Question

Three salespeople are paid commissions in proportion to the amount of their sales, which total \(\$25,000\),\( \$40,000,\) and \(\$60,000\), respectively. If a total of \(\$20,000\) is allocated for these three commissions, what is the amount of the largest commission paid?

• A. $8,000
• B. $8,400
• C. $9,600
• D. $10,000
• E. $12,000

Hint:

In ratio problems, you can often simplify the numbers before calculating the final proportions. The sales ratio is 25,000 : 40,000 : 60,000, which simplifies to 25:40:60, and further to 5:8:12. The total parts are \(5+8+12=25\). The largest share is 12 parts out of 25, so the commission is \( \frac{12}{25} \times $20,000 \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:  
This is a problem about proportional distribution. The total commission amount must be divided among the three salespeople according to the ratio of their individual sales. 
Step 2: Key Formula or Approach: 
1. Find the total sales amount by summing the individual sales. 
2. Determine the proportion of the total sales that each salesperson contributed. 
3. The largest commission will go to the salesperson with the highest sales. Calculate this commission by multiplying their proportion of sales by the total commission pool. 
Step 3: Detailed Explanation: 
First, find the total sales from all three salespeople. 
\[ \text{Total Sales} = 25,000 + 40,000 + 60,000 = 125,000 \] The total commission pool is $20,000. 
The largest commission will be paid to the salesperson with the highest sales, which is $60,000. 
Now, we find the proportion of this salesperson's sales relative to the total sales. 
\[ \text{Proportion of largest sale} = \frac{\text{Highest sales}}{\text{Total sales}} = \frac{60,000}{125,000} \] We can simplify this fraction by dividing both the numerator and denominator by common factors. 
\[ \frac{60,000}{125,000} = \frac{60}{125} = \frac{12 \times 5}{25 \times 5} = \frac{12}{25} \] The largest commission is this proportion of the total commission pool. 
\[ \text{Largest Commission} = \text{Proportion} \times \text{Total Commission} \] \[ \text{Largest Commission} = \frac{12}{25} \times 20,000 \] \[ \text{Largest Commission} = 12 \times \frac{20,000}{25} = 12 \times 800 = 9,600 \] Step 4: Final Answer: 
The amount of the largest commission paid is $9,600.