The annual stockholders' report for Corporation X stated that profits were up 10 percent over the previous year, although profits...

1. Translate the problem requirements: Clarify what "profits up 10%" and "profits as percent of sales down 10%" mean mathematically, and identify that we need to find the ratio of current year sales to previous year sales
2. Set up algebraic relationships using simple variables: Define variables for previous year's profits and sales, then express current year's profits in terms of these variables using the given percentage changes
3. Apply the profit margin relationship: Use the fact that profit margin equals profits divided by sales to create an equation linking the current year's sales to known quantities
4. Solve for the sales ratio: Manipulate the equation to find what percent current year sales represent compared to previous year sales

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this business scenario is telling us in plain English:

"Profits were up 10 percent over the previous year" means the company made more money this year. If they made \(\$100\) last year, they made \(\$110\) this year.

"Profits as a percent of sales were down 10 percent" is talking about profit margin - how much profit the company keeps from each dollar of sales. This is a bit trickier. If their profit margin was \(20\%\) last year, it dropped to \(18\%\) this year (that's a \(10\%\) decrease from \(20\%\)).

The question asks: What percent were this year's total sales compared to last year's sales? If last year's sales were \(\$1000\) and this year's were \(\$1220\), then this year's sales would be \(122\%\) of last year's sales.

Process Skill: TRANSLATE - Converting business language into mathematical relationships

2. Set up algebraic relationships using simple variables

Now let's assign simple variables to make our work easier:

Let's say last year:

• Profits = \(\mathrm{P}\)
• Sales = \(\mathrm{S}\)
• Profit margin = \(\mathrm{P/S}\)

This year, based on what we translated:

• Profits = \(\mathrm{1.1P}\) (10% increase)
• Sales = ? (this is what we need to find)
• Profit margin = \(\mathrm{0.9 \times (P/S)}\) (10% decrease from last year's margin)

So this year's profit margin is \(\mathrm{0.9P/S}\)

3. Apply the profit margin relationship

Here's the key insight: profit margin equals profits divided by sales, no matter which year we're looking at.

For this year, we know two ways to express the profit margin:

Method 1: This year's profits ÷ This year's sales = \(\mathrm{1.1P}\) ÷ (This year's sales)

Method 2: 10% less than last year's margin = \(\mathrm{0.9 \times (P/S) = 0.9P/S}\)

Since these must be equal:

\(\mathrm{\frac{1.1P}{\text{This year's sales}} = \frac{0.9P}{S}}\)

Process Skill: INFER - Recognizing that the same profit margin can be calculated in two different ways

4. Solve for the sales ratio

Now we can solve for this year's sales in terms of last year's sales:

Starting with: \(\mathrm{\frac{1.1P}{\text{This year's sales}} = \frac{0.9P}{S}}\)

We can cancel out P from both sides:

\(\mathrm{\frac{1.1}{\text{This year's sales}} = \frac{0.9}{S}}\)

Cross multiply to get:

\(\mathrm{1.1 \times S = 0.9 \times (\text{This year's sales})}\)

Solving for this year's sales:

This year's sales = \(\mathrm{\frac{1.1S}{0.9} = \frac{1.1}{0.9} \times S}\)

Let's calculate \(\mathrm{1.1 ÷ 0.9}\):

\(\mathrm{1.1 ÷ 0.9 = \frac{11}{10} ÷ \frac{9}{10} = \frac{11}{10} \times \frac{10}{9} = \frac{11}{9} ≈ 1.22}\)

So this year's sales = \(\mathrm{1.22 \times S}\), which means this year's sales are \(122\%\) of last year's sales.

5. Final Answer

This year's sales were approximately 122% of last year's sales.

The answer is D. 122%

Quick verification: If sales increased by 22% but profit margin decreased by 10%, it makes sense that overall profits could still increase by 10%, since the sales increase more than compensated for the margin decrease.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "profits as a percent of sales were down 10 percent"

Students often confuse this with "profits were down 10 percent." The key distinction is that this refers to profit margin (profit/sales ratio) decreasing by 10%, not the absolute profit amount. If last year's profit margin was 20%, then this year's margin is 18% (which is 20% - 10% of 20% = 20% - 2% = 18%), not 10%.

2. Setting up incorrect relationships between the variables

Students may struggle to connect that the same profit margin can be expressed in two ways: (1) this year's profits ÷ this year's sales, and (2) 90% of last year's margin. Missing this key insight makes it impossible to set up the crucial equation that links all the given information.

3. Confusion about what the question is actually asking

Students might think the question asks for the absolute difference in sales or the percentage increase in sales, rather than expressing this year's sales as a percentage of last year's sales. The question asks "what percent of sales for the previous year," which means if this year's sales are 1.22 times last year's, the answer is 122%.

Errors while executing the approach

1. Arithmetic errors when calculating \(\mathrm{1.1 ÷ 0.9}\)

Students may struggle with the division \(\mathrm{1.1 ÷ 0.9}\), especially when converting to fractions. The correct calculation is \(\mathrm{\frac{11}{10} ÷ \frac{9}{10} = \frac{11}{10} \times \frac{10}{9} = \frac{11}{9} ≈ 1.22}\). Common errors include getting 1.02 or other incorrect decimal values.

2. Incorrectly canceling or cross-multiplying the equation

When working with the equation \(\mathrm{\frac{1.1P}{\text{This year's sales}} = \frac{0.9P}{S}}\), students might make algebraic errors such as not properly canceling the P terms or making mistakes during cross-multiplication, leading to incorrect relationships between the sales figures.

Errors while selecting the answer

1. Selecting 22% instead of 122%

After correctly calculating that this year's sales are 1.22 times last year's sales, students might incorrectly conclude that the answer is 22% (the increase) rather than 122% (this year's sales as a percent of last year's sales). The question asks for this year's sales as a percentage of last year's sales, not the percentage increase.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient values for previous year

Let's set convenient values that will make our calculations clean:

• Previous year profits = \(\$100\)
• Previous year sales = \(\$1,000\)

This gives us a previous year profit margin of \(\mathrm{\frac{100}{1,000} = 10\%}\)

Step 2: Calculate current year profits

Profits are up 10%, so:

Current year profits = \(\mathrm{\$100 \times 1.10 = \$110}\)

Step 3: Determine current year profit margin

Profits as a percent of sales are down 10%, meaning the profit margin decreased by 10%:

Current year profit margin = \(\mathrm{10\% \times (1 - 0.10) = 10\% \times 0.90 = 9\%}\)

Step 4: Calculate current year sales

Using the relationship: Profit Margin = Profits ÷ Sales

\(\mathrm{9\% = \frac{\$110}{\text{Current year sales}}}\)

\(\mathrm{0.09 = \frac{110}{\text{Current year sales}}}\)

Current year sales = \(\mathrm{\frac{110}{0.09} = \$1,222.22}\)

Step 5: Find the percentage relationship

Current year sales as a percent of previous year sales:

\(\mathrm{\frac{\$1,222.22}{\$1,000} \times 100\% = 122.22\% ≈ 122\%}\)

Answer: D. 122%

The smart numbers approach works well here because we can choose round, convenient values that satisfy the percentage relationships while keeping calculations manageable.