In this explainer, we will learn how to apply the properties of the division operation as a strategy to divide rational numbers.
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 โportionsโ) the whole is split into, while the numerator is the number of these shares that make the part we are considering.
Let us look at a division of a fraction by a fraction. For instance, let us find the number given that (of a given whole) is of . This is mathematically described with
By dividing both sides by , we get
So, we need to divide by to find the value of . We are going to use two different methods to find the value of , both of which will allow us to state a general method to divide by fractions.
First method
We can also multiply both sides of our first equation by . We get which is the same as and since , we find that
Combining both equations, we find that
Second method
We can also find the value of by reasoning with diagrams as shown below.
The top diagram shows of a given whole. We know that it is of . So, in the first stage, we need to split our in 4 shares of value (second diagram):
The number is made of 5 of these shares (third diagram):
Number is then of our given whole, represented by the larger rectangle in our diagrams.
The two stages to find the number were
- dividing by 4,
- multiplying the result by 5.
Thus, we have found that
The order in which the division by 4 and the multiplication by 5 are carried out does not matter. This double-operation (multiplying by 5 and dividing by 4) is then equivalent to multiplying by the fraction . We get
With both methods, we have found that
This is a general result that can be used whenever we divide by a fraction: dividing by a fraction is equivalent to multiplying by its multiplicative inverse or reciprocal.
How To: Dividing by Fractions
Dividing by a fraction is equivalent to multiplying by its multiplicative inverse or reciprocal. Hence, when a fraction is divided by another fraction, we can write
We are going to prove this rule algebraically. First, we write our division in the form of a fraction:
Then, we multiply both the numerator and the denominator by the multiplicative inverse of the divisor, that is, by :
The denominator is by definition equal to 1. So,
Now, the fractions can be either positive or negative. As with multiplications, the sign of a quotient depends on the signs of the dividend and divisor:
- If the dividend and divisor are of the same sign, then the quotient is positive.
- If the dividend and divisor are of opposite signs, then the quotient is negative.
Let us look at some questions involving division by a fraction.
Example 1: Dividing Negative Fractions
Evaluate .
Answer
Here, we have a negative fraction divided by another negative fraction. Let us first determine the sign of the result. Both fractions are negative, so the quotient is positive. To find now the result of this division, we apply the rule for dividing by a fraction, namely, that it is equivalent to multiplying by the inverse of the fraction. Hence, we have
Note here that the negative signs could be removed given that we know that the quotient is positive. We now simply multiply the numerators together and the denominators together, given that there is no simplification to do between the numerators and the denominators. We get
Example 2: Understanding Dividing by One-Half
Which of the following numbers gives a result less than when divided by ?
Answer
Here, we are asked to find which of the given numbers gives a result less than when divided by . Let us think about what happens to a number when it is divided by . Dividing by is, for instance, finding how many halves there are in a given number: it is twice the number. Indeed, dividing by is equivalent to multiplying by 2. This means that, in the question here, we need to find which number is less than when doubled. This, in turn, means that the number must be less than .
Going through the numbers given, we find that only is less than . We can check it easily by realizing that . So, as the denominator in is greater than that in (), the value of each share is smaller (the whole has been split into 17 instead of 16 equal shares), and so .
The answer is that gives a result less than when divided by .
Let us look now at a word problem.
Example 3: Dividing Fractions in a Word Problem
Bassem cuts melons into portions, each of which is of the melon. How many portions does he get?
Answer
In this problem, we are told that David cuts melons into portions that are each of the melon. We want to answer the question โhow many portions of size of a melon are there in melons?โ The answer is the number of pieces, and it is given by dividing by . Before carrying out the division, we need to write as a fraction. Since 1 is two halves, we find that .
Hence,
Recall that dividing by a fraction is equivalent to multiplying by its multiplicative inverse or reciprocal, so
We find that David cut 4 portions of melon. We can check our answer by checking that 4 portions of size of a melon give melons:
We are going to look at some examples involving mixed numbers. Remember that mixed numbers are just a particular way of writing so-called improper fractions, that is, fractions that have a numerator greater than the denominator. In this case, the part we are considering is bigger than the whole. This can be envisioned, for instance, with two cakes that have been cut into 8 equal pieces each, and there are 11 pieces left. The fraction of cake that is left is , or one full cake plus three pieces (eighths), written as .
Let us look at some examples of dividing mixed numbers.
Example 4: Dividing Mixed Numbers
Find given that and .
Answer
We need to compute . For this, we are going to write the mixed numbers as improper fractions. We find that and . Hence,
Recall that dividing by a fraction is equivalent to multiplying by its multiplicative inverse or reciprocal, so
This fraction cannot be simplified, so this is our answer.
Example 5: Dividing Two Mixed Numbers
Which of the following division expressions has a quotient of ?
Answer
We want to find which of the given five divisions gives . We could of course compute all of them and see which of the results is . We can also think that is slightly larger than 1.
Therefore, we are looking for a division where the dividend is slightly larger than the divisor. Let us go now through the five divisions and see if we can already exclude some of them:
- can be approximated to , giving a result greater than 2, so it can be excluded.
- can be approximated to , giving a result greater than 2, so it can be excluded.
- can be approximated to , giving a result smaller than 1, so it can be excluded.
- can be approximated to , giving a result slightly greater than 1, so it must be kept as an option.
- can be approximated to , giving a result greater than 4, so it can be excluded.
We are left with only one option, . Let us check that the result of this division is indeed . To carry out the division, we first need to write the mixed numbers as improper fractions.
We find that and
Hence, and we know that dividing by a fraction is equivalent to multiplying by its multiplicative inverse or reciprocal, so
Since , we can simplify the expression on the right-hand side by 7, and we get
Hence, the answer is that the division has a quotient of .
Example 6: Consecutive Divisions by a Fraction and a Mixed Number
Calculate . Give your answer in its simplest form.
Answer
Convert the mixed numbers to improper fractions. Then divide. Perform the division inside the brackets first.
Remember that dividing by a fraction is the same as multiplying by the reciprocal of the fraction.
Key Points
- Dividing by a fraction is equivalent to multiplying by its multiplicative inverse or reciprocal. Hence, when a fraction is divided by another fraction, we can write
- This rule can be proven algebraically.
- First, we write our division in the form of a fraction:
- Then, we multiply both the numerator and the denominator by the multiplicative inverse of the divisor, that is, by :
- The denominator is by definition equal to 1. So,
- As with multiplications, the sign of a quotient depends on the signs of the dividend and divisor.
If the dividend and divisor are of the same sign, then the quotient is positive.
If the dividend and divisor are of opposite signs, then the quotient is negative.
