Explainer: Applying the Properties of Division When Dividing Rational Numbers

In this explainer, we will learn how to apply the properties of the division operation as a strategy to divide rational numbers.

Recall that a fraction compares a part to a whole and describes what we call a proportion. The denominator of the fraction is the number of equal shares (or β€œportions”) the whole is split into, while the numerator is the number of these shares that make the part we are considering.

Let us look at a division of a fraction by a fraction. For instance, let us find the number 𝑏 given that 23 (of a given whole) is 45 of 𝑏. This is mathematically described with 23=45⋅𝑏.

By dividing both sides by 45, we get 23Γ·45=𝑏.

So, we need to divide 23 by 45 to find the value of 𝑏. We are going to use two different methods to find the value of 𝑏, both of which will allow us to state a general method to divide by fractions.

First method

We can also multiply both sides of our first equation by 54. We get 23β‹…54=45⋅𝑏⋅54, which is the same as 23β‹…54=45β‹…54⋅𝑏, and since 45β‹…54=1, we find that 23β‹…54=𝑏.

Combining both equations, we find that 𝑏=23Γ·45=23β‹…54.

Second method

We can also find the value of 𝑏 by reasoning with diagrams as shown below.

The top diagram shows 23 of a given whole. We know that it is 45 of 𝑏. So, in the first stage, we need to split our 23 in 4 shares of value π‘Ž (second diagram): π‘Ž=23Γ·4=16.

The number 𝑏 is made of 5 of these shares (third diagram): 𝑏=5Γ—16=56.

Number 𝑏 is then 56 of our given whole, represented by the larger rectangle in our diagrams.

The two stages to find the number 𝑏 were

  • dividing 23 by 4,
  • multiplying the result by 5.

Thus, we have found that 𝑏=23Γ·4Γ—5.

The order in which the division by 4 and the multiplication by 5 are carried out does not matter. This double-operation (multiplying by 5 and dividing by 4) is then equivalent to multiplying by the fraction 54. We get 𝑏=23Γ—54=56.

With both methods, we have found that 23Γ·45=23β‹…54.

This is a general result that can be used whenever we divide by a fraction: dividing by a fraction is equivalent to multiplying by its multiplicative inverse or reciprocal.

How to Divide by Fractions

Dividing by a fraction is equivalent to multiplying by its multiplicative inverse or reciprocal. Hence, when a fraction is divided by another fraction, we can write π‘Žπ‘Γ·π‘π‘‘=π‘Žπ‘Γ—π‘‘π‘.

We are going to prove this rule algebraically. First, we write our division in the form of a fraction: π‘Žπ‘Γ·π‘π‘‘=π‘Žπ‘π‘π‘‘.

Then, we multiply both the numerator and the denominator by the multiplicative inverse of the divisor, that is, by 𝑑𝑐: π‘Žπ‘Γ·π‘π‘‘=π‘Žπ‘Γ—π‘‘π‘π‘π‘‘Γ—π‘‘π‘.

The denominator 𝑐𝑑×𝑑𝑐 is by definition equal to 1. So, π‘Žπ‘Γ·π‘π‘‘=π‘Žπ‘Γ—π‘‘π‘.

Now, the fractions can be either positive or negative. As with multiplications, the sign of a quotient depends on the signs of the dividend and divisor:

  • If the dividend and divisor are of the same sign, then the quotient is positive.
  • If the dividend and divisor are of opposite signs, then the quotient is negative.

Let us look at some questions involving division by a fraction.

Example 1: Dividing Negative Fractions

Evaluate βˆ’52Γ·ο€Όβˆ’23.

Answer

Here, we have a negative fraction divided by another negative fraction. Let us first determine the sign of the result. Both fractions are negative, so the quotient is positive. To find now the result of this division, we apply the rule for dividing by a fraction, namely, that it is equivalent to multiplying by the inverse of the fraction. Hence, we have βˆ’52Γ·ο€Όβˆ’23=52β‹…32.

Note here that the negative signs could be removed given that we know that the quotient is positive. We now simply multiply the numerators together and the denominators together, given that there is no simplification to do between the numerators and the denominators. We get βˆ’52Γ·ο€Όβˆ’23=154.

Example 2: Understanding Dividing by One-Half

Which of the following numbers gives a result less than 12 when divided by 12?

  1. 417
  2. 58
  3. 14
  4. 12
  5. 1517

Answer

Here, we are asked to find which of the given numbers gives a result less than 12 when divided by 12. Let us think about what happens to a number when it is divided by 12. Dividing by 12 is, for instance, finding how many halves there are in a given number: it is twice the number. Indeed, dividing by 12 is equivalent to multiplying by 2. This means that, in the question here, we need to find which number is less than 12 when doubled. This, in turn, means that the number must be less than 14.

Going through the numbers given, we find that only 417 is less than 14. We can check it easily by realizing that 416=14. So, as the denominator in 417 is greater than that in 416 (17>16), the value of each share is smaller (the whole has been split into 17 instead of 16 equal shares), and so 417<416.

The answer is that 417 gives a result less than 12 when divided by 12.

Let us look now at a word problem.

Example 3: Dividing Fractions in a Word Problem

James cuts 112 melons into portions, each of which is 38 of the melon. How many portions does he get?

Answer

In this problem, we are told that David cuts 112 melons into portions that are each 38 of the melon. We want to answer the question β€œhow many portions of size 38 of a melon are there in 112 melons?” The answer is the number of pieces, and it is given by dividing 112 by 38. Before carrying out the division, we need to write 112 as a fraction. Since 1 is two halves, we find that 112=32.

Hence, 112Γ·38=32Γ·38.

Recall that dividing by a fraction is equivalent to multiplying by its multiplicative inverse or reciprocal, so 32Γ·38=32Γ—83=82=4.

We find that David cut 4 portions of melon. We can check our answer by checking that 4 portions of size 38 of a melon give 112 melons: 4Γ—38=32=112.

We are going to look at some examples involving mixed numbers. Remember that mixed numbers are just a particular way of writing so-called improper fractions, that is, fractions that have a numerator greater than the denominator. In this case, the part we are considering is bigger than the whole. This can be envisioned, for instance, with two cakes that have been cut into 8 equal pieces each, and there are 11 pieces left. The fraction of cake that is left is 118, or one full cake plus three pieces (eighths), written as 138.

Let us look at some examples of dividing mixed numbers.

Example 4: Dividing Mixed Numbers

Find π‘₯÷𝑦 given that π‘₯=367 and 𝑦=657.

Answer

We need to compute 367Γ·657. For this, we are going to write the mixed numbers as improper fractions. We find that 367=217+67=277 and 657=427+57=477. Hence, 367Γ·657=277Γ·477.

Recall that dividing by a fraction is equivalent to multiplying by its multiplicative inverse or reciprocal, so 277Γ·477=277Γ—747=2747.

This fraction cannot be simplified, so this is our answer.

Example 5: Dividing Two Mixed Numbers

Which of the following division expressions has a quotient of 113?

  1. 927Γ·345
  2. 659Γ·289
  3. 814Γ·967
  4. 423Γ·312
  5. 935Γ·216

Answer

We want to find which of the given five divisions gives 113. We could of course compute all of them and see which of the results is 113. We can also think that 113 is slightly larger than 1.

Therefore, we are looking for a division where the dividend is slightly larger than the divisor. Let us go now through the five divisions and see if we can already exclude some of them:

  • 927Γ·345 can be approximated to 9Γ·4, giving a result greater than 2, so it can be excluded.
  • 659Γ·289 can be approximated to 6.5Γ·3, giving a result greater than 2, so it can be excluded.
  • 814Γ·967 can be approximated to 8Γ·10, giving a result smaller than 1, so it can be excluded.
  • 423Γ·312 can be approximated to 5Γ·3.5, giving a result slightly greater than 1, so it must be kept as an option.
  • 935Γ·216 can be approximated to 9.5Γ·2, giving a result greater than 4, so it can be excluded.

We are left with only one option, 423Γ·312. Let us check that the result of this division is indeed 113. To carry out the division, we first need to write the mixed numbers as improper fractions.

We find that 423=4+23=123+23=143 and 312=3+12=62+12=72.

Hence, 423Γ·312=143Γ·72, and we know that dividing by a fraction is equivalent to multiplying by its multiplicative inverse or reciprocal, so 423Γ·312=143Γ—27.

Since 14=2Γ—7, we can simplify the expression on the right-hand side by 7, and we get 423Γ·312=23Γ—21=43=113.

Hence, the answer is that the division 423Γ·312 has a quotient of 113.

Example 6: Consecutive Divisions by a Fraction and a Mixed Number

Calculate ο€Ό135Γ·45÷578. Give your answer in its simplest form.

Answer

Convert the mixed numbers to improper fractions. Then divide. Perform the division inside the brackets first.

Remember that dividing by a fraction is the same as multiplying by the reciprocal of the fraction. ο€Ό135Γ·45÷578=ο€Ό85Γ·45÷478=ο€Ό85Γ—54÷478=ο€Ό4020÷478=2Γ·478=2Γ—847=1647.

Key Points

  1. Dividing by a fraction is equivalent to multiplying by its multiplicative inverse or reciprocal. Hence, when a fraction is divided by another fraction, we can write π‘Žπ‘Γ·π‘π‘‘=π‘Žπ‘Γ—π‘‘π‘.
  2. This rule can be proven algebraically.
    1. First, we write our division in the form of a fraction: π‘Žπ‘Γ·π‘π‘‘=π‘Žπ‘π‘π‘‘.
    2. Then, we multiply both the numerator and the denominator by the multiplicative inverse of the divisor, that is, by 𝑑𝑐: π‘Žπ‘Γ·π‘π‘‘=π‘Žπ‘Γ—π‘‘π‘π‘π‘‘Γ—π‘‘π‘.
    3. The denominator 𝑐𝑑×𝑑𝑐 is by definition equal to 1. So, π‘Žπ‘Γ·π‘π‘‘=π‘Žπ‘Γ—π‘‘π‘.
  3. As with multiplications, the sign of a quotient depends on the signs of the dividend and divisor.
    If the dividend and divisor are of the same sign, then the quotient is positive.
    If the dividend and divisor are of opposite signs, then the quotient is negative.

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