Division

You and two friends find a bag of 12 sweets and want to split them so nobody feels hard done by. Sharing a pizza, dealing out cards, putting the same number of buns in each lunchbox — any time something is split up fairly, you are dividing.

Division splits a total into equal parts. If you have a pile of things and you want everyone to get a fair share — the same amount each, nobody more, nobody less — that is division. We write it with the division sign \div:

a \div b = c

You read 12 \div 3 as "twelve divided by three". It means twelve shared into three equal groups — and each group gets four:

strawberry | | = 4 each

Twelve strawberries, split into three equal piles of four: 12 \div 3 = 4 Press play below to deal a pile of dots out one at a time, round and round the groups, until the whole total is shared out evenly. Replay it for a fresh fact each time.

Two ways to picture division

The very same sum, 12 \div 3, can be read in two ways. Both give the answer 4 — they are just two stories about the same picture.

Sharing: "Share 12 among 3." You know how many groups (3) and you want to know how many in each. Deal the 12 out, one to each group, round and round, until they run out — each group ends up with 4.
Grouping: "How many 3s are in 12?" Now you know how big each group is (3) and you want to know how many groups you can make. Scoop the 12 into piles of 3: you can make 4 piles.

Both land on 4, but they ask different questions — one finds how many each, the other finds how many groups. Whenever you meet a division, decide which story fits, then deal or scoop accordingly.

Three hungry frogs have found 6 strawberries and want to share them so everyone gets the same. Hand them out one at a time — one to each frog, again and again — until they are gone. Each frog ends up with 2: 6 \div 3 = 2. That is the sharing story: we knew there were 3 frogs and found 2 strawberries each.

frog strawberry frog frog

We have 8 bananas and every monkey eats exactly 2. This time we don't know how many monkeys — we know each gets 2, and we want to know how many groups of 2 the 8 bananas make. Scoop them into pairs: that's 4 pairs, so we can feed 4 monkeys: 8 \div 2 = 4. That is the grouping story.

monkey banana banana monkey banana banana monkey banana banana monkey banana banana

See it: deal the counters into groups

Here is the sharing story drawn as counters. A pile of dots has been dealt out evenly into a few equal rings — one for each group. Count the dots in any single ring to see how many each group gets, and read the division fact underneath. Press Refresh for a brand new share.

Division undoes multiplication

Division is the inverse of multiplication. Saying 12 \div 3 = 4 is the same as saying 4 \times 3 = 12 — three groups of four make twelve. Every division fact hides a multiplication fact, so knowing your times tables means you already know how to divide: if you can answer "3 times what is 12?" you have just divided.

You can also picture division as repeated subtraction: keep taking away one group of b until nothing is left, and count how many times you could do it. From 12, take away 3 four times to reach 0 — so 12 \div 3 = 4. That is exactly the grouping story again.

Three divisions to try

Once you can share and group, every division works the same way:

15 \div 5 = 3 — share 15 grapes among 5 plates and each plate gets 3; or check it the other way round: 3 \times 5 = 15.
10 \div 2 = 5 — how many 2s are in 10? Count in twos: 2, 4, 6, 8, 10 — that's 5 jumps, so five groups of two.
7 \div 1 = 7 — sharing among just one group means that group keeps everything. Dividing any number by 1 leaves it unchanged.

Two traps that catch out new dividers:

Order matters! Unlike multiplication, you cannot swap the numbers. 12 \div 3 = 4, but 3 \div 12 is much smaller than one — sharing 3 strawberries among 12 friends gives everyone less than a whole one. So 12 \div 3 \neq 3 \div 12: the bigger number (the total you are sharing) goes first.
The groups must be equal. Putting 12 into a pile of 7 and a pile of 5 is not dividing by 2 — a fair share means every group holds exactly the same amount.

Try sharing 7 cookies among 2 owls. Deal them out: each owl gets 3, and there is 1 cookie left over that won't split fairly. That leftover is called a remainder. Every division on this page shares out exactly, with nothing left — remainders are a story for a little later.

owl cookie cookie cookie owl cookie cookie cookie + cookie left over

Khan Academy walks through division here: