Lesson Explainer: Multiplying Rational Numbers Mathematics • 7th Grade

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 5×13, we want to add 13 to itself such that it appears 5 times in the sum. This is equivalent to saying that 5×13 is five repeated displacements of 13 from the 0 on a number line. We have 5×13=13+13+13+13+13.

All of these fractions have the same denominator, so we can add them together to get 5×13=1+1+1+1+13=53.

We note that the numerator could also be written as 5×1, so we have 5×13=5×13.

We extend this to negative numbers by instead using a repeated subtraction. So, (5)×13 is subtracting 13 from itself so it appears 5 times in the subtraction: (5)×13=1313131313=53.

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 𝑛 is a nonzero integer and 𝑎𝑏 is a rational number, then 𝑛×𝑎𝑏 is the sum (or difference) of 𝑎𝑏 with itself such that it appears 𝑛 times in the expression. We note that 𝑎𝑏+𝑎𝑏++𝑎𝑏=𝑎+𝑎++𝑎𝑏=|𝑛|×𝑎𝑏.||terms

If we remove the absolute value operator from 𝑛, then we change the repeated sum into a repeated difference and the result still holds true. Hence, 𝑛×𝑎𝑏=𝑛×𝑎𝑏.

It is worth noting that we cannot use this reasoning if 𝑛 is zero; however, multiplication by 0 is always just equal to 0. So, this result still holds true if 𝑛=0. We have shown the following result.

Definition: Product of an Integer and a Rational Number

If 𝑛 and 𝑎𝑏, then 𝑛×𝑎𝑏=𝑛×𝑎𝑏.

This result allows us to show a lot of useful results. For example, for any rational number 𝑎𝑏,

  • 𝑏×𝑎𝑏=𝑏×𝑎𝑏=𝑎,
  • 𝑎𝑏=𝑎×1𝑏.

Let’s now consider what this means on a number line. For example, to represent 5×13 on a number line, we could do this in several different ways. We could write 5×13 as 13+13+13+13+13, split the number line into thirds, and then mark the fifth increment. However, it is easier to write 5×13 as 53 and then split 5 into 3 sections of equal size and then mark the first increment as shown.

In this case, multiplying a number 𝑛 by 13 means we split 𝑛 into 3 equal sections and then take the first increment. In general, 𝑛×𝑎𝑏 will split 𝑛 into |𝑏| sections, and then we take |𝑎| increments in the direction dictated by the sign. We can extend this to rational numbers by allowing 𝑛.

Let’s do this by considering an example; we will consider 12×23 by using a number line. We can find 12 on a number line by splitting 1 into halves. To multiply this by 23, we need to split each of these sections into thirds and take the second increment as shown.

We have found the point representing 12×23 on the number line, and we can use this to find a simplified fraction for this number. We note that we find increments of 12 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 16 onto the number line as shown.

We see that our point is 2 increments of 16 from 0, so 12×23=26.

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 2×3=6 sections. Similarly, choosing the first section of 12 and then the second increment inside this section is the same as just choosing increment 1×2=2 of 16.

We can apply this same process in general to show the following result.

Definition: Product of Rational Numbers

If 𝑎𝑏,𝑐𝑑, then 𝑎𝑏×𝑐𝑑=𝑎×𝑐𝑏×𝑑.

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 𝑎×𝑐𝑏×𝑑, 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 75×34.

Answer

We first recall that if 𝑎𝑏,𝑐𝑑, then 𝑎𝑏×𝑐𝑑=𝑎×𝑐𝑏×𝑑.

We can write this product in this form by noting that 75=75.

Hence, 75×34=75×34=(7)×35×4=2120.

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 0.4×54, 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 𝑎𝑏,𝑐𝑑, then 𝑎𝑏×𝑐𝑑=𝑎×𝑐𝑏×𝑑.

We can use this to multiply the two given rational numbers if we convert 0.4 into a fraction. We see that 0.4=410=25. Substituting these fractions into the formula gives 0.4×54=25×54=2×55×4=1020.

We can cancel the shared factor of 10 in the numerator and denominator to get 1020=1×102×10=12.

Finally, we know that 12=0.5.

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 25×16×0.08. 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 𝑎𝑏,𝑐𝑑, then 𝑎𝑏×𝑐𝑑=𝑎×𝑐𝑏×𝑑.

We then note that 25×16=256. Then, we can convert 0.08 into a fraction as follows: 0.08=8100=2×425×4=225.

Substituting these into the expression and evaluating gives 25×16×0.08=256×225=25×26×25.

We can cancel the shared factor of 25 in the numerator and denominator to get 25×26×25=26.

We can then cancel the shared factor of 2 to get 26=1×23×2=13.

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 13 the weight of sand of the original weight of the larger bucket. This means there is 13×2.8 kg of sand in the smaller bucket. We can evaluate this expression by converting 2.8 into a fraction. We note that 2.8=2810=2×142×5=145.

We can then recall that we can multiply rational numbers given as fractions by using the fact that if 𝑎𝑏,𝑐𝑑, then 𝑎𝑏×𝑐𝑑=𝑎×𝑐𝑏×𝑑.

We can use this to evaluate the expression as follows: 13×2.8=13×145=1×143×5=1415.

Hence, the weight of the small bucket of sand is 1415 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 𝑥𝑦+𝑧 if 𝑥=734, 𝑦=913, and 𝑧=437.

Answer

We first recall that we can multiply rational numbers given as fractions by using the fact that if 𝑎𝑏,𝑐𝑑, then 𝑎𝑏×𝑐𝑑=𝑎×𝑐𝑏×𝑑.

We will start by rewriting each value in the product as a fraction; we have 734=7×4+34=314,913=9×3+13=283.

Thus, 𝑥𝑦=734×913=314×283=31×284×3.

Instead of evaluating this, we can cancel the shared factor of 4 in the numerator and denominator to get 31×284×3=31×7×44×3=31×73=2173.

We can write this as a mixed fraction as follows: 2173=72×3+13=7213.

We can now substitute these values into the expression: 𝑥𝑦+𝑧=734×913+437=7213+437.

We can add mixed fractions by adding their integer and fractional parts separately.

First, 72+4=76.

Second, 13+37=1×73×7+3×37×3=721+921=7+921=1621.

Hence, 7213+437=761621.

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 𝑥𝑦=734×913=7+34×9+13=(9×7)+9×34+7×13+13×34.

We can then evaluate each term where we note that 13×34=1×33×4=14. This gives us (9×7)+9×34+7×13+13×34=63+274+73+14.

We can add the fractions with denominators of 4 and simplify to get 63+274+73+14=63+27+14+73=63+284+73=63+7+73=70+73.

We can write this as a mixed fraction by noting that 73=213. So, 70+73=7213.

We can now substitute these values into the expression as follows: 𝑥𝑦+𝑧=734×913+437=7213+437.

Evaluating this gives 7213+437=761621.

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 𝑥=213, 𝑦=34, and 𝑧=0.7, evaluate 𝑥𝑧𝑦.

Answer

We first recall that we can multiply rational numbers given as fractions by using the fact that if 𝑎𝑏,𝑐𝑑, then 𝑎𝑏×𝑐𝑑=𝑎×𝑐𝑏×𝑑.

We will start by rewriting each value in the product as a fraction; we have 213=2×3+13=6+13=73,0.7=710.

Thus, 𝑥𝑧=213×0.7=73×710=7×73×10=4930.

We can then substitute these values into the expression to get 𝑥𝑧𝑦=213×0.734=493034.

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 493034=49×230×23×154×15=98604560=984560=5360.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • For any integers 𝑎, 𝑏, 𝑐, and 𝑑, where 𝑏 and 𝑑 are nonzero, we have 𝑎𝑏×𝑐𝑑=𝑎×𝑐𝑏×𝑑.
  • 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.

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