Question:5a = b - 2cThe given equation relates the variables a, b, and c. Which equation correctly expresses c in...
GMAT Advanced Math : (Adv_Math) Questions
\(5\mathrm{a} = \mathrm{b} - 2\mathrm{c}\)
The given equation relates the variables \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{c}\). Which equation correctly expresses \(\mathrm{c}\) in terms of \(\mathrm{a}\) and \(\mathrm{b}\)?
- \(\mathrm{c} = \frac{\mathrm{b} - 5\mathrm{a}}{2}\)
- \(\mathrm{c} = \frac{5\mathrm{a} - \mathrm{b}}{2}\)
- \(\mathrm{c} = \frac{\mathrm{b}}{2} - 5\mathrm{a}\)
- \(\mathrm{c} = 2(\mathrm{b} - 5\mathrm{a})\)
1. TRANSLATE the problem information
- Given equation: \(\mathrm{5a = b - 2c}\)
- Goal: Express c in terms of a and b (isolate c)
2. INFER the solution strategy
- To isolate c, I need to get the term containing c by itself on one side
- Since c has coefficient \(\mathrm{-2}\), I'll need to handle that negative coefficient carefully
- My plan: Move terms to get \(\mathrm{2c}\) alone, then divide by 2
3. SIMPLIFY by moving the c term
- Add \(\mathrm{2c}\) to both sides: \(\mathrm{5a + 2c = b}\)
- This gets the c term on the same side as \(\mathrm{5a}\)
4. SIMPLIFY by isolating the c term
- Subtract \(\mathrm{5a}\) from both sides: \(\mathrm{2c = b - 5a}\)
- Now the term with c is isolated
5. SIMPLIFY by solving for c
- Divide both sides by 2: \(\mathrm{c = \frac{b - 5a}{2}}\)
- This gives c in terms of a and b
Answer: A. \(\mathrm{c = \frac{b - 5a}{2}}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make sign errors when handling negative coefficients or when moving terms between sides of the equation.
For example, a student might try an alternative approach: subtract b from both sides to get \(\mathrm{5a - b = -2c}\), then divide by \(\mathrm{-2}\). But they forget to handle the negative sign correctly, getting \(\mathrm{c = \frac{5a - b}{2}}\) instead of \(\mathrm{c = \frac{b - 5a}{2}}\).
This may lead them to select Choice B (\(\mathrm{c = \frac{5a - b}{2}}\))
Second Most Common Error:
Poor SIMPLIFY reasoning: Students don't apply division consistently to all terms in an expression.
When they reach \(\mathrm{2c = b - 5a}\), they might think "divide by 2" but only divide the \(\mathrm{5a}\) term, getting \(\mathrm{c = \frac{b}{2} - 5a}\) instead of \(\mathrm{c = \frac{b - 5a}{2}}\).
This may lead them to select Choice C (\(\mathrm{c = \frac{b}{2} - 5a}\))
The Bottom Line:
Literal equations require careful attention to sign changes and consistent application of operations. Each algebraic step must be executed precisely to avoid the systematic errors that lead to wrong answer choices.