In this explainer, we will learn how to divide rational numbers, including fractions and decimals.
To divide rational numbers, we first recall that division is defined to be the inverse operation of multiplication. For example, dividing 1 by 2 gives a result of . A key point to note is that when we multiply this result by 2, we get back to our original number 1:
Since 2 multiplies by to give the multiplicative identity, we call these numbers multiplicative inverses of each other. This allows us to easily explain division of rational numbers as multiplying by the multiplicative inverse.
For example, we can consider . Multiplying this result by 2 must give us our original number . We can note that
So, is the number that when multiplied by 2 gives . So, must be the same as .
Now, let’s move onto the more general case. First, we note that by definition, the result of is the unique number that when multiplied by gives , where is nonzero.
We can find this for rational numbers and by considering multiplicative inverses in the same way we did above. We note that provided ,
So, the multiplicative inverse of a nonzero rational number is ; this is also called its reciprocal.
We can now note that multiplying by the reciprocal is the same as division by noting that
Now, we can divide both sides of the equation by to observe the relationship
We have therefore shown the following result.
Rule: Division of Rational Numbers
If , so , and and , and we also have that is nonzero, then
In other words, dividing by a rational number is equivalent to multiplying by the reciprocal of that rational number.
Let’s now see an example of how to apply this process to divide two rational numbers given as decimals.
Example 1: Dividing Two Rational Decimals
We first recall that division is defined as the inverse operation of multiplication. In general, if are rational numbers and , then
In other words, to divide by , we multiply by its reciprocal.
We can use this to evaluate this division by converting 0.8 and 0.4 into fractions. We have
Hence, we can substitute these values into the expression to get
Next, we instead multiply by the reciprocal to get
Finally, we evaluate the product:
In our next example, we will evaluate an expression involving both the multiplication and division of rational numbers.
Example 2: Evaluating a Numerical Expression Involving the Division of Fractions
Evaluate giving the answer in its simplest form.
We can evaluate this expression in steps. We note that the order of operations tells us to first evaluate the expression inside the parentheses. We have
We now recall that dividing by a rational number is the same as multiplying by its reciprocal.
Since division of rational numbers is equivalent to multiplying by the reciprocal and we know that multiplication of rational numbers is associative, we have just proved that the order of operations does not matter for this form of question. In other words, if an expression only involves multiplication and division of rational numbers, we can evaluate these operations in any order.
In our next example, we will evaluate the division of rational numbers where one is given as a recurring decimal.
Example 3: Dividing a Recurring Decimal by a Fraction
Evaluate giving the answer in its simplest form.
We first recall that we can divide by a rational number given as a fraction by multiplying by its reciprocal. To use this idea, we will first write both numbers as fractions.
We can rewrite as a fraction using a calculator. This gives us . However, as a brief aside, we can prove this result as follows.
First, we have
We factor out on the right-hand side of the equation, giving us
We can also evaluate the left-hand side of the equation to get
Then, we divide the equation through by 9 to get
Second, we recall that taking the absolute value of a negative number returns the positive value of this number. So,
We can now rewrite this division operation as multiplication using the reciprocal of , giving
In our next example, we will use the idea of dividing by rational numbers to solve a number problem.
Example 4: Solving a Number Problem Involving Fractions
The product of two rational numbers is . If one of the numbers is , find the other number.
We are told that the product of an unknown rational number with is equal to . If we call the unknown rational number , we can write this as the following equation:
We can solve this by multiplying and rearranging. However, we can also divide both sides of the equation by . We have
We recall that dividing by a fraction is the same as multiplying by its multiplicative inverse. Hence, we can rewrite the equation as
By applying the associativity property to the left-hand side of the equation and evaluating, we get
Finally, we evaluate the right-hand side of the equation:
Hence, the missing number is . We can verify this by finding its product with . We have
Confirming that the missing number is .
In the previous example, we showed a useful property of the division of rational numbers. Since division by a nonzero rational number is the same as multiplying by its multiplicative inverse, we have
In particular, we can use this to solve equations. We saw that we could solve the equation by dividing through by . In particular, we can cancel the division as follows:
It is worth noting that this works for rational numbers given in any form.
In our final two examples, we will use this process to divide by rational numbers and solve real-world problems.
Example 5: Solving a Word Problem Involving Division of Rational Numbers
Each box of oranges weighs kilograms. How many boxes are there if they weigh a total of 50.25 kilograms?
The total weight of the boxes will be the number of boxes multiplied by the weight of each box. So, if we call the number of boxes of oranges , then we can form the following equation:
We can divide both sides of the equation through by to find an expression for . We get
We can now recall that to divide by a rational number, we multiply by its reciprocal. So, we will first convert both of these numbers into fractions. We have
Rewriting the division as multiplication by the reciprocal then gives
We can factor 201 into primes to note that . Substituting this in and canceling then gives us
Hence, there are 6 boxes of oranges.
Example 6: Solving a Real-World Problem Involving Fractions
Maged constructs of a wall in days. How many days will he need to construct the wall?
We will start by calling the number of days Maged needs to construct the wall . Since we are told that Maged constructs of the wall in days, we know that
We can solve for by dividing both sides of the equation through by . This gives
To divide by a rational number, we want to multiply by the reciprocal. So, we will rewrite as a proper fraction to get , giving
Now, we instead multiply by the reciprocal of and evaluate to get
Since the original value in days was given as a mixed fraction, we will also give our answer as a mixed number. We have .
Hence, it will take Maged days in total to build the wall.
Let’s finish by recapping some of the important points from this explainer.
- If , then the division of the two rational numbers and is a rational number. In general,
- We call the reciprocal of , otherwise known as the multiplicative inverse. This means that dividing by a rational number is the same as multiplying by its reciprocal.
- We can more easily divide rational numbers given in any form (such as decimals or mixed numbers) by converting into fractions.