Lesson Explainer: Multiplying Rational Numbers | Nagwa Lesson Explainer: Multiplying Rational Numbers | Nagwa

Lesson Explainer: Multiplying Rational Numbers Mathematics • First Year of Preparatory School

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

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

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

We extend this to negative numbers by instead using a repeated subtraction. So, (βˆ’5)Γ—13 is subtracting 13 from itself so it appears 5 times in the subtraction: (βˆ’5)Γ—13=βˆ’13βˆ’13βˆ’13βˆ’13βˆ’13=βˆ’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 π‘Žπ‘ is a rational number, then π‘›Γ—π‘Žπ‘ is the sum (or difference) of π‘Žπ‘ with itself such that it appears 𝑛 times in the expression. We note that π‘Žπ‘+π‘Žπ‘+β‹―+π‘Žπ‘ο‡Œο†²ο†²ο†²ο†²ο‡ο†²ο†²ο†²ο†²ο‡Ž=π‘Ž+π‘Ž+β‹―+π‘Žπ‘=|𝑛|Γ—π‘Žπ‘.||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, π‘›Γ—π‘Žπ‘=π‘›Γ—π‘Žπ‘.

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 π‘›βˆˆβ„€ and π‘Žπ‘βˆˆβ„š, then π‘›Γ—π‘Žπ‘=π‘›Γ—π‘Žπ‘.

This result allows us to show a lot of useful results. For example, for any rational number π‘Žπ‘,

  • π‘Γ—π‘Žπ‘=π‘Γ—π‘Žπ‘=π‘Ž,
  • π‘Žπ‘=π‘ŽΓ—1𝑏.

Let’s now consider what this means on a number line. For example, to represent 5Γ—13 on a number line, we could do this in several different ways. We could write 5Γ—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 5Γ—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, π‘›Γ—π‘Žπ‘ will split 𝑛 into |𝑏| sections, and then we take |π‘Ž| increments in the direction dictated by the sign. We can extend this to rational numbers by allowing π‘›βˆˆβ„š.

Let’s do this by considering an example; we will consider 12Γ—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 12Γ—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 12Γ—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 2Γ—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 1Γ—2=2 of 16.

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

Definition: Product of Rational Numbers

If π‘Žπ‘,π‘π‘‘βˆˆβ„š, then π‘Žπ‘Γ—π‘π‘‘=π‘ŽΓ—π‘π‘Γ—π‘‘.

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 π‘ŽΓ—π‘π‘Γ—π‘‘βˆˆβ„š, 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 βˆ’75Γ—34.

Answer

We first recall that if π‘Žπ‘,π‘π‘‘βˆˆβ„š, then π‘Žπ‘Γ—π‘π‘‘=π‘ŽΓ—π‘π‘Γ—π‘‘.

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

Hence, βˆ’75Γ—34=ο€Όβˆ’75οˆΓ—34=(βˆ’7)Γ—35Γ—4=βˆ’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 0.4Γ—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 π‘Žπ‘,π‘π‘‘βˆˆβ„š, then π‘Žπ‘Γ—π‘π‘‘=π‘ŽΓ—π‘π‘Γ—π‘‘.

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 0.4Γ—54=25Γ—54=2Γ—55Γ—4=1020.

We can cancel the shared factor of 10 in the numerator and denominator to get 1020=1Γ—102Γ—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 25Γ—16Γ—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 π‘Žπ‘,π‘π‘‘βˆˆβ„š, then π‘Žπ‘Γ—π‘π‘‘=π‘ŽΓ—π‘π‘Γ—π‘‘.

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

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

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

We can then cancel the shared factor of 2 to get 26=1Γ—23Γ—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 13Γ—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=2Γ—142Γ—5=145.

We can then recall that we can multiply rational numbers given as fractions by using the fact that if π‘Žπ‘,π‘π‘‘βˆˆβ„š, then π‘Žπ‘Γ—π‘π‘‘=π‘ŽΓ—π‘π‘Γ—π‘‘.

We can use this to evaluate the expression as follows: 13Γ—2.8=13Γ—145=1Γ—143Γ—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 π‘₯𝑦+𝑧 if π‘₯=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 π‘Žπ‘,π‘π‘‘βˆˆβ„š, then π‘Žπ‘Γ—π‘π‘‘=π‘ŽΓ—π‘π‘Γ—π‘‘.

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

Thus, π‘₯𝑦=734Γ—913=314Γ—283=31Γ—284Γ—3.

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

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

We can now substitute these values into the expression: π‘₯𝑦+𝑧=734Γ—913+437=7213+437.

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

First, 72+4=76.

Second, 13+37=1Γ—73Γ—7+3Γ—37Γ—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 π‘₯𝑦=734Γ—913=ο€Ό7+34οˆΓ—ο€Ό9+13=(9Γ—7)+ο€Ό9Γ—34+ο€Ό7Γ—13+ο€Ό13Γ—34.

We can then evaluate each term where we note that 13Γ—34=1Γ—33Γ—4=14. This gives us (9Γ—7)+ο€Ό9Γ—34+ο€Ό7Γ—13+ο€Ό13Γ—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: π‘₯𝑦+𝑧=734Γ—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 π‘₯=213, 𝑦=34, and 𝑧=0.7, evaluate π‘₯π‘§βˆ’π‘¦.

Answer

We first recall that we can multiply rational numbers given as fractions by using the fact that if π‘Žπ‘,π‘π‘‘βˆˆβ„š, then π‘Žπ‘Γ—π‘π‘‘=π‘ŽΓ—π‘π‘Γ—π‘‘.

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

Thus, π‘₯𝑧=213Γ—0.7=73Γ—710=7Γ—73Γ—10=4930.

We can then substitute these values into the expression to get π‘₯π‘§βˆ’π‘¦=213Γ—0.7βˆ’34=4930βˆ’34.

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 4930βˆ’34=49Γ—230Γ—2βˆ’3Γ—154Γ—15=9860βˆ’4560=98βˆ’4560=5360.

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

Key Points

  • For any integers π‘Ž, 𝑏, 𝑐, and 𝑑, where 𝑏 and 𝑑 are nonzero, we have π‘Žπ‘Γ—π‘π‘‘=π‘ŽΓ—π‘π‘Γ—π‘‘.
  • 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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