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If Bill's salary is \(\frac{11}{14}\) of Elizabeth's salary, then Bill's salary is what percent less than the sum of both salaries?
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:
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
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!
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.
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.
Bill's salary is 56% less than the sum of both salaries.
The answer is E. 56%
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%).
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.
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.
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.
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."
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."
This problem can be effectively solved using smart numbers by choosing a convenient value for Elizabeth's salary that makes the fraction calculations clean.
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).
Bill's salary = \(\frac{11}{14} \times \$14 = \$11\)
Sum = Bill's salary + Elizabeth's salary = \(\$11 + \$14 = \$25\)
Difference = Sum - Bill's salary = \(\$25 - \$11 = \$14\)
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.