- Translate the problem requirements: Break down the commission structure into its components - percentage-based commission (\(\mathrm{x\%}\) of total sales) plus fixed bonuses (\(\$20\) for first 10 printers, \(\$30\) for additional printers). We need to find the percentage rate x.
- Calculate the fixed bonus component: Determine the total fixed bonus earned from selling 12 printers based on the tiered structure.
- Isolate the percentage-based commission: Subtract the fixed bonus from total commission to find how much came from the percentage of sales.
- Solve for the percentage rate: Use the percentage commission amount and total selling price to calculate x.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know about this salesperson's commission structure in plain English:
- The salesperson gets x percent of whatever they sell (this is the percentage-based part)
- Plus they get a bonus for each printer: \(\$20\) for each of the first 10 printers, then \(\$30\) for any additional printers
What happened last month:
- Sold 12 printers
- Total selling price was \(\$1,800\)
- Total commission earned was \(\$305\)
We need to find the value of x (the percentage rate).
Process Skill: TRANSLATE - Converting the commission structure into clear, manageable components
2. Calculate the fixed bonus component
Now let's figure out how much the salesperson earned just from the fixed bonuses (not counting the percentage part yet):
Since they sold 12 printers:
- First 10 printers: \(10 \times \$20 = \$200\)
- Remaining 2 printers: \(2 \times \$30 = \$60\)
- Total fixed bonus = \(\$200 + \$60 = \$260\)
So out of the \(\$305\) total commission, \(\$260\) came from these fixed bonuses.
3. Isolate the percentage-based commission
Now we can find how much came from the percentage of sales:
Percentage-based commission = Total commission - Fixed bonuses
Percentage-based commission = \(\$305 - \$260 = \$45\)
This means the salesperson earned \(\$45\) from getting \(\mathrm{x\%}\) of the \(\$1,800\) in sales.
4. Solve for the percentage rate
Now we can find x. We know that:
\(\mathrm{x\%}\) of \(\$1,800 = \$45\)
In other words: \(\frac{\mathrm{x}}{100} \times \$1,800 = \$45\)
To solve for x:
\(\mathrm{x} = \frac{\$45 \times 100}{\$1,800}\)
\(\mathrm{x} = \frac{\$4,500}{\$1,800}\)
\(\mathrm{x} = 2.5\)
Let's verify: \(2.5\%\) of \(\$1,800 = 0.025 \times \$1,800 = \$45\) ✓
5. Final Answer
The value of x is 2.5.
Answer: D. 2.5
Verification: Commission = \(2.5\% \times \$1,800 + \$260\) (bonuses) = \(\$45 + \$260 = \$305\) ✓
Common Faltering Points
Errors while devising the approach
- Misunderstanding the tiered bonus structure: Students often misread the commission structure and think it's \(\$20\) for ALL printers sold or \(\$30\) for ALL printers sold, rather than understanding it's \(\$20\) for the first 10 printers and THEN \(\$30\) for each additional printer beyond 10.
- Overlooking the dual commission components: Students may focus only on the percentage-based commission (\(\mathrm{x\%}\) of sales) and completely forget about the fixed bonus component, or vice versa. They fail to recognize that the total commission has TWO separate parts that must be added together.
- Setting up the equation incorrectly: Students may try to solve for x without first isolating the percentage-based portion, attempting to work with the entire \(\$305\) commission amount instead of subtracting out the fixed bonuses first.
Errors while executing the approach
- Calculation errors in the tiered bonus: Students correctly understand the structure but make arithmetic mistakes when calculating \(10 \times \$20 + 2 \times \$30\), either getting the multiplication wrong or the addition wrong, leading to an incorrect fixed bonus amount.
- Percentage calculation errors: When solving \(\frac{\mathrm{x}}{100} \times \$1,800 = \$45\) for x, students may forget to multiply by 100 or divide incorrectly, getting \(\mathrm{x} = 0.025\) instead of \(\mathrm{x} = 2.5\), or making other arithmetic errors in the division \(\$4,500 \div \$1,800\).
- Incorrect isolation of percentage component: Students may subtract the wrong amount from \(\$305\) due to errors in calculating the fixed bonus portion, leading them to work with an incorrect dollar amount when solving for the percentage rate.
Errors while selecting the answer
- Converting between decimal and percentage forms: Students may correctly calculate that the decimal rate is 0.025 but then select answer choice A (1.0) instead of D (2.5), confusing the decimal form (0.025) with the percentage form (2.5%).
- Failing to verify the final answer: Students may arrive at an answer but not check it against the original conditions, missing the opportunity to catch calculation errors that would be obvious when plugging the value back into the commission formula.
