1.04 The distributive property
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$72−48.
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)$72−48=24×(3−2)
Finally, we multiply the two integers to find our answer.
$24\times\left(3-2\right)=24\times1$24×(3−2)=24×1
$24\times1=24$24×1=24
So, $72-48=24$72−48=24.
And there you go! Another way to find the sum or difference between to numbers.
Practice questions
Question 1
Consider the difference $88-20$88−20 :
Find the greatest common factor of $88$88 and $20$20.
Complete the gaps such that $88-20$88−20 is rewritten as an equivalent multiplication using the distributive property.
$88-20$88−20
$=$= $4\cdot\left(\editable{}-5\right)$4·(−5)
$=$= $4\cdot\editable{}$4·
Question 2
Consider $11\left(8-3\right)$11(8−3).
Using the distributive property complete the gap so that $11\left(8-3\right)$11(8−3) is rewritten as the difference of two integers.
$11\left(8-3\right)=88-\editable{}$11(8−3)=88−
Question 3
Hermione and Yuri both earn $\$11$$11 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).
Difference in wages = $\editable{}\left(\editable{}-\editable{}\right)$(−)