In this explainer, we will learn how to divide proper fractions by proper fractions.
For this purpose, we are going to start with dividing fractions by whole numbers and vice
versa before going into dividing fractions by fractions. We are going to look at the two
different meanings such divisions can have, and, with the help of diagrams, we will find how
to compute them. This will lead us to a general method to compute divisions by fractions.
Let us first recall that a fraction compares a part to a whole and describes what we call a
proportion. The denominator of the fraction is the number of equal shares (or “portions”) the
whole is split into, while the numerator is the number of these shares that make the part we
are considering.
Now we are going to start with a very simple example of a division of a fraction by a whole
number.
Example 1: Dividing a Fraction by a Whole Number
Calculate . Give your answer in its simplest
form.
Answer
We are considering the fraction here; that is, a whole has
been divided into three equal shares, and the part we are considering is made of two of
these shares.
We now want to divide the part (the shaded area in the diagram) into three parts. By
dividing each third in three parts, we have our whole divided into 9 equal shares, and we
see that is two of them, that is, two-ninths:
.
Notice here how dividing a fraction is further splitting the whole; the denominator, which is
the number of shares, is indeed multiplied! So, dividing a half in three is creating sixths of
the whole.
Let us look at another example where we can use this.
Example 2: Identifying When a Fraction Is Half of Another Fraction
Which of the following division expressions has a quotient of
?
Answer
We are looking for a division whose result is one-half; in other words, we are looking
for a division where the dividend is half the divisor. In all the options given, the
dividends and divisors are fractions. So, we are looking for a fraction divided by another
fraction that is double its value.
Let us take two examples to see how a fraction can be one-half of another. First,
is half of as both fractions
have the same denominator but the numerator of the first is half that of the second.
And, second, is half of since
both have the same numerator but the denominator has been divided by two in the second
fraction, meaning that the number of shares has been halved, so their size has been indeed
doubled.
Therefore, we are looking for one of these situations here: either the denominators are
the same and the numerator of the dividend is half that of the divisor
, or the numerators are
the same and the denominator of the dividend is double that of the divisor
.
Hence, the answer is (option D).
In the next example, we are looking at the division of a whole number by a fraction.
Example 3: Dividing a Whole Number by a Fraction
Evaluate .
Answer
We are asked here to find the result of dividing 10 by . This
can be understood as finding how many times there is in
10.
There are two halves in 1, so there will be 10 times as many in 10; that is, .
Hence, .
Let us look now at dividing a fraction by a fraction. For instance, we want to find
, that is, how many
are in . We can use a diagram to
help us visualize this.
We see that, in this case, because both fractions have the same denominator, this division is
simply 12 divided by 3: there are 4 times in
.
That is, we have .
From this, we see that a strategy to divide a fraction by another fraction with a different
denominator is to rewrite the fractions so that they have the same denominator and then divide
the numerators. It is a good strategy, for example, to compute since can be easily rewritten as
, and so .
In the following, we are going to discover a general method equivalent to rewriting the
fractions with the same denominator and dividing the numerators, but simpler.
So far, we have interpreted the division by a fraction as a measurement division: how many of
this fraction are there in a given number?
There is another way to envision such a division. For instance, imagine that Oscar has
written 12 pages for his assignment and he says that it is three-quarters of what is required. How can we find how many pages he is required to write for his assignment?
Let us use a diagram to help us.
We know that 12 is three-quarters of what is
required. So, if we start with a rectangle to represent what is required, we need to split it
in 4 and the 12 pages already written are three of these shares.
Now, let us see how to find the number of pages required. First, we find the value of each
quarter by dividing the 12 by 3. This gives 4. The number of pages required is 4 times this
value (which is 4 quarters); that is, .
Now, we were told that 12 pages are three-quarters of what is required. We can write that as
Now, think of the complementarity between division and multiplication sentences. For
instance, (two groups of 5 make 10) means that (10 can be split in 2 groups of 5). Hence, we can rewrite the above equation
as
What we have found by reasoning on the diagram was nothing else as the result of . And the stages were
divide 12 by 3,
multiply the result by 4.
We see that . This equation can
be rewritten as
We have found here a very important result: dividing by a fraction is mathematically the same
as multiplying by the multiplicative inverse, or reciprocal, of the fraction (i.e., the
numerator and denominator have been swapped over).
Let us look at a division of a fraction by a fraction. For instance, let us find a number
knowing that four-fifths of this unknown number
equals . That is, . For this, we are going to use three diagrams.
In the top diagram, is represented in dark green, and the bigger
rectangle represents our whole. We know that the dark green rectangle represents
of the unknown number. So, in as the first stage, we need to
split our in 4 shares of value (second
diagram):
Note that, without any diagram, we would have probably written and then simplified this fraction to . This is of course completely correct.
The number we are looking for, called , is made of 5 of these shares
(third diagram):
The unknown number is found to be five-sixths of our whole.
Here, again, the two stages of the division by were
divide by 4,
multiply the result by 5.
We have found that .
Let us now interpret this division as a measurement division; that is, how many
are there in ?
For this, we have already seen that it is helpful to rewrite the fractions with the same
denominator. The lowest common multiple of 3 and 5 is 15. As and , we have
This step of rewriting the dividend and divisor with the same denominator can be visualized
in the diagram.
Now that both fractions have the same denominator, finding how many
there are in is simply finding
how many 12s there are in 10: it is given by . This
number is smaller than 1 because 12 is bigger than 10.
We have found that . Let us look
back at how we got these numbers, 10 and 12, from when we renamed the fractions with the same denominator. The 10 was
given by and the 12 by . So, we did
find that
that is,
We can write the general method we have found for dividing fractions by fractions.
How To: Dividing by Fractions
Dividing by a fraction is equivalent to dividing by the numerator of the fraction and then
multiplying by its denominator:
In other words, it is equivalent to multiplying by its multiplicative inverse or
reciprocal. Hence, when a fraction is divided by another fraction, we can write
Let us look at an example where we are going to use our understanding of division by
fractions.
Example 4: Dividing a Fraction by a Fraction
Find .
Answer
To find the result of dividing by a fraction, we use the fact that dividing by a fraction
is equivalent to multiplying by its inverse (i.e., a fraction where the numerator and the
denominator have been swapped). Hence, we have
Now, we just need to multiply the numerators together and the denominators together. We
find
We have found out that one half is three-quarters of two-thirds and two-thirds of
three-quarters!
Our last example is a word problem.
Example 5: Using Division of Fractions in a Word Problem
Sally and Ramy went out to get some ice cream. Sally had
pt
of chocolate chip ice cream, while Ramy had
pt
of strawberry-flavored ice cream. Determine how many times as much ice cream Ramy had as Sally.
Answer
Both Sally and Ramy have got a fraction of a pint of ice cream. We want
to find how many times as much ice cream Ramy had as Sally. This is given
mathematically by dividing the amount of ice cream Ramy had by the amount of ice
cream Sally had, that is,
We know that when we divide by a fraction, this is equivalent to multiplying by its
reciprocal, so
This fraction is an improper fraction: the numerator is larger than the denominator. Hence, we will express it as a mixed number:
Ramy had times as much ice cream as Sally had.
Key Points
A fraction compares a part to a whole. The denominator of the fraction is the number of
equal shares (or “portions”) the whole is split into, while the numerator is
the number of these shares that make the part we are considering.
If we know that a number is a fraction of another number, for instance, 12 is three
quarters of an unknown number, we can write . As we know, for instance, that is
equivalent to , we can write that .
Reasoning with
the diagram, we find that, to find this unknown number, we need to divide 12 by 3 to find
the value of one quarter and then multiply this by 4 to find the unknown number. So, we
have since dividing by 3 and
multiplying by 4 is the same as multiplying by .
Dividing a fraction by a like fraction, for instance, , can be interpreted as answering the question “how many
groups of can be made with
?” Then, it is the same as finding how many groups of
3 can be made with 12. So, when we divide a fraction by a like fraction, we simply divide
the numerators be each other, ignoring the common denominator. We have
.
Following the previous point, to divide a fraction by a fraction we can rename both
fractions with the same denominator and then simply divide their numerators, ignoring the
common denominator. Consider, for instance,
and let us rename both fractions with the same denominator, 15: We now have a fraction divided by a like fraction, so we can simply
divide the numerators: which can be written in fraction form:
Using the commutativity of multiplication
to swap the 3 and the 4 in the denominator in the right-hand side of the equation and
splitting the fraction in the product of two fractions, we find that
The three previous points lead to a general property of division by a fraction: it is
equivalent to multiplication by the inverse of the fraction, that is, a fraction where the
numerator and the denominator have been swapped:
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