frac{4}{11} = 0.\dot{3}\dot{6} \]
Convert a fraction with denominator 11 to recurring decimal.
Sevenths to recurring decimal
\[ \frac{3}{7} = 0.\dot{4}2857\dot{1} \]
Convert a fraction with denominator 7 to recurring decimal.
Twelfths to recurring decimal
\[ \frac{7}{12} = 0.58\dot{3} \]
Convert a fraction with denominator 12 to recurring decimal.
Other unit fractions
\[ \frac{1}{15} = 0.0\dot{6} \]
Convert other unit fractions to recurring decimals.
Non-unit fractions
\[ \frac{5}{11} = 0.\dot{4}\dot{5} \]
Convert non-unit fractions to recurring decimals.
Mixed numbers
\[ 2\frac{1}{3} = 2.\dot{3} \]
Convert a mixed number to a recurring decimal.
Recurring to Fraction: Immediate Recurrence
Pattern recognition: ninths
\[ 0.\dot{4} = \frac{4}{9} \]
Single recurring digit gives denominator 9.
Pattern recognition: ninety-ninths
\[ 0.\dot{2}\dot{7} = \frac{3}{11} \]
Two recurring digits give denominator 99.
Pattern recognition: 999ths
\[ 0.\dot{1}2\dot{5} = \frac{125}{999} \]
Three recurring digits give denominator 999.
Recurring to Fraction: Algebraic Method
Set up equation for immediate recurrence
\[ x = 0.\dot{2}\dot{3} \rightarrow \times ? \]
Identify the correct multiplier to use.
Single recurring digit (algebraic)
\[ 0.\dot{7} \rightarrow \frac{7}{9} \]
Use algebra for one recurring digit.
Two recurring digits (algebraic)
\[ 0.\dot{1}\dot{8} \rightarrow \frac{2}{11} \]
Use algebra for two recurring digits.
Three recurring digits (algebraic)
\[ 0.\dot{1}2\dot{3} \rightarrow \frac{41}{333} \]
Use algebra for three recurring digits.
Recurring to Fraction: Delayed Recurrence
Identify multipliers for delayed recurrence
\[ 0.1\dot{6} \rightarrow \times 10, \times 100 \]
Identify both multipliers needed.
One non-recurring, one recurring
\[ 0.1\dot{6} = \frac{1}{6} \]
Convert decimals like 0.16̇ to fractions.
One non-recurring, two recurring
\[ 0.4\dot{1}\dot{6} \rightarrow \text{fraction} \]
Convert decimals like 0.416̇ to fractions.
Two non-recurring, one recurring
\[ 0.08\dot{3} = \frac{1}{12} \]
Convert decimals like 0.083̇ to fractions.
Mixed recurring pattern
\[ 0.2\dot{7} \rightarrow \text{fraction} \]
Convert various delayed recurrence patterns.
Special Cases
Recurring percentage to fraction
\[ 33.\dot{3}\% = \frac{1}{3} \]
Convert recurring percentages like 33.3̇% to fractions.
Recurring percentage: non-benchmark
\[ 8.\dot{3}\% = \frac{1}{12} \]
Convert non-benchmark recurring percentages.
Round a recurring decimal
\[ 0.\dot{1}\dot{8} \approx 0.182 \text{ (3 d.p.)} \]
Round a recurring decimal to specified decimal places.
Order recurring and terminating decimals
\[ 0.3, \; 0.\dot{3}, \; 0.34 \]
Order a mix of recurring and terminating decimals.
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