Solution - Fraction to decimal (repeating)

$$\frac{82251}{270725}$$

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

$$\frac{82251}{270725}$$

This cycle contributes one new decimal digit.

$$10335$$

This cycle contributes one new decimal digit.

$$103350$$

This cycle contributes one new decimal digit.

$$221325$$

This cycle contributes one new decimal digit.

$$47450$$

This cycle contributes one new decimal digit.

$$203775$$

This cycle contributes one new decimal digit.

$$142675$$

This cycle contributes one new decimal digit.

$$73125$$

This cycle contributes one new decimal digit.

$$189800$$

This cycle contributes one new decimal digit.

$$2925$$

This cycle contributes one new decimal digit.

$$29250$$

This cycle contributes one new decimal digit.

$$21775$$

This cycle contributes one new decimal digit.

$$217750$$

This cycle contributes one new decimal digit.

$$11700$$

This cycle contributes one new decimal digit.

$$117000$$

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

$$14$$

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

$$0.30\left(381752701080432172869147659063625450180072028811524609843937575030012004801920768307322929171668667466986794717887154861944777911164465786314525810324129651860744297719087635054021608643457382953181272509003601440576230492196878751500600240096038415366146458583433373349339735894357743097238895558223289315726290516206482593037214885954\right)$$

A repeated remainder appears, so the repeating block is $$381752701080432172869147659063625450180072028811524609843937575030012004801920768307322929171668667466986794717887154861944777911164465786314525810324129651860744297719087635054021608643457382953181272509003601440576230492196878751500600240096038415366146458583433373349339735894357743097238895558223289315726290516206482593037214885954$$.

$$0.30\left(381752701080432172869147659063625450180072028811524609843937575030012004801920768307322929171668667466986794717887154861944777911164465786314525810324129651860744297719087635054021608643457382953181272509003601440576230492196878751500600240096038415366146458583433373349339735894357743097238895558223289315726290516206482593037214885954\right)$$

Conclusion: the fraction converts to a repeating decimal.

$$0.30\left(381752701080432172869147659063625450180072028811524609843937575030012004801920768307322929171668667466986794717887154861944777911164465786314525810324129651860744297719087635054021608643457382953181272509003601440576230492196878751500600240096038415366146458583433373349339735894357743097238895558223289315726290516206482593037214885954\right)$$

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

$$0.30\left(381752701080432172869147659063625450180072028811524609843937575030012004801920768307322929171668667466986794717887154861944777911164465786314525810324129651860744297719087635054021608643457382953181272509003601440576230492196878751500600240096038415366146458583433373349339735894357743097238895558223289315726290516206482593037214885954\right)$$

Write the repeating decimal with parentheses: $$0.30\left(381752701080432172869147659063625450180072028811524609843937575030012004801920768307322929171668667466986794717887154861944777911164465786314525810324129651860744297719087635054021608643457382953181272509003601440576230492196878751500600240096038415366146458583433373349339735894357743097238895558223289315726290516206482593037214885954\right)$$.