Question:In physics and chemistry, the density, d, of an object is given by the formula d = m/v, where m...
GMAT Advanced Math : (Adv_Math) Questions
In physics and chemistry, the density, d, of an object is given by the formula \(\mathrm{d = \frac{m}{v}}\), where m is the mass of the object and v is the volume of the object. Which of the following equations correctly gives the volume, v, in terms of d and m?
- \(\mathrm{v = \frac{d}{m}}\)
- \(\mathrm{v = \frac{m}{d}}\)
- \(\mathrm{v = md}\)
- \(\mathrm{v = m - d}\)
1. TRANSLATE the problem requirements
- Given information:
- Density formula: \(\mathrm{d = m/v}\)
- Need to solve for volume v in terms of d and m
- This means we need to rearrange the equation so v appears alone on one side
2. SIMPLIFY to isolate the variable v
- Start with: \(\mathrm{d = m/v}\)
- To get v out of the denominator, multiply both sides by v:
\(\mathrm{v × d = (m/v) × v}\)
\(\mathrm{vd = m}\)
- Now divide both sides by d to isolate v:
\(\mathrm{vd/d = m/d}\)
\(\mathrm{v = m/d}\)
Answer: B (\(\mathrm{v = m/d}\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students get confused about how to handle the fraction and incorrectly think they should "flip" the relationship.
They might reason: "Since \(\mathrm{d = m/v}\), then v should equal \(\mathrm{d/m}\)" without properly applying algebraic steps.
This leads them to select Choice A (\(\mathrm{v = d/m}\))
Second Most Common Error:
Poor TRANSLATE reasoning: Students misunderstand what algebraic operations to perform when "solving for v."
They might think solving means just rearranging the letters without following proper algebraic rules, leading to multiplying everything together.
This may lead them to select Choice C (\(\mathrm{v = md}\))
The Bottom Line:
This problem requires systematic algebraic thinking rather than intuitive rearrangement. Students need to follow proper equation-solving steps: eliminate the denominator first, then isolate the desired variable.