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Solution - Fraction to decimal (repeating)

Final Result Steps Terms and topics

0. (181102362204724409448818897637795275590551)

0.(181102362204724409448818897637795275590551)

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Step-by-step explanation

1. Parse fraction to decimal repeating input

$\frac{46}{254}$

Operation detected: fraction to decimal repeating conversion.

$\frac{46}{254}$

Normalize numerator and denominator from the input.

$\frac{46}{254}$

Rewrite the fraction as a long-division setup.

$46 \div 254$

Plan: generate digits while tracking remainders to detect repetition.

$46 \div 254$

Parse the fraction and set up long division before generating decimal digits.

2. Execute long-division cycle transcript

$\frac{46}{254}$

At each cycle: multiply remainder by 10, divide by denominator, record digit and new remainder.

$\frac{46}{254}$

$\frac{460}{254} = d i g i t 1, r e m a \in d e r 206$

This cycle contributes one new decimal digit.

$206$

$\frac{2060}{254} = d i g i t 8, r e m a \in d e r 28$

This cycle contributes one new decimal digit.

$28$

$\frac{280}{254} = d i g i t 1, r e m a \in d e r 26$

This cycle contributes one new decimal digit.

$26$

$\frac{260}{254} = d i g i t 1, r e m a \in d e r 6$

This cycle contributes one new decimal digit.

$6$

$\frac{60}{254} = d i g i t 0, r e m a \in d e r 60$

This cycle contributes one new decimal digit.

$60$

$\frac{600}{254} = d i g i t 2, r e m a \in d e r 92$

This cycle contributes one new decimal digit.

$92$

$\frac{920}{254} = d i g i t 3, r e m a \in d e r 158$

This cycle contributes one new decimal digit.

$158$

$\frac{1580}{254} = d i g i t 6, r e m a \in d e r 56$

This cycle contributes one new decimal digit.

$56$

$\frac{560}{254} = d i g i t 2, r e m a \in d e r 52$

This cycle contributes one new decimal digit.

$52$

$\frac{520}{254} = d i g i t 2, r e m a \in d e r 12$

This cycle contributes one new decimal digit.

$12$

$\frac{120}{254} = d i g i t 0, r e m a \in d e r 120$

This cycle contributes one new decimal digit.

$120$

$\frac{1200}{254} = d i g i t 4, r e m a \in d e r 184$

This cycle contributes one new decimal digit.

$184$

$\frac{1840}{254} = d i g i t 7, r e m a \in d e r 62$

This cycle contributes one new decimal digit.

$62$

$\frac{620}{254} = d i g i t 2, r e m a \in d e r 112$

This cycle contributes one new decimal digit.

$112$

Cycle transcript shortened for readability; continue using the same pattern.

$14$

A remainder repeated, so digits from that point form the repeating cycle.

$0. (181102362204724409448818897637795275590551)$

A repeated remainder appears, so the repeating block is $181102362204724409448818897637795275590551$ .

3. Conclude decimal representation

$46 / 254 = 0. (181102362204724409448818897637795275590551)$

$0. (181102362204724409448818897637795275590551)$

Conclusion: the fraction converts to a repeating decimal.

$0. (181102362204724409448818897637795275590551)$

Study tip: keep parentheses around repeating digits to preserve exact value.

$0. (181102362204724409448818897637795275590551)$

Write the repeating decimal with parentheses: $0. (181102362204724409448818897637795275590551)$ .

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Why learn this

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Repeating decimals are common in algebra and statistics. Detecting cycles helps with exact arithmetic and checking answers.

Terms and topics

Fraction to decimal (repeating)