 # 1.04 The distributive property

Lesson

Let's look at a way to find the sum of or difference between two numbers that uses the greatest common factor (GCF) and the distributive property.

#### Exploration

For example, let's say we wanted to evaluate $72-48$7248.

First, we can find the greatest common factor (GCF) between the two numbers.

The factors of $48$48 are:

$1,2,3,4,6,8,12,16,24,48$1,2,3,4,6,8,12,16,24,48

The factors of $72$72 are:

$1,2,3,4,6,8,9,12,18,24,36,72$1,2,3,4,6,8,9,12,18,24,36,72

The numbers that appear in both factor lists are:

$1,2,3,4,6,8,12,24$1,2,3,4,6,8,12,24

The largest number in this list is the GCF, $24$24.

Now, we can rewrite the expression as an equivalent multiplication by using the distributive property.

$48=24\times2$48=24×2

$72=24\times3$72=24×3

$72-48=24\times\left(3-2\right)$7248=24×(32)

Finally, we multiply the two integers to find our answer.

$24\times\left(3-2\right)=24\times1$24×(32)=24×1

$24\times1=24$24×1=24

So, $72-48=24$7248=24.

And there you go! Another way to find the sum or difference between to numbers.

#### Practice questions

##### Question 1

Consider the difference $96-80$9680 :

1. Find the GCF of $96$96 and $80$80.

2. Complete the gaps such that $96-80$9680 is rewritten as an equivalent multiplication using the distributive property.

 $96-80$96−80 $=$= $16\times\left(\editable{}-5\right)$16×(−5) $=$= $16\times\editable{}$16×

##### Question 2

Consider $11\left(8-3\right)$11(83).

1. Using the distributive property complete the gap so that $11\left(8-3\right)$11(83) is rewritten as the difference of two integers.

$11\left(8-3\right)=88-\editable{}$11(83)=88

##### Question 3

Hermione and Yuri both earn $\$1111 per hour in their casual job. In a day where one works for $7$7 hours and the other works for $2$2 hours, complete the number sentence that can be used to evaluate the difference in their wages (with the difference expressed as a positive quantity).

1. Difference in wages = $\editable{}\left(\editable{}-\editable{}\right)$()

### Outcomes

#### MGSE6.NS.4

Find the common multiples of two whole numbers less than or equal to 12 and the common factors of two whole numbers less than or equal to 100.

#### MGSE6.NS.4a

Find the greatest common factor of 2 whole numbers and use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factors. (GCF)

#### MGSE6.NS.4b

Apply the least common multiple of two whole numbers less than or equal to 12 to solve real- world problems.