Fractions of a Quantity
There are 12 sweets in the bag and you are allowed a quarter of them — how many can you take?
Finding a fraction of an amount answers everyday questions like this, from splitting sweets to
working out your share of the money.
You already know a fraction
is equal parts of a whole. But the "whole" doesn't have to be one pizza — it can be a
quantity, like 12 sweets or 20 pounds. Finding a fraction of an
amount just means splitting that amount into equal parts and keeping some of them.
Start with the easiest kind, a unit fraction — a fraction with a
1 on top, like \frac{1}{4}. Take
\frac{1}{4} of 12. The bottom number,
the denominator, says split into four equal groups — and splitting
into equal groups is exactly
division:
12 \div 4 = 3
So \frac{1}{4} of 12 is
3 — one of the four equal groups. Picture sharing 12 marbles
between 4 friends: each friend gets 3, and one friend's share is one quarter.
Taking more than one group
To find \frac{3}{4} of 12, do the same
split into four groups, then keep three of those groups. The top number, the
numerator, tells you how many groups to take, so you
multiply:
12 \div 4 = 3, \qquad 3 \times 3 = 9
\frac{3}{4} \text{ of } 12 = 9
That is the whole rule in one line:
divide by the bottom, then times by the top. Divide by the denominator to
find one group, then multiply by the numerator to take that many groups.
Three worked examples
The same two steps work every time — divide, then multiply:
-
\frac{1}{3} of 15:
split 15 into 3 groups, 15 \div 3 = 5. It is a unit fraction
(a 1 on top), so we just take one group:
\frac{1}{3} of 15 is
5.
-
\frac{2}{5} of 20:
split 20 into 5 groups, 20 \div 5 = 4 in each. Take 2 groups:
4 \times 2 = 8. So
\frac{2}{5} of 20 is
8.
-
\frac{3}{4} of 8:
split 8 into 4 groups, 8 \div 4 = 2 in each. Take 3 groups:
2 \times 3 = 6. So
\frac{3}{4} of 8 is
6.
The two traps that catch everyone:
-
Divide by the bottom, times by the top — not the other way round. For
\frac{3}{4} you divide by 4 and
multiply by 3. Swapping them gives a silly answer.
-
\frac{1}{4} of 12 is
3, not 4. The
4 is how many groups you make — the answer is how many are in
one group.
You have 12 cookies and want to give \frac{2}{3}
of them to your friends. Share the cookies into 3 equal piles — that is
12 \div 3 = 4 in each pile — then take 2 of the
piles: 4 \times 2 = 8 cookies.
The two bright piles are the 8 cookies you give away; the faded
pile is the \frac{1}{3} you keep.
There are 8 ducks on the pond and \frac{1}{4}
of them are asleep. Share the ducks into 4 equal groups —
8 \div 4 = 2 — and one group is asleep. So
2 ducks are napping.
See it built
Watch a set of objects get shared into the denominator's number of equal groups, then the
numerator's number of groups taken. Counting the taken objects gives the answer. Step
through it, then press Refresh for a brand-new example.
See it explained
Sal Khan works through finding a fraction of a whole number using two approaches.