Equivalent Expressions & the Distributive Property - Instruction - Level F

Two rooms in a tiny house share a wall. The area of the living room is 54 square feet, and the area of the bedroom is 42 square feet. The expression [tex]54 + 42[/tex] represents the total area in square feet.

What is the GCF of 54 and 42?

The GCF of 54 and 42 is [tex]6[/tex].

Which expression shows each term with the GCF as one of the factors?

A. [tex](6 \cdot 9) + (6 \cdot 7)[/tex]

B. [tex](9 \cdot 6) + (9 \cdot 7)[/tex]

C. [tex](6 \cdot 9) + 7[/tex]

We start with the areas of the two rooms:

- Living room area: [tex]$54$[/tex] square feet
- Bedroom area: [tex]$42$[/tex] square feet

Step 1. Find the greatest common factor (GCF).

The GCF of [tex]$54$[/tex] and [tex]$42$[/tex] is [tex]$6$[/tex].

Step 2. Express each area as a product involving the GCF.

Divide each area by [tex]$6$[/tex]:

- For the living room:
[tex]$$54 \div 6 = 9,$$[/tex]
so [tex]$54$[/tex] can be written as [tex]$6 \cdot 9$[/tex].

- For the bedroom:
[tex]$$42 \div 6 = 7,$$[/tex]
so [tex]$42$[/tex] can be written as [tex]$6 \cdot 7$[/tex].

Step 3. Write the expression for the total area factoring out the GCF.

Substitute the factored forms into the sum:
[tex]$$
54 + 42 = (6 \cdot 9) + (6 \cdot 7).
$$[/tex]

Thus, the expression that shows each term with the GCF as one of the factors is:
[tex]$$
(6 \cdot 9) + (6 \cdot 7).
$$[/tex]