In a certain furniture store, each week Nancy earns a salary of $240 plus 5% of the amount of her...

1. Translate the problem requirements: Nancy earns \(\$240\) base salary plus \(5\%\) commission on any sales above \(\$800\). We need to find her total sales when her total earnings were \(\$450\).
2. Identify the commission portion: Calculate how much of Nancy's \(\$450\) total came from commission by subtracting her base salary.
3. Determine sales above threshold: Since commission is \(5\%\) of sales exceeding \(\$800\), work backwards from the commission amount to find the excess sales.
4. Calculate total sales: Add the threshold amount (\(\$800\)) to the excess sales to get Nancy's total sales for the week.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what Nancy's earnings look like in everyday language. Nancy has two sources of income each week:

- A guaranteed base salary of \(\$240\) (she gets this no matter what)
- A bonus commission of \(5\%\) on any sales she makes above \(\$800\)

Think of it like this: If Nancy sells exactly \(\$800\), she only gets her \(\$240\) base salary. But for every dollar she sells above \(\$800\), she gets an extra 5 cents as commission.

We know that one particular week, Nancy's total earnings were \(\$450\). We need to figure out what her total sales were that week.

Process Skill: TRANSLATE - Converting the problem's language into clear mathematical understanding

2. Identify the commission portion

Since Nancy earned \(\$450\) total and her base salary is always \(\$240\), let's figure out how much came from commission:

\(\mathrm{Commission\,earned} = \mathrm{Total\,earnings} - \mathrm{Base\,salary}\)
\(\mathrm{Commission\,earned} = \$450 - \$240 = \$210\)

So Nancy made \(\$210\) in commission that week. This \(\$210\) represents \(5\%\) of her sales that exceeded \(\$800\).

3. Determine sales above threshold

Now we need to work backwards. If \(\$210\) represents \(5\%\) of her sales above \(\$800\), what amount of sales above \(\$800\) would generate \(\$210\) in commission?

Let's think about this: If \(5\%\) of something equals \(\$210\), then that "something" must be much larger.

Since \(\mathrm{Commission} = 5\% \times (\mathrm{Sales\,above\,\$800})\)
\(\$210 = 0.05 \times (\mathrm{Sales\,above\,\$800})\)

To find the sales above \(\$800\):
\(\mathrm{Sales\,above\,\$800} = \$210 \div 0.05 = \$210 \times 20 = \$4,200\)

So Nancy sold \(\$4,200\) worth of merchandise above the \(\$800\) threshold.

4. Calculate total sales

Now we can find Nancy's total sales for the week:

\(\mathrm{Total\,sales} = \mathrm{Threshold\,amount} + \mathrm{Sales\,above\,threshold}\)
\(\mathrm{Total\,sales} = \$800 + \$4,200 = \$5,000\)

Let's verify this makes sense:

- Base salary: \(\$240\)
- Commission on sales above \(\$800\): \(5\% \times \$4,200 = \$210\)
- Total earnings: \(\$240 + \$210 = \$450\) ✓

Final Answer

Nancy's total sales that week were \(\$5,000\).

Looking at our answer choices, this matches choice E: \(\$5,000\).

Common Faltering Points

Errors while devising the approach

Students often think Nancy earns \(5\%\) commission on ALL her sales, rather than only on sales that exceed \(\$800\). This leads them to set up the equation as: \(\$450 = \$240 + 0.05 \times (\mathrm{total\,sales})\), which would give an incorrect total sales amount.

Some students might think the \(5\%\) applies to her total earnings or to the amount above her base salary, rather than specifically to the sales amount above \(\$800\). This conceptual confusion leads to incorrect equation setup from the start.

Errors while executing the approach

When calculating sales above threshold, students might incorrectly convert \(5\%\) to \(0.5\) instead of \(0.05\), or make errors when dividing \(\$210\) by \(0.05\), potentially calculating \(\$210 \div 0.05\) as \(\$42\) instead of \(\$4,200\).

After correctly calculating that sales above \(\$800\) equal \(\$4,200\), students might forget the final step of adding the \(\$800\) threshold back to get total sales, stopping at \(\$4,200\) as their final answer instead of \(\$5,000\).

Errors while selecting the answer

Students who calculated the sales above threshold correctly (\(\$4,200\)) might mistakenly select choice C (\(\$4,200\)) as their final answer, forgetting that this represents only the portion above \(\$800\), not the total sales amount.