What does it mean to divide two fractions?

That you could convert any pair of fractions to have a common denominator via cross multiplying. For illustration: 3/four and three/5 come to be: three/four * 5 and three/5 * four => 15/20 and 12/20 You could add these two fractions together to get a sum of 27/20. In this case the lowest long-established Denominator is 20, since you can't scale down both fractions. However in some cases you might. For example: three/4 and 3/eight end up three/four * 8 and 3/8 * 4 => 24/32 and 12/32 These fractions may also be lowered dividing the entire numerators and denominators by using 4 to get 6/eight and three/eight. In this case, 8 is the bottom fashioned Denominator.

There are a lot of parallels between addition and multiplication.

For every x, y, and z:

ADDITIVE ASSOCIATIVITY: (x + y) + z = x + (y + z).

ADDITIVE COMMUTATIVITY: x + y = y + x.

ADDITIVE IDENTITY: x + 0 = x.

MULTIPLICATIVE ASSOCIATIVITY: (x • y) • z = x • (y • z).

MULTIPLICATIVE COMMUTATIVITY: x • y = y • x.

MULTIPLICATIVE IDENTITY: x • 1 = x.

Notice how symmetrical these properties are. There is a slight unsymmetry when we get to inverses.

ADDITITVE INVERSE: For every x there is a ⁻x such that x + (⁻x) = 0. ⁻x is referred to as the opposite of x.

MULTIPLICATIVE INVERSE: For every x ≠ 0, there is a 1/x such that x • (1/x) = 1. 1/x is referred to as the reciprocal of x.

The unsymmetry is that 0 does not have a reciprocal. This is necessary because there is no number 1/x such that 0 • (1/x) = 1. This fact drives a lot of students crazy but it is what it is.

Having said all of this, now we can answer your question.

DEFINITION OF SUBTRACTION: x - y is defined as x + (⁻y)

DEFINITION OF DIVISION: If y ≠ 0, then x/y = x • (1/y)

So, BY DEFINITION, x/y is x times the reciprocal of y.

So (a/b) / (c/d) is a/b times the reciprocal of c/d

So (a/b) / (c/d) = (a/b) • (d/c)

you have half a pie and divide it into pieces the size of 1/3 of a pie. how many such pieces will you get?

1/2 ÷ 1/3 =

3/6 ÷ 2/6 =

1½ ÷ 1

cut your half pie into 3 equal pieces (each is equal to 1/6 of the pie), and take 2 of them, which is 2/6 = 1/3 of the pie. the 1 piece left over is half the size of your piece.

or try it with money. you have $4.50 (in quarters) and divide it into $0.75 piles. how many piles?

4½ ÷ ¾ = 9/2 • 4/3 = 36/6 = 6 complete piles.

division by anything is always multiplying by 1/<that something>. fractions are not special in this regard.