A survey of employers found that during 1993 employment costs rose 3.5%, where employment costs consist of salary costs and...

1. Translate the problem requirements: We need to find what fraction fringe benefits represented of total employment costs at the start of 1993. We know total costs rose 3.5%, salary costs rose 3%, and fringe costs rose 5.5%.
2. Set up the weighted average relationship: Recognize that the overall 3.5% increase is a weighted average of the 3% salary increase and 5.5% fringe increase, where the weights are the proportions of each cost type.
3. Apply the weighted average formula intuitively: If S is the salary proportion and F is the fringe proportion, then \(\mathrm{S + F = 1}\), and the weighted average gives us \(\mathrm{3S + 5.5F = 3.5}\).
4. Solve for the fringe proportion: Substitute \(\mathrm{S = 1 - F}\) into the equation and solve for F to get the percentage that fringe benefits represented.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we know and what we need to find.

We know:
• Total employment costs rose by 3.5% during 1993
• Employment costs = salary costs + fringe-benefit costs
• Salary costs rose by 3% during 1993
• Fringe-benefit costs rose by 5.5% during 1993

We need to find: What percent of employment costs did fringe-benefit costs represent at the beginning of 1993?

Think of it this way: we're looking for the proportion or "slice" that fringe benefits made up of the total employment cost pie at the start of the year.

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

2. Set up the weighted average relationship

Here's the key insight: when different parts of something grow at different rates, the overall growth rate is a weighted average of the individual growth rates.

Imagine you have a basket with two types of fruit - apples and oranges. If apple prices go up 3% and orange prices go up 5.5%, and your overall fruit bill goes up 3.5%, then the 3.5% is somewhere between 3% and 5.5%. The exact value depends on how much of your original basket was apples versus oranges.

In our case:
• "Apples" = salary costs (grew 3%)
• "Oranges" = fringe costs (grew 5.5%)
• Overall basket growth = 3.5%

The overall 3.5% increase must be a weighted average of the 3% and 5.5% increases, where the weights are the original proportions of salary and fringe costs.

3. Apply the weighted average formula intuitively

Let's say fringe benefits represented some fraction of the total costs at the beginning. We'll call this fraction F.

Then salary costs must represent the remaining fraction, which is (1 - F).

Now, here's how the weighted average works:
(Fraction of salary) × (Salary growth) + (Fraction of fringe) × (Fringe growth) = Overall growth

In plain English: \(\mathrm{(1 - F) \times 3\% + F \times 5.5\% = 3.5\%}\)

4. Solve for the fringe proportion

Now let's solve this equation step by step:

\(\mathrm{(1 - F) \times 3 + F \times 5.5 = 3.5}\)

Expanding the left side:
\(\mathrm{3 - 3F + 5.5F = 3.5}\)

Combining like terms:
\(\mathrm{3 + 2.5F = 3.5}\)

Subtracting 3 from both sides:
\(\mathrm{2.5F = 0.5}\)

Dividing both sides by 2.5:
\(\mathrm{F = 0.5 ÷ 2.5 = 0.5 ÷ (5/2) = 0.5 \times (2/5) = 1/5 = 0.2}\)

So F = 0.2, which means fringe benefits represented 20% of employment costs at the beginning of 1993.

Final Answer

Fringe-benefit costs represented 20% of employment costs at the beginning of 1993.

Verification: If fringe benefits were 20% and salaries were 80% of total costs, then:
\(\mathrm{0.8 \times 3\% + 0.2 \times 5.5\% = 2.4\% + 1.1\% = 3.5\%}\) ✓

The answer is (B) 20%.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what the question is asking for

Students often confuse whether they need to find the percentage at the beginning or end of 1993. The question asks for fringe-benefit costs as a percent of employment costs "at the beginning of 1993," but students might try to calculate the percentage after the increases have occurred.

2. Not recognizing this as a weighted average problem

Many students don't realize that when different components grow at different rates, the overall growth rate is a weighted average. They might try to set up equations based on absolute dollar amounts or attempt to work backwards from the final percentages, making the problem much more complicated than necessary.

3. Setting up the wrong relationship between salary and fringe costs

Students might incorrectly assume that salary costs and fringe costs are given as specific percentages of the total, rather than understanding that these two components together make up 100% of employment costs. This leads to setting up equations that don't properly account for the fact that salary fraction + fringe fraction = 1.

Errors while executing the approach

1. Algebraic manipulation errors

When solving the equation \(\mathrm{(1-F) \times 3 + F \times 5.5 = 3.5}\), students commonly make errors when expanding and combining like terms, particularly when dealing with the negative term (-3F) and combining it with the positive term (+5.5F) to get +2.5F.

2. Decimal and fraction conversion mistakes

Students often struggle with the division 0.5 ÷ 2.5, either making basic arithmetic errors or getting confused when converting between decimals and fractions. Some might incorrectly calculate this as 0.25 instead of 0.2.

3. Using wrong percentage values in calculations

Students might use the percentage values (3%, 5.5%, 3.5%) as whole numbers (3, 5.5, 3.5) in their weighted average equation, or conversely, use decimal forms (0.03, 0.055, 0.035) when they should use the whole number forms, leading to incorrect final answers.

Errors while selecting the answer

1. Forgetting to convert the final answer to percentage form

After correctly calculating F = 0.2, students might select 0.2 or look for an answer choice with 0.2, forgetting that the question asks for the answer as a percentage, so they need to convert 0.2 to 20%.

2. Selecting the salary percentage instead of fringe percentage

If students correctly find that fringe benefits represent 20% of costs, they might mistakenly think this means salary costs are 20% and select an answer choice around 80%, confusing which component the question is asking about.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose convenient starting values for employment costs

Let's assume total employment costs at the beginning of 1993 were $100. This makes percentage calculations straightforward and intuitive.

Step 2: Set up variables using our smart number

Let fringe-benefit costs = $F at the beginning of 1993
Then salary costs = $(100 - F) at the beginning of 1993

Step 3: Calculate costs after the increases

After 1993 increases:
• Salary costs became: \(\mathrm{\$(100 - F) \times 1.03}\)
• Fringe-benefit costs became: \(\mathrm{\$F \times 1.055}\)
• Total employment costs became: \(\mathrm{\$100 \times 1.035 = \$103.50}\)

Step 4: Set up the equation

Since the sum of individual cost increases equals the total cost increase:
\(\mathrm{(100 - F) \times 1.03 + F \times 1.055 = 103.50}\)

Step 5: Solve for F

\(\mathrm{103 - 1.03F + 1.055F = 103.50}\)
\(\mathrm{103 + 0.025F = 103.50}\)
\(\mathrm{0.025F = 0.50}\)
\(\mathrm{F = 20}\)

Step 6: Interpret the result

Since we started with $100 total employment costs and found F = $20, fringe-benefit costs represented 20% of employment costs at the beginning of 1993.

Verification:
• Salary costs: \(\mathrm{\$80 → \$80 \times 1.03 = \$82.40}\)
• Fringe costs: \(\mathrm{\$20 → \$20 \times 1.055 = \$21.10}\)
• Total: \(\mathrm{\$82.40 + \$21.10 = \$103.50}\) ✓
• Overall increase: \(\mathrm{\$103.50/\$100 = 3.5\%}\) ✓