If Bill's salary is 11/14 of Elizabeth's salary, then Bill's salary is what percent less than the sum of both...

1. Translate the problem requirements: We need to find what percent Bill's salary is less than the sum of both Bill's and Elizabeth's salaries. This means we're comparing Bill's salary to the total of both salaries combined.
2. Express both salaries using a common reference: Since Bill's salary is given as a fraction of Elizabeth's salary, we'll use Elizabeth's salary as our base unit to avoid complex fractions.
3. Calculate the sum of both salaries: Add Bill's salary and Elizabeth's salary, both expressed in terms of Elizabeth's salary.
4. Find the difference and calculate the percentage: Determine how much less Bill's salary is compared to the sum, then convert this to a percentage.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is asking in plain English. We have two people: Bill and Elizabeth. Bill earns \(\frac{11}{14}\) of what Elizabeth earns. The question wants to know: "Bill's salary is what percent less than the sum of both salaries?"

This means we need to:

• Find the total of both their salaries combined
• Compare Bill's individual salary to this total
• Express how much smaller Bill's salary is as a percentage

Think of it this way: if you and your friend had a combined income, and we wanted to know how much less your individual income is compared to your combined income, that's exactly what we're calculating here.

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

2. Express both salaries using a common reference

Since Bill's salary is described as a fraction of Elizabeth's salary, let's use Elizabeth's salary as our reference point. This will keep our numbers simple and avoid complex calculations.

Let's say Elizabeth's salary = 14 units (choosing 14 because it works nicely with the fraction \(\frac{11}{14}\))

Then Bill's salary = \(\frac{11}{14} \times 14 = 11\) units

Why did we choose 14 as our reference? Because when Bill's salary is \(\frac{11}{14}\) of Elizabeth's, and Elizabeth earns 14 units, then Bill earns exactly 11 units. No messy fractions to deal with!

3. Calculate the sum of both salaries

Now we can easily add up their combined salaries:

Elizabeth's salary: 14 units
Bill's salary: 11 units
Sum of both salaries: \(14 + 11 = 25\) units

So their combined income is 25 units total.

4. Find the difference and calculate the percentage

Now we need to find how much less Bill's salary is compared to the sum of both salaries.

Bill's salary: 11 units
Sum of both salaries: 25 units
Difference: \(25 - 11 = 14\) units

To find what percent less Bill's salary is:
Percentage = (Difference ÷ Sum) × 100%
Percentage = \(\frac{14}{25} \times 100\% = 0.56 \times 100\% = 56\%\)

Let's verify this makes sense: Bill earns 11 units out of a total of 25 units. So he earns \(\frac{11}{25} = 44\%\) of the combined salary. This means his salary is \(100\% - 44\% = 56\%\) less than the combined total.

Final Answer

Bill's salary is 56% less than the sum of both salaries.

The answer is E. 56%

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "percent less than" means

Students often confuse "Bill's salary is what percent less than the sum" with "Bill's salary is what percent of the sum." This leads them to calculate \(\frac{11}{25} = 44\%\) instead of recognizing they need to find how much smaller Bill's salary is compared to the total (which is 56%).

2. Comparing Bill's salary to Elizabeth's salary instead of the sum

Since the problem starts by relating Bill's salary to Elizabeth's salary (\(\frac{11}{14}\)), students may mistakenly think they need to find what percent less Bill's salary is compared to Elizabeth's salary alone, rather than compared to their combined salaries.

3. Setting up the wrong reference point

Students might choose Elizabeth's salary as 11 units instead of 14 units, which would make Bill's salary a fraction again (\(\frac{11}{14} \times 11\)), leading to unnecessarily complex calculations and potential errors.

Errors while executing the approach

1. Incorrect fraction arithmetic

When calculating \(\frac{11}{14} \times 14\), students might make basic arithmetic errors, especially if they don't recognize that the 14s cancel out to give exactly 11.

2. Wrong percentage calculation formula

Students might use the formula \(\frac{\text{Bill's salary}}{\text{Sum}} \times 100\% = \frac{11}{25} \times 100\% = 44\%\) and think this is the final answer, forgetting they need to subtract from 100% to get "percent less than."

Errors while selecting the answer

1. Selecting the complement percentage

After correctly calculating that Bill earns 44% of the combined salary, students might select 44% as their final answer if that were an option, instead of recognizing that "44% of the sum" means "56% less than the sum."

Alternate Solutions

Smart Numbers Approach

This problem can be effectively solved using smart numbers by choosing a convenient value for Elizabeth's salary that makes the fraction calculations clean.

Step 1: Choose a smart number for Elizabeth's salary

Since Bill's salary is \(\frac{11}{14}\) of Elizabeth's salary, let's choose Elizabeth's salary = $14 (this makes the fraction calculation straightforward since 14 is the denominator).

Step 2: Calculate Bill's salary

Bill's salary = \(\frac{11}{14} \times \$14 = \$11\)

Step 3: Find the sum of both salaries

Sum = Bill's salary + Elizabeth's salary = \(\$11 + \$14 = \$25\)

Step 4: Calculate how much less Bill's salary is than the sum

Difference = Sum - Bill's salary = \(\$25 - \$11 = \$14\)

Step 5: Convert to percentage

Percentage less = (Difference ÷ Sum) × 100%
= \(\frac{\$14}{\$25} \times 100\%\)
= \(\frac{14}{25} \times 100\%\)
= \(0.56 \times 100\% = 56\%\)

Answer: E. 56%

Why this smart number works: Choosing Elizabeth's salary as $14 eliminates fractions when calculating Bill's salary, making all subsequent calculations with whole numbers. This approach is logically sound because any positive value for Elizabeth's salary will yield the same percentage result.