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.
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 ?
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
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, .
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
We can write the general method we have found for dividing fractions by fractions.
How to Divide 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
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
Natalie and Mason went out to get some ice cream. Natalie had pt of chocolate chip ice cream, while Mason had pt of strawberry-flavored ice cream. Determine how many times as much ice cream Mason had as Natalie.
Both Natalie and Mason have got a fraction of a pint of ice cream. We want to find how many times as much ice cream Mason had as Natalie. This is given mathematically by dividing the amount of ice cream Mason had by the amount of ice cream Natalie 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:
Mason had times as much ice cream as Natalie had.
- 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: