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Solution - Fraction to decimal (repeating)

Final Result Steps Terms and topics

0. (530920060331825037707390648567119155354449472096)

0.(530920060331825037707390648567119155354449472096)

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Step-by-step explanation

1. Parse fraction to decimal repeating input

$\frac{352}{663}$

Operation detected: fraction to decimal repeating conversion.

$\frac{352}{663}$

Normalize numerator and denominator from the input.

$\frac{352}{663}$

Rewrite the fraction as a long-division setup.

$352 \div 663$

Plan: generate digits while tracking remainders to detect repetition.

$352 \div 663$

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

2. Execute long-division cycle transcript

$\frac{352}{663}$

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

$\frac{352}{663}$

$\frac{3520}{663} = d i g i t 5, r e m a \in d e r 205$

This cycle contributes one new decimal digit.

$205$

$\frac{2050}{663} = d i g i t 3, r e m a \in d e r 61$

This cycle contributes one new decimal digit.

$61$

$\frac{610}{663} = d i g i t 0, r e m a \in d e r 610$

This cycle contributes one new decimal digit.

$610$

$\frac{6100}{663} = d i g i t 9, r e m a \in d e r 133$

This cycle contributes one new decimal digit.

$133$

$\frac{1330}{663} = d i g i t 2, r e m a \in d e r 4$

This cycle contributes one new decimal digit.

$4$

$\frac{40}{663} = d i g i t 0, r e m a \in d e r 40$

This cycle contributes one new decimal digit.

$40$

$\frac{400}{663} = d i g i t 0, r e m a \in d e r 400$

This cycle contributes one new decimal digit.

$400$

$\frac{4000}{663} = d i g i t 6, r e m a \in d e r 22$

This cycle contributes one new decimal digit.

$22$

$\frac{220}{663} = d i g i t 0, r e m a \in d e r 220$

This cycle contributes one new decimal digit.

$220$

$\frac{2200}{663} = d i g i t 3, r e m a \in d e r 211$

This cycle contributes one new decimal digit.

$211$

$\frac{2110}{663} = d i g i t 3, r e m a \in d e r 121$

This cycle contributes one new decimal digit.

$121$

$\frac{1210}{663} = d i g i t 1, r e m a \in d e r 547$

This cycle contributes one new decimal digit.

$547$

$\frac{5470}{663} = d i g i t 8, r e m a \in d e r 166$

This cycle contributes one new decimal digit.

$166$

$\frac{1660}{663} = d i g i t 2, r e m a \in d e r 334$

This cycle contributes one new decimal digit.

$334$

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

$14$

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

$0. (530920060331825037707390648567119155354449472096)$

A repeated remainder appears, so the repeating block is $530920060331825037707390648567119155354449472096$ .

3. Conclude decimal representation

$352 / 663 = 0. (530920060331825037707390648567119155354449472096)$

$0. (530920060331825037707390648567119155354449472096)$

Conclusion: the fraction converts to a repeating decimal.

$0. (530920060331825037707390648567119155354449472096)$

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

$0. (530920060331825037707390648567119155354449472096)$

Write the repeating decimal with parentheses: $0. (530920060331825037707390648567119155354449472096)$ .

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Why learn this

Learn more with Tiger

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

Terms and topics

Fraction to decimal (repeating)