Dividing fractions

Quotient of two rational numbers

The quotient of the integer −6 by the integer 2 is the integer −3, since: −6=2⋅(−3)

This exercise of multiplying integers can be written as a division:(−6):2=−3↗↑↖dividend  divisor quotient

In the same way, the rational number320 can be expressed as the product of the rational number 34 by another one. Which rational is this another one?

We can prove that this rational is 15:320=34⋅15=3⋅14⋅5And then we say that the quotient of the division of 320 by 34 is equal to 15. In the same way as done with integers, the exercise of multiplying 320=34⋅ can be written as a division:320:34=?.

Calculating the quotient of two rational numbers

  1. The exercise: 320:34=? can be written as: ?⋅34=320

  2. Multiplying both terms of the equality by the inverse of the divisor: (?⋅34)⋅43=(320⋅43)

  3. Bearing in mind the properties of the product of fractions, we obtain: ?⋅34⋅43=320⋅43 And as 34⋅43=1, we have?⋅1=320⋅43=3⋅420⋅3=420=15Therefore:320:34=320⋅43=15

Namely, to find the quotient of two rational numbers ab (dividend) and cd (divisor), the divisor being other than zero, it is necessary to multiply the dividend by the inverse of the divisor:ab:cd=ab⋅dc

Example

Calculate the quotient of −45 by −32:−45:(−32)We multiply the dividend −45 by the inverse of the divisor−32, that is −23:−45:(−32)=−45⋅(−23)=−45⋅−23=815

In the same way as with integers, when we have an expression with sums, subtractions, multiplications and divisions of fractions, we must operate first on the brackets, later the multiplications and the divisions and, finally, on the sums and subtractions.

Quotient of a rational number and an integer

To divide an integer a by a rational number mn we have to express the integer a as a1 and proceed as in the previous case:a:mn=a1:mn=a1⋅nm

And, in the same way, to divide a rational number mn by an integer a, we do: mn:a=mn:a1=nm⋅1a