In this explainer, we will learn how to multiply rational numbers, including fractions and decimals.
To understand how to multiply rational numbers, we first need to consider what is actually meant by the product of rational numbers. First, we recall that multiplication by integers is defined by repeated addition or subtraction. We can extend this to rational numbers in the same way. For example, to evaluate , we want to add to itself such that it appears 5 times in the sum. This is equivalent to saying that is five repeated displacements of from the 0 on a number line. We have
All of these fractions have the same denominator, so we can add them together to get
We note that the numerator could also be written as , so we have
We extend this to negative numbers by instead using a repeated subtraction. So, is subtracting from itself so it appears 5 times in the subtraction:
This allows us to find the product of any nonzero integer and rational number. We can use this same reasoning to write this result in general. If is a nonzero integer and is a rational number, then is the sum (or difference) of with itself such that it appears times in the expression. We note that
If we remove the absolute value operator from , then we change the repeated sum into a repeated difference and the result still holds true. Hence,
It is worth noting that we cannot use this reasoning if is zero; however, multiplication by 0 is always just equal to 0. So, this result still holds true if . We have shown the following result.
Definition: Product of an Integer and a Rational Number
If and , then
This result allows us to show a lot of useful results. For example, for any rational number ,
- ,
- .
Let’s now consider what this means on a number line. For example, to represent on a number line, we could do this in several different ways. We could write as , split the number line into thirds, and then mark the fifth increment. However, it is easier to write as and then split 5 into 3 sections of equal size and then mark the first increment as shown.
In this case, multiplying a number by means we split into 3 equal sections and then take the first increment. In general, will split into sections, and then we take increments in the direction dictated by the sign. We can extend this to rational numbers by allowing .
Let’s do this by considering an example; we will consider by using a number line. We can find on a number line by splitting 1 into halves. To multiply this by , we need to split each of these sections into thirds and take the second increment as shown.
We have found the point representing on the number line, and we can use this to find a simplified fraction for this number. We note that we find increments of by splitting the integers into 2 equal sections. If we then split these into 3 sections, this will be the same as splitting the integers in 6 equal sections. Therefore, we can add increments of onto the number line as shown.
We see that our point is 2 increments of from 0, so
We can note that this seems to imply that we can multiply the numerator and denominator separately. Indeed, we can generalize this process by noting that splitting the integers into 2 sections and these into 3 sections is the same as just splitting the integers into sections. Similarly, choosing the first section of and then the second increment inside this section is the same as just choosing increment of .
We can apply this same process in general to show the following result.
Definition: Product of Rational Numbers
If , then
This result allows us to multiply any two rational numbers given as fractions by multiplying the numerators and denominators separately. It is also worth noting that , so the product of two rational numbers is also a rational number.
Let’s now see an example of applying definition to find the product of two rational numbers.
Example 1: Multiplying Fractions with Unlike Denominators
Evaluate .
Answer
We first recall that if , then
We can write this product in this form by noting that .
Hence,
In our next example, we will find the product of two rational numbers giving the answer as a decimal.
Example 2: Multiplying Rational Numbers Including a Decimal
Evaluate , giving your answer as a decimal.
Answer
We first recall that we can multiply rational numbers given as fractions by using the fact that if , then
We can use this to multiply the two given rational numbers if we convert 0.4 into a fraction. We see that . Substituting these fractions into the formula gives
We can cancel the shared factor of 10 in the numerator and denominator to get
Finally, we know that .
In our next example, we will find the product of three rational numbers where we need to give our answer as a fraction in its simplest form.
Example 3: Multiplying Whole Numbers by Fractions by Decimal Numbers
Calculate . Give your answer as a fraction in its simplest form.
Answer
We first recall that we can multiply rational numbers given as fractions by using the fact that if , then
We then note that . Then, we can convert 0.08 into a fraction as follows:
Substituting these into the expression and evaluating gives
We can cancel the shared factor of 25 in the numerator and denominator to get
We can then cancel the shared factor of 2 to get
In our next example, we will use this technique for multiplying rational numbers to solve a real-world problem.
Example 4: Solving a Real-World Problem Using the Product of Rational Numbers
Karim has a large bucket that contains 2.8 kg of sand. He takes one-third of the sand out of the bucket and fills another small bucket with this sand. What is the weight of the sand in the small bucket?
Answer
We first note that the smaller bucket has the weight of sand of the original weight of the larger bucket. This means there is kg of sand in the smaller bucket. We can evaluate this expression by converting 2.8 into a fraction. We note that
We can then recall that we can multiply rational numbers given as fractions by using the fact that if , then
We can use this to evaluate the expression as follows:
Hence, the weight of the small bucket of sand is kg.
In our next example, we will evaluate an algebraic expression that involves the product of mixed fractions.
Example 5: Evaluating an Algebraic Expression Involving Multiplication of Rational Numbers
Evaluate if , , and .
Answer
We first recall that we can multiply rational numbers given as fractions by using the fact that if , then
We will start by rewriting each value in the product as a fraction; we have
Thus,
Instead of evaluating this, we can cancel the shared factor of 4 in the numerator and denominator to get
We can write this as a mixed fraction as follows:
We can now substitute these values into the expression:
We can add mixed fractions by adding their integer and fractional parts separately.
First,
Second,
Hence,
We can also answer this question by multiplying the mixed fractions by using an area model.
We can multiply the mixed fractions by adding the areas of each rectangle. We have
We can then evaluate each term where we note that . This gives us
We can add the fractions with denominators of 4 and simplify to get
We can write this as a mixed fraction by noting that . So,
We can now substitute these values into the expression as follows:
Evaluating this gives
In our final example, we will evaluate an algebraic expression that involves the product of decimals and mixed fractions.
Example 6: Evaluating an Algebraic Expression Involving Multiplication of a Mixed Fraction and a Decimal
If , , and , evaluate .
Answer
We first recall that we can multiply rational numbers given as fractions by using the fact that if , then
We will start by rewriting each value in the product as a fraction; we have
Thus,
We can then substitute these values into the expression to get
We can subtract fractions by rewriting them to have the same denominator. The lowest common multiple of 4 and 30 is 60. So, we will rewrite both fractions to have a denominator of 60. This gives
Let’s finish by recapping some of the important points from this explainer.
Key Points
- For any integers , , , and , where and are nonzero, we have
- The product of any two rational numbers is a rational number.
- We can multiply rational numbers in different forms by first converting them into fractions.
