For real numbers a and b, define a @ b = (3a - b)/4. What is the value of 6...

1. TRANSLATE the problem information

• Given operation definition: \(\mathrm{a @ b = \frac{3a - b}{4}}\)
• Need to find: \(\mathrm{6 @ 2}\)
• This means we substitute \(\mathrm{a = 6}\) and \(\mathrm{b = 2}\) into the definition

2. TRANSLATE the specific calculation

• Replace \(\mathrm{a}\) with \(\mathrm{6}\) and \(\mathrm{b}\) with \(\mathrm{2}\) in the expression \(\mathrm{\frac{3a - b}{4}}\)
• This gives us: \(\mathrm{6 @ 2 = \frac{3(6) - 2}{4}}\)

3. SIMPLIFY using order of operations

• First, multiply inside the parentheses: \(\mathrm{3(6) = 18}\)
• Expression becomes: \(\mathrm{\frac{18 - 2}{4}}\)
• Next, subtract inside parentheses: \(\mathrm{18 - 2 = 16}\)
• Expression becomes: \(\mathrm{\frac{16}{4}}\)
• Finally, divide: \(\mathrm{16 ÷ 4 = 4}\)

Answer: B. 4

Why Students Usually Falter on This Problem

Most Common Error Path:

SIMPLIFY execution error: Students make arithmetic mistakes during calculation, most commonly getting \(\mathrm{18 - 2 = 20}\) instead of \(\mathrm{16}\).

When they calculate \(\mathrm{\frac{18 - 2}{4}}\) as \(\mathrm{\frac{20}{4}}\), they get \(\mathrm{\frac{20}{4} = 5}\).

This leads them to select Choice C (5).

Second Most Common Error:

TRANSLATE reasoning error: Students misread the operation definition as addition instead of subtraction, thinking \(\mathrm{a @ b = \frac{3a + b}{4}}\).

Using this incorrect definition: \(\mathrm{\frac{3(6) + 2}{4} = \frac{18 + 2}{4} = \frac{20}{4} = 5}\).

This also leads them to select Choice C (5).

The Bottom Line:

This problem tests careful attention to detail in both reading the definition accurately and executing basic arithmetic correctly. The key is methodical substitution followed by precise calculation.