Lesson Explainer: Dividing Rational Numbers Mathematics • 7th Grade

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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 12. A key point to note is that when we multiply this result by 2, we get back to our original number 1: 12Γ—2=1.

Since 2 multiplies by 12 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 π‘₯2. Multiplying this result by 2 must give us our original number π‘₯. We can note that ο€Όπ‘₯Γ—12οˆΓ—2=π‘₯Γ—ο€Ό12Γ—2=π‘₯.

So, π‘₯Γ—12 is the number that when multiplied by 2 gives π‘₯. So, π‘₯Γ—12 must be the same as π‘₯2.

Now, let’s move onto the more general case. First, we note that by definition, the result of ο€»π‘Žπ‘ο‡Γ·ο€»π‘π‘‘ο‡ is the unique number that when multiplied by 𝑐𝑑 gives π‘Žπ‘, where 𝑐 is nonzero.

We can find this for rational numbers π‘Žπ‘ and 𝑐𝑑 by considering multiplicative inverses in the same way we did above. We note that provided 𝑐≠0, 𝑐𝑑×𝑑𝑐=𝑐×𝑑𝑑×𝑐=1.

So, the multiplicative inverse of a nonzero rational number 𝑐𝑑 is 𝑑𝑐; this is also called its reciprocal.

We can now note that multiplying by the reciprocal is the same as division by noting that ο€½π‘Žπ‘Γ—π‘‘π‘ο‰Γ—π‘π‘‘=π‘Žπ‘Γ—ο€½π‘‘π‘Γ—π‘π‘‘ο‰=π‘Žπ‘Γ—1=π‘Žπ‘.

Thus, ο€½π‘Žπ‘Γ—π‘‘π‘ο‰Γ—π‘π‘‘=π‘Žπ‘.

Now, we can divide both sides of the equation by 𝑐𝑑 to observe the relationship π‘Žπ‘Γ—π‘‘π‘=π‘Žπ‘Γ·π‘π‘‘.

We have therefore shown the following result.

Rule: Division of Rational Numbers

If π‘Žπ‘,π‘π‘‘βˆˆβ„š, so π‘Ž,𝑏,𝑐, and π‘‘βˆˆβ„€ and 𝑏,𝑑≠0, and we also have that 𝑐 is nonzero, then π‘Žπ‘Γ·π‘π‘‘=π‘Žπ‘Γ—π‘‘π‘=π‘Žπ‘‘π‘π‘.

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

Evaluate 0.8Γ·0.4.

Answer

We first recall that division is defined as the inverse operation of multiplication. In general, if π‘Žπ‘,𝑐𝑑 are rational numbers and 𝑐≠0, then π‘Žπ‘Γ·π‘π‘‘=π‘Žπ‘Γ—π‘‘π‘.

In other words, to divide by 𝑐𝑑, we multiply by its reciprocal.

We can use this to evaluate this division by converting 0.8 and 0.4 into fractions. We have 0.8=810=45,0.4=410=25.

Hence, we can substitute these values into the expression to get 0.8Γ·0.4=45Γ·25.

Next, we instead multiply by the reciprocal to get 45Γ·25=45Γ—52.

Finally, we evaluate the product: 45Γ—52=4Γ—55Γ—2=4Γ—55Γ—2=42=2.

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 ο€Ό34Γ—βˆ’23÷15 giving the answer in its simplest form.

Answer

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 34Γ—βˆ’23=3Γ—(βˆ’2)4Γ—3=3Γ—(βˆ’2)2Γ—2Γ—3=βˆ’12.

Thus, ο€Ό34Γ—βˆ’23÷15=ο€Όβˆ’12÷15.

We now recall that dividing by a rational number is the same as multiplying by its reciprocal.

Hence, ο€Όβˆ’12÷15=ο€Όβˆ’12οˆΓ—51=βˆ’52.

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 0.Μ‡8Γ·|||βˆ’54||| giving the answer in its simplest form.

Answer

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 0.Μ‡8 as a fraction using a calculator. This gives us 0.Μ‡8=89. However, as a brief aside, we can prove this result as follows.

First, we have 10Γ—0.Μ‡8=8.Μ‡8.

Thus, 8.Μ‡8βˆ’0.Μ‡8=10Γ—0.Μ‡8βˆ’0.Μ‡8.

We factor out 0.Μ‡8 on the right-hand side of the equation, giving us 8.Μ‡8βˆ’0.Μ‡8=(10βˆ’1)Γ—0.Μ‡88.Μ‡8βˆ’0.Μ‡8=9Γ—0.Μ‡8.

We can also evaluate the left-hand side of the equation to get 8=9Γ—0.Μ‡8.

Then, we divide the equation through by 9 to get 89=0.Μ‡8.

Second, we recall that taking the absolute value of a negative number returns the positive value of this number. So, |||βˆ’54|||=54.

Therefore, 0.Μ‡8Γ·|||βˆ’54|||=89Γ·54.

We can now rewrite this division operation as multiplication using the reciprocal of 54, giving 89Γ·54=89Γ—45=8Γ—49Γ—5=3245.

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 βˆ’169. If one of the numbers is βˆ’43, find the other number.

Answer

We are told that the product of an unknown rational number with βˆ’43 is equal to βˆ’169. If we call the unknown rational number π‘Žπ‘, we can write this as the following equation: π‘Žπ‘Γ—ο€Όβˆ’43=βˆ’169.

We can solve this by multiplying and rearranging. However, we can also divide both sides of the equation by βˆ’43. We have π‘Žπ‘Γ—ο€Όβˆ’43οˆΓ·ο€Όβˆ’43=βˆ’169Γ·ο€Όβˆ’43.

We recall that dividing by a fraction is the same as multiplying by its multiplicative inverse. Hence, we can rewrite the equation as π‘Žπ‘Γ—ο€Όβˆ’43οˆΓ—ο€Όβˆ’34=βˆ’169Γ—ο€Όβˆ’34.

By applying the associativity property to the left-hand side of the equation and evaluating, we get ο€»π‘Žπ‘ο‡Γ—ο€Όβˆ’43οˆΓ—ο€Όβˆ’34=βˆ’169Γ—ο€Όβˆ’34οˆο€»π‘Žπ‘ο‡Γ—ο€Όο€Όβˆ’43οˆΓ—ο€Όβˆ’34=βˆ’169Γ—ο€Όβˆ’34οˆο€»π‘Žπ‘ο‡Γ—1=βˆ’169Γ—ο€Όβˆ’34οˆπ‘Žπ‘=βˆ’169Γ—ο€Όβˆ’34.

Finally, we evaluate the right-hand side of the equation: π‘Žπ‘=βˆ’169Γ—ο€Όβˆ’34=(βˆ’16)Γ—(βˆ’3)9Γ—4=4Γ—4Γ—33Γ—3Γ—4=4Γ—4Γ—33Γ—3Γ—4=43.

Hence, the missing number is 43. We can verify this by finding its product with βˆ’43. We have 43Γ—ο€Όβˆ’43=4Γ—(βˆ’4)3Γ—3=βˆ’169.

Confirming that the missing number is 43.

In the previous example, we showed a useful property of the division of rational numbers. Since division by a nonzero rational number π‘Žπ‘ is the same as multiplying by its multiplicative inverse, we have π‘Žπ‘Γ·π‘Žπ‘=π‘Žπ‘Γ—π‘π‘Ž=1.

In particular, we can use this to solve equations. We saw that we could solve the equation π‘Žπ‘Γ—ο€Όβˆ’43=βˆ’169, by dividing through by βˆ’43. In particular, we can cancel the division as follows: ο€Όπ‘Žπ‘Γ—ο€Όβˆ’43οˆοˆΓ·ο€Όβˆ’43=βˆ’169Γ·ο€Όβˆ’43οˆπ‘Žπ‘=βˆ’169Γ·ο€Όβˆ’43.

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 838 kilograms. How many boxes are there if they weigh a total of 50.25 kilograms?

Answer

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 𝑏, then we can form the following equation: 𝑏×838=50.25.

We can divide both sides of the equation through by 838 to find an expression for 𝑏. We get 𝑏=50.25Γ·838.

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 50.25=50+14=2004+14=2014,838=648+38=678.

Thus, 𝑏=50.25Γ·838=2014Γ·678.

Rewriting the division as multiplication by the reciprocal then gives 𝑏=2014Γ—867.

We can factor 201 into primes to note that 201=3Γ—67. Substituting this in and canceling then gives us 𝑏=3Γ—674Γ—867=3Γ—21=6.

Hence, there are 6 boxes of oranges.

Example 6: Solving a Real-World Problem Involving Fractions

Maged constructs 34 of a wall in 123 days. How many days will he need to construct the wall?

Answer

We will start by calling the number of daysΒ Maged needs to construct the wall 𝑑. Since we are told that Maged constructs 34 of the wall in 123 days, we know that 34×𝑑=123.

We can solve for 𝑑 by dividing both sides of the equation through by 34. This gives 𝑑=123Γ·34.

To divide by a rational number, we want to multiply by the reciprocal. So, we will rewrite 123 as a proper fraction to get 123=1Γ—3+23=53, giving 𝑑=53Γ·34.

Now, we instead multiply by the reciprocal of 34 and evaluate to get 53Γ·34=53Γ—43=5Γ—43Γ—3=209.

Since the original value in days was given as a mixed fraction, we will also give our answer as a mixed number. We have 209=18+29=229.

Hence, it will take MagedΒ 229 days in total to build the wall.

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

Key Points

  • If 𝑐≠0, then the division of the two rational numbers π‘Žπ‘ and 𝑐𝑑 is a rational number. In general, π‘Žπ‘Γ·π‘π‘‘=π‘Žπ‘Γ—π‘‘π‘=π‘Žπ‘‘π‘π‘.
  • We call 𝑑𝑐 the reciprocal of 𝑐𝑑, 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.

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