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Solution - Fraction to decimal (repeating)

Final Result Steps Terms and topics

15. (0344827586206896551724137931)

15.(0344827586206896551724137931)

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Step-by-step explanation

1. Parse fraction to decimal repeating input

$\frac{436}{29}$

Operation detected: fraction to decimal repeating conversion.

$\frac{436}{29}$

Normalize numerator and denominator from the input.

$\frac{436}{29}$

Rewrite the fraction as a long-division setup.

$436 \div 29$

Plan: generate digits while tracking remainders to detect repetition.

$436 \div 29$

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

2. Execute long-division cycle transcript

$\frac{436}{29}$

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

$\frac{436}{29}$

$\frac{10}{29} = d i g i t 0, r e m a \in d e r 10$

This cycle contributes one new decimal digit.

$10$

$\frac{100}{29} = d i g i t 3, r e m a \in d e r 13$

This cycle contributes one new decimal digit.

$13$

$\frac{130}{29} = d i g i t 4, r e m a \in d e r 14$

This cycle contributes one new decimal digit.

$14$

$\frac{140}{29} = d i g i t 4, r e m a \in d e r 24$

This cycle contributes one new decimal digit.

$24$

$\frac{240}{29} = d i g i t 8, r e m a \in d e r 8$

This cycle contributes one new decimal digit.

$8$

$\frac{80}{29} = d i g i t 2, r e m a \in d e r 22$

This cycle contributes one new decimal digit.

$22$

$\frac{220}{29} = d i g i t 7, r e m a \in d e r 17$

This cycle contributes one new decimal digit.

$17$

$\frac{170}{29} = d i g i t 5, r e m a \in d e r 25$

This cycle contributes one new decimal digit.

$25$

$\frac{250}{29} = d i g i t 8, r e m a \in d e r 18$

This cycle contributes one new decimal digit.

$18$

$\frac{180}{29} = d i g i t 6, r e m a \in d e r 6$

This cycle contributes one new decimal digit.

$6$

$\frac{60}{29} = d i g i t 2, r e m a \in d e r 2$

This cycle contributes one new decimal digit.

$2$

$\frac{20}{29} = d i g i t 0, r e m a \in d e r 20$

This cycle contributes one new decimal digit.

$20$

$\frac{200}{29} = d i g i t 6, r e m a \in d e r 26$

This cycle contributes one new decimal digit.

$26$

$\frac{260}{29} = d i g i t 8, r e m a \in d e r 28$

This cycle contributes one new decimal digit.

$28$

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

$14$

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

$15. (0344827586206896551724137931)$

A repeated remainder appears, so the repeating block is $0344827586206896551724137931$ .

3. Conclude decimal representation

$436 / 29 = 15. (0344827586206896551724137931)$

$15. (0344827586206896551724137931)$

Conclusion: the fraction converts to a repeating decimal.

$15. (0344827586206896551724137931)$

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

$15. (0344827586206896551724137931)$

Write the repeating decimal with parentheses: $15. (0344827586206896551724137931)$ .

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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)