Question

Will anybody explain the steps of diving a fraction?

Use this equation to explain: 8/9 ÷ 3/8

Answer 1

64/27

Explanation:

Take the reciprocal of 3/8: 8/3

Multiply this number by 8/9: 8/9 * 8/3 = 64/27

Therefore, 8/9 ÷ 3/8 = 64/27

Answer 2

64/27

Explanation:

You want the steps of dividing fractions.

Division

Division by anything is the same as multiplying by its multiplicative inverse. The multiplicative inverse is also called the reciprocal. The reciprocal of a fraction is found by swapping the numerator and denominator.

`(8)/(9)/(3)/(8)=(8)/(9)*(8)/(3)`

The multiplication is carried out in the usual way. The numerator is the product of the numerators, and the denominator is the product of the denominators.

`(8)/(9)*(8)/(3)=(8*8)/(9*3)=\boxed{(64)/(27)}`

Alternate methods

When the fractions have the same denominator, their ratio is the ratio of their numerators.

`(\left((8)/(9)\right))/(\left((3)/(8)\right))=(\left((64)/(72)\right))/(\left((27)/(72)\right))=\boxed{(64)/(27)}`

When the fractions have the same numerator, their ratio is the inverse of the ratio of their denominators.

`(\left((8)/(9)\right))/(\left((3)/(8)\right))=(\left((24)/(27)\right))/(\left((24)/(64)\right))=\boxed{(64)/(27)}`

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Additional comment

The method is variously described as "invert and multiply" or "dot swap". In the latter case, the "dot" refers to the multiplication symbol, and the "swap" refers to swapping the numerator and denominator of the divisor fraction.

`(8)/(9)/(3)/(8)=(8)/(9)\cdot(8)/(3)`

If you divide the fractions shown in the alternate methods using "dot swap", you find the common numbers cancel, confirming these methods give the same result.

Usually, it is only worthwhile to use one of the alternate methods if the numerators or denominators are already the same. We have gone to the trouble to express the fractions here with the same numerators or denominators simply to show the method.

Many calculators and spreadsheets will deal with fractions directly.

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