4.4: Dividing fractions

In the following, “A/B” represents the fraction “A over B.” For example, “2/3” refers to the fraction “2 over 3” or “two-thirds.” In the fraction, A/B, “A” is called the numerator and “B” is called the denominator.

Method 1: divide numerators and divide denominators [not recommended]

3.3: Multiplying fractions showed that to multiply two fractions together, we can multiply the numerators and the denominators separately. Can we do something similar when dividing fractions? In theory we can, although in practice this method doesn’t always work out neatly. For example, here’s one fraction division problem that works out neatly using this method:

• 3/8 ÷ 3/4 (three-eighths divided by three-quarters).
• = (3÷3) / (8÷4)    [divide numerators and denominators separately]
• = 1/2

However, here’s another fraction division problem that doesn’t work out so neatly:

• 1/3 ÷ 3/5 (one-third divided by three-fifths).
• = (1÷3) / (3÷5)    [divide numerators and denominators separately]
• = now what?

Method 2: express fractions using common denominators and divide numerators

Here’s a better method that always works and makes sense intuitively. To divide two fractions using this method, express the fractions using common denominators and divide the numerators. For example,

• 3/8 ÷ 3/4 = ?
• A common denominator for 8 and 4 is 8.
• Express 3/4 as (3×2)/(4×2) = 6/8.
• So, 3/8 ÷ 3/4 = 3/8 ÷ 6/8.
• Now, divide the numerators: 3/8 ÷ 3/4 = 3/8 ÷ 6/8 = 3 ÷ 6 = 1/2.
• This works because we can use the “how-many-groups” interpretation of division to represent “3/8 ÷ 6/8” as, “How many times does 6/8 go into 3/8?” which is the same as asking “How many times does 6 go into 3?” Answer: 1/2 (one-half).

Here’s another example:

• 1/3 ÷ 3/5 = ?
• A common denominator for 3 and 5 is 15.
• Express 1/3 as (1×5)/(3×5) = 5/15.
• Express 3/5 as (3×3)/(5×3) = 9/15.
• So, 1/3 ÷ 3/5 = 5/15 ÷ 9/15 = 5 ÷ 9 = 5/9.

Method 3: invert and multiply (also known as “keep, change, flip”)

To divide two fractions using this method, invert the divisor fraction and multiply. Equivalently, keep the dividend fraction, change division to multiplication, and flip the divisor fraction. For example,

• 3/8 ÷ 3/4
• = 3/8 x 4/3    [invert the divisor fraction and multiply]
• = (3×4) / (8×3)    [multiply numerators and denominators separately]
• = 12/24
• = 1/2    [simplify]
• This works because we can use the “how-many-units-in-one-group” interpretation of division to represent “3/8 ÷ 3/4” as, “How many units are in one group if 3/8 units are divided into 3/4 groups?” To scale 3/4 groups to one group, multiply by 4/3, so also scale 3/8 units by multiplying by 4/3.

Here’s another example:

• 1/3 ÷ 3/5
• = 1/3 x 5/3
• = (1×5) / (3×3)
• = 5/9

The video below works through some examples of dividing fractions.