The average (arithmetic mean) of two numbers is 75. One of the numbers is 50. What is the value of...

1. TRANSLATE the problem information

• Given information:
  • The arithmetic mean of two numbers is 75
  • One number is 50
  • Need to find the other number (call it x)

• What this tells us: We can write \(\frac{50 + \mathrm{x}}{2} = 75\)

2. SIMPLIFY to solve for the unknown

• Start with: \(\frac{50 + \mathrm{x}}{2} = 75\)
• Multiply both sides by 2: \(50 + \mathrm{x} = 150\)
• Subtract 50 from both sides: \(\mathrm{x} = 100\)

3. Verify the answer

• Check: \(\frac{50 + 100}{2} = \frac{150}{2} = 75\) ✓

Answer: D (100)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand what "arithmetic mean" means and think it refers to the difference between numbers rather than their average.

They might calculate: \(75 - 50 = 25\), thinking the mean tells us how far apart the numbers are.

This may lead them to select Choice A (25)

Second Most Common Error:

Incomplete INFER reasoning: Students recognize they need to use the average formula but confuse which value they're solving for.

They might think: "The average is 75, so the other number must also be 75" without considering that different numbers can have the same average.

This may lead them to select Choice C (75)

The Bottom Line:

This problem tests whether students truly understand that arithmetic mean represents the sum divided by count, not just a "middle value" or difference between numbers.