Division can mean splitting into equal parts, or asking how many of one amount fit inside another. Both ideas help us understand why dividing by a fraction turns into multiplying by its reciprocal.
Start with a whole-number divisor
Dividing by 2 means taking one half
Start with 12 objects. We want to divide them by 2.
Step 1: begin with all 12
↓
Step 2: split the 12 into 2 equal groups
6
6
The two groups are the same size, so each group contains 6.
↓
Step 3: one of those two equal groups is one half of 12
12 of 12 = 6
So asking for 12 ÷ 2 and asking for one half of 12 give the same amount.
12 ÷ 2=12 × 12= 6
Divide by 2 = take one half.
More generally, dividing by a whole number n is the same as taking one of n equal parts — multiplying by 1n.
Mini-trainer · 6 examples
Division and the matching fraction
For each example, read the division and the equivalent fraction statement together.
Now divide by a fraction
How many halves fit inside 3?
This time, think of division as asking how many pieces of size 12 fit into 3 wholes.
½½
1 whole
½½
1 whole
½½
1 whole
There are 6 halves in 3.
3 ÷ 12 = 6
Unit fractions first
Dividing by a smaller piece gives more pieces
If we divide by one quarter, we are asking how many quarters fit. Four quarters fit inside each whole.
1 ÷ 14
There are 4 quarters in one whole.
= 4
2 ÷ 14
There are 4 quarters in each whole.
= 8
3 ÷ 15
There are 5 fifths in each whole.
= 15
Dividing by 1n has the same effect as multiplying by n.
Watch the division turn into multiplication
a÷1n
a×1n
a×n1=an
÷ changes to
×,
and 1n flips to
n1.
Mini-trainer · 6 examples
Dividing by a unit fraction
Ask how many halves, thirds, quarters or fifths fit.
A general fraction
How many 23s fit into 4?
Watch this example unfold
4÷×23=122=6
Only the operation and divisor change first:
÷ becomes
×, and
23 flips to
32.
Then the calculation continues to
122 = 6.
First count how many thirds are in 4 wholes. Then group those thirds in pairs, because each divisor is two thirds.
Step 1: turn 4 wholes into thirds
4 × 3 = 12 thirds
Step 2: group the thirds in twos
12 ÷ 2 = 6 groups
4 ÷ 23 = 6
Mini-trainer · 6 examples
Whole number ÷ fraction
Turn wholes into the denominator-sized pieces, then group them.
See what happened
Multiply by the denominator, then divide by the numerator
For 23, we multiplied by 3 and divided by 2.
4× 3→12÷ 2→6
4 × 32 = 6
The fraction 32 is the reciprocal of 23: the numerator and denominator swap places.
Now both numbers can be fractions
34 ÷ 25
The rule is exactly the same. Keep the first fraction still. Change ÷ to ×, and swap the numerator and denominator of the divisor.
Watch the divisor flip
34÷×25=158
34 stays unchanged.
÷ becomes
×, and only
25 swaps to
52.
Then multiply: 3 × 54 × 2 =
158.
Mini-trainer · 6 examples
Fraction ÷ fraction
Keep the first fraction, use the reciprocal of the divisor, then multiply.
Watch out
Only flip the number you are dividing by
34
÷
25
=
34
×
52
The first fraction stays as it is. The divisor changes to its reciprocal.
Interactive lab
Which reciprocal do we need?
Choose the reciprocal of the divisor.
Choose an answer.
Combined Trainer
Bring all the division ideas together
Try five fresh questions. Choose guided working or type your final answer. Results feed into your Practice Lab log on this browser/device.
How would you like to answer?
Question 1 of 5 · 0 answered
Type a fraction using /, a mixed number such as 2_1/7 (where _ denotes a press of the space bar), a whole number or a decimal. Equivalent answers are accepted.
1. Reciprocal of the divisor
??
2. Multiply the numerators
×=
3. Multiply the denominators
×=
Result
??
If division facts themselves feel uncertain, use
Times Tables: Reverse
to reconnect multiplication and division.
You have learned
Dividing by a fraction asks how many pieces of that size fit into the amount.
To divide by a fraction, multiply by its reciprocal: keep the first number, flip the divisor, then multiply.
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