Dividing fractions asks how much of one quantity fits into another, or finds a quotient that reverses multiplication. Keep the first fraction and multiply by the reciprocal of the divisor. The divisor must not be zero, and every original denominator must also be nonzero.
Why only the divisor is inverted
Multiplying by a nonzero number and then by its reciprocal returns the original quantity. That makes reciprocal multiplication the inverse operation to multiplying by the divisor. Inverting both fractions or only the first computes a different quantity.
In this notation b and d cannot be zero because they are denominators. c also cannot be zero because it is the numerator of the divisor. The dividend a may be zero: zero divided by a nonzero fraction is zero.
Whole numbers have denominator one
Write a whole number as itself over one before taking its reciprocal. This matters whether the whole number is the dividend or divisor. The following examples divide 3 by four thirds and five quarters by 5.
The first quotient exceeds 1; the second is one quarter. Division does not always make a positive number smaller: dividing by a positive number below 1 makes it larger.
Worked example: three fifths divided by seven thirds
Leave three fifths unchanged. Replace the divisor seven thirds by its reciprocal three sevenths, then multiply numerators and denominators. The resulting nine thirty-fifths is already reduced because 9 and 35 share no factor greater than 1.
Several divisions must respect their grouping
Without brackets, evaluate divisions from left to right. For three quarters divided by five sevenths and then by four fifths, you may multiply the first by the reciprocals of the next two. This is a left-associated calculation, not a general permission to ignore brackets.
Dividing by a grouped quotient is different: work out that divisor first or invert the whole grouped fraction. To check an answer to one division, multiply the quotient by the original divisor and recover the dividend. These results belong to the rational numbers whenever the inputs are rational and division is defined.
