Division Undoes Multiplication
Say you buy 3 packs of trading cards with
4 cards in each — that is 12 cards in
all. Later you want to deal those 12 cards evenly into
3 piles to swap with friends; how many go in each pile? Working
that out is dividing, and it is really your multiplication run backwards.
Division is
the inverse of
multiplication
— it undoes it. If you multiply to get a total, dividing takes you straight
back to where you started:
a \times b = c \quad\Longrightarrow\quad c \div b = a
Multiplying by b and then dividing by
b lands you back on a,
like walking forwards and then back again. Press play to build an array by
multiplying, then split it back into equal groups to divide.
One array, four facts: the fact family
Because the two operations undo each other, one multiplication fact comes
with a whole fact family — four number sentences that all
describe the same picture. Take an array of
3 rows with 4 cookies
in each row:
There are 12 cookies in all. The very same array
answers four questions at once:
- 3 \times 4 = 12 — three rows of four.
- 4 \times 3 = 12 — read it as four columns of three.
- 12 \div 4 = 3 — twelve split into rows of four makes three rows.
- 12 \div 3 = 4 — twelve split into three rows puts four in each.
In general, from a \times b = c you instantly get
the other three members of the family:
b \times a = c \qquad c \div b = a \qquad c \div a = b
So every time you learn one times-table fact, you also learn two divisions
for free.
A fact family is really just one array looked at from different sides.
Count it in rows and you are multiplying; cover up an answer and ask
"how many in each row?" and you are dividing. Tip the page sideways and
the rows become columns — that is why 3 \times 4
and 4 \times 3 give the same total. Nothing was
added or taken away; you just chose what to call a row.
Use a times-table fact you already know
This is why your
times
tables are the key to dividing quickly. You don't have to share
dots out one by one — just turn the division back into a multiplication
question. Here are three worked examples.
Example 1 — work out 12 \div 3.
Ask "three times what makes twelve?" You know
3 \times 4 = 12, so
12 \div 3 = 4. Picture the twelve cookies split
into three rows: four land in each.
Example 2 — work out 20 \div 5.
Ask "five times what makes twenty?" Since
5 \times 4 = 20, the answer is
20 \div 5 = 4.
Example 3 — work out 56 \div 8.
Ask "eight times what makes fifty-six?" You know
8 \times 7 = 56, so
56 \div 8 = 7. Knowing the multiplication fact
is knowing the division.
-
Always check by multiplying back. To trust
12 \div 4 = 3, ask: does
4 \times 3 = 12? Yes — so the division is
right. If the multiply-back doesn't return your starting number, the
answer is wrong.
-
The two divisions in a family are different facts.
12 \div 4 = 3 but
12 \div 3 = 4 — the answers swap. Make sure
you divide by the number you actually mean.
-
Order matters for division (but not for multiplication):
12 \div 4 is not the same as
4 \div 12. The big total always goes first.
You have 10 grapes to share equally between
2 friends:
|
Rather than dealing them out one at a time, remember
2 \times 5 = 10. So
10 \div 2 = 5 — five grapes each — and the
multiply-back check, 2 \times 5 = 10, confirms
it instantly.
See it: a fresh fact family every time
Here is a random array of dots. Count the rows and columns to get the two
multiplications, then split it back the two ways to get the two divisions.
Press Refresh for a brand-new array and a brand-new family.
Khan Academy relates multiplication and division here: