In a drawing for a certain company's recent holiday party door prizes, 12 employees were awarded a total of $120,...

Let's visualize this problem to make it crystal clear...

Understanding the Problem

We have a door prize distribution scenario:

4 employees × $A each = 4A dollars
8 employees × $B each = 8B dollars
                Total = $120

This gives us our key equation: \(4\mathrm{A} + 8\mathrm{B} = 120\)

Simplifying the Relationship

Let's simplify by dividing everything by 4:
\(\mathrm{A} + 2\mathrm{B} = 30\)

Rearranging to express A in terms of B:
\(\mathrm{A} = 30 - 2\mathrm{B}\)

Finding Valid Values

Now we'll check each possible B value from our choices to see if it produces a valid A value that's also in our choices.

Testing B = 4:

• \(\mathrm{A} = 30 - 2(4) = 30 - 8 = 22\)
• Is 22 in our choices? No, continue.

Testing B = 5:

• \(\mathrm{A} = 30 - 2(5) = 30 - 10 = 20\)
• Is 20 in our choices? Yes! ✓

? Stop here - we found our answer.

Verification

Let's quickly verify:

• 4 employees × $20 = $80
• 8 employees × $5 = $40
• Total = \(\$80 + \$40 = \$120\) ✓

Final Answer

\(\mathrm{A} = 20\) (the amount each of the 4 employees receives)
\(\mathrm{B} = 5\) (the amount each of the 8 employees receives)