John's annual salary of x dollars is twice Lee's annual salary and half of Sarah's annual salary. Which of the...

1. Translate the problem requirements: We need to clarify what "John's salary is twice Lee's and half of Sarah's" means in mathematical terms, and understand that we're finding the arithmetic mean of all three salaries.
2. Express all salaries in terms of x: Since John's salary is x, we'll determine Lee's and Sarah's salaries by working backwards from the given relationships.
3. Apply the average formula: Add all three salaries and divide by 3 to find the arithmetic mean.
4. Simplify to match answer choices: Combine like terms and reduce the fraction to its simplest form.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the problem is telling us in everyday language first. We have three people - John, Lee, and Sarah - and we know some relationships between their salaries.

The key phrase is "John's annual salary of x dollars is twice Lee's annual salary and half of Sarah's annual salary." Let's think about what this means:

• If John makes twice what Lee makes, then Lee makes half of what John makes
• If John makes half of what Sarah makes, then Sarah makes twice what John makes

So if John makes x dollars:

• Lee makes half that amount
• Sarah makes double that amount

We need to find the average (arithmetic mean) of all three salaries.

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

2. Express all salaries in terms of x

Now let's put actual numbers to these relationships. We know John's salary is x dollars.

For Lee's salary: Since John's salary is twice Lee's, we can say:
John = 2 × Lee
\(\mathrm{x = 2 \times Lee}\)
Therefore: \(\mathrm{Lee = \frac{x}{2}}\)

For Sarah's salary: Since John's salary is half of Sarah's, we can say:
John = Sarah/2
\(\mathrm{x = \frac{Sarah}{2}}\)
Therefore: \(\mathrm{Sarah = 2x}\)

Let's verify our relationships make sense:

• John's salary: x
• Lee's salary: \(\mathrm{\frac{x}{2}}\) (John makes twice this: \(\mathrm{2 \times \frac{x}{2} = x}\) ✓)
• Sarah's salary: \(\mathrm{2x}\) (John makes half this: \(\mathrm{\frac{2x}{2} = x}\) ✓)

3. Apply the average formula

To find the arithmetic mean (average) of three numbers, we add them up and divide by 3.

\(\mathrm{Average = \frac{John's\,salary + Lee's\,salary + Sarah's\,salary}{3}}\)

Substituting our expressions:
\(\mathrm{Average = \frac{x + \frac{x}{2} + 2x}{3}}\)

Let's add the numerator step by step:

• We have: \(\mathrm{x + \frac{x}{2} + 2x}\)
• First, let's combine the whole number terms: \(\mathrm{x + 2x = 3x}\)
• So we have: \(\mathrm{3x + \frac{x}{2}}\)
• To add these, we need a common denominator of 2: \(\mathrm{3x = \frac{6x}{2}}\)
• Therefore: \(\mathrm{\frac{6x}{2} + \frac{x}{2} = \frac{7x}{2}}\)

So our average is: \(\mathrm{\frac{7x}{2} \div 3}\)

4. Simplify to match answer choices

When we divide a fraction by a whole number, we multiply by the reciprocal:

\(\mathrm{\frac{7x}{2} \div 3 = \frac{7x}{2} \times \frac{1}{3} = \frac{7x}{2 \times 3} = \frac{7x}{6}}\)

Looking at our answer choices, this matches choice C: \(\mathrm{\frac{7x}{6}}\).

Let's verify with a concrete example: If John makes \$60,000 (x = 60,000):

• Lee makes \$30,000 (half of John's)
• Sarah makes \$120,000 (double John's)
• Average = (60,000 + 30,000 + 120,000) ÷ 3 = 210,000 ÷ 3 = \$70,000

Using our formula: \(\mathrm{\frac{7x}{6} = \frac{7(60,000)}{6} = \frac{420,000}{6} = \$70,000}\) ✓

Final Answer

The average of the three salaries is \(\mathrm{\frac{7x}{6}}\), which corresponds to answer choice C.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the relationship direction

Students often confuse "John's salary is twice Lee's salary" with "Lee's salary is twice John's salary." This leads them to incorrectly set Lee = 2x instead of \(\mathrm{Lee = \frac{x}{2}}\). The word "twice" can be tricky - students need to carefully identify who is making more money in each comparison.

2. Mixing up "half of" relationships

When the problem states "John's salary is half of Sarah's salary," students sometimes incorrectly conclude that \(\mathrm{Sarah = \frac{x}{2}}\) instead of \(\mathrm{Sarah = 2x}\). They fail to recognize that if John makes half of what Sarah makes, then Sarah must make twice what John makes.

3. Setting up inconsistent variable expressions

Students may correctly identify one relationship but then create contradictory expressions for the other. For example, they might correctly get \(\mathrm{Lee = \frac{x}{2}}\) but then incorrectly write \(\mathrm{Sarah = \frac{x}{2}}\) as well, failing to maintain the logical consistency between all three salary relationships.

Errors while executing the approach

1. Common denominator errors when adding fractions

When calculating \(\mathrm{x + \frac{x}{2} + 2x}\), students often struggle with combining whole numbers and fractions. They may incorrectly add to get \(\mathrm{\frac{4x}{2}}\) instead of properly converting to get \(\mathrm{\frac{6x + x}{2} = \frac{7x}{2}}\). The key error is not recognizing that whole numbers need to be expressed with the common denominator.

2. Division by 3 arithmetic mistake

After correctly getting \(\mathrm{\frac{7x}{2}}\) as the sum, students may incorrectly divide by 3. Common errors include writing \(\mathrm{\frac{7x}{2} \div 3 = \frac{7x}{6}}\) but then second-guessing themselves or making computational mistakes like getting \(\mathrm{\frac{7x}{5}}\) or \(\mathrm{\frac{21x}{2}}\).

3. Incorrect fraction multiplication

Some students may attempt to divide \(\mathrm{\frac{7x}{2} \div 3}\) by incorrectly multiplying: \(\mathrm{\frac{7x}{2} \times 3}\) instead of \(\mathrm{\frac{7x}{2} \times \frac{1}{3}}\). This fundamental error with fraction division leads them to get \(\mathrm{\frac{21x}{2}}\) instead of the correct \(\mathrm{\frac{7x}{6}}\).

Errors while selecting the answer

No likely faltering points - the final expression \(\mathrm{\frac{7x}{6}}\) directly matches answer choice C, and there's no additional conversion or selection logic required that would commonly trip up students.

Alternate Solutions

Smart Numbers Approach

We can solve this problem by choosing a convenient value for John's salary that makes the calculations clean and straightforward.

Step 1: Choose a smart value for John's salary

Let's set John's salary x = \$12. This number works well because:

• It's easily divisible by 2 (for finding Lee's salary)
• It can be easily doubled (for finding Sarah's salary)
• It will give us clean numbers for averaging

Step 2: Calculate the other salaries

Given that John's salary is x = \$12:

• Lee's salary: Since John's salary is twice Lee's salary:
  12 = 2 × Lee's salary
  Lee's salary = 12 ÷ 2 = \$6
• Sarah's salary: Since John's salary is half of Sarah's salary:
  12 = Sarah's salary ÷ 2
  Sarah's salary = 12 × 2 = \$24

Step 3: Calculate the average of the three salaries

Average = (John's + Lee's + Sarah's) ÷ 3

Average = (12 + 6 + 24) ÷ 3 = 42 ÷ 3 = \$14

Step 4: Express the result in terms of x

We found that when x = 12, the average is 14.

Let's check which answer choice gives us 14 when x = 12:

• A. \(\mathrm{\frac{5x}{6} = \frac{5(12)}{6} = \frac{60}{6} = 10}\) ❌
• B. x = 12 ❌
• C. \(\mathrm{\frac{7x}{6} = \frac{7(12)}{6} = \frac{84}{6} = 14}\) ✓
• D. \(\mathrm{\frac{6x}{5} = \frac{6(12)}{5} = \frac{72}{5} = 14.4}\) ❌
• E. \(\mathrm{\frac{5x}{4} = \frac{5(12)}{4} = \frac{60}{4} = 15}\) ❌

Answer: C. \(\mathrm{\frac{7x}{6}}\)

Verification with the relationship

This smart numbers approach confirms our algebraic solution. By using x = 12, we clearly see that:

• Lee earns \$6 (half of John's \$12)
• Sarah earns \$24 (double John's \$12)
• The average is (6 + 12 + 24) ÷ 3 = 14 = \(\mathrm{\frac{7x}{6}}\)