A factory sets a production goal of at least 21 widgets per shift. A machine produces widgets at an average...
GMAT Algebra : (Alg) Questions
A factory sets a production goal of at least \(\mathrm{21}\) widgets per shift. A machine produces widgets at an average rate of \(\mathrm{4}\) widgets per hour. What is the minimum number of hours the machine must operate to meet the production goal?
Solution
Step 1: Calculate the minimum hours needed
To find the minimum number of hours required, divide the production goal by the production rate:
\(\mathrm{Hours\ needed = \frac{21}{4} = 5.25}\)
Step 2: Round up to the nearest whole hour
Since the machine cannot operate for a fractional hour in practical terms, and we need at least \(\mathrm{21}\) widgets, we must round up to ensure the production goal is met:
\(\mathrm{5.25}\) hours rounds up to \(\mathrm{6}\) hours
Step 3: Verify the answer
At \(\mathrm{4}\) widgets per hour for \(\mathrm{6}\) hours:
\(\mathrm{4 \times 6 = 24\ widgets}\)
Since \(\mathrm{24 \geq 21}\), the production goal is met.
Summary
- Production goal: \(\mathrm{21}\) widgets per shift
- Production rate: \(\mathrm{4}\) widgets per hour
- Calculation: \(\mathrm{21 \div 4 = 5.25}\) hours
- Minimum whole hours needed: \(\mathrm{6}\) hours (rounded up)
- Verification: \(\mathrm{4 \times 6 = 24}\) widgets, which meets the goal of at least \(\mathrm{21}\) widgets
Answer: The machine must operate for a minimum of \(\mathrm{6}\) hours to meet the production goal.
4
5
6
21
1. TRANSLATE the problem information
- Given information:
- Production goal: at least 21 widgets per shift
- Machine rate: 4 widgets per hour
- Need: minimum number of hours
- What this tells us: We need to find the smallest whole number of hours that produces \(\geq\mathrm{21\ widgets}\)
2. INFER the mathematical approach
- Since \(\mathrm{Work = Rate \times Time}\), we can rearrange to get \(\mathrm{Time = Work \div Rate}\)
- We need: \(\mathrm{Time = 21\ widgets \div 4\ widgets\ per\ hour = 5.25\ hours}\)
- But machines operate in whole hours, so we need to consider what happens at different whole-hour intervals
3. APPLY CONSTRAINTS to determine the practical answer
- At 5 hours: \(\mathrm{5 \times 4 = 20\ widgets}\) (falls short of the "at least 21" requirement)
- At 6 hours: \(\mathrm{6 \times 4 = 24\ widgets}\) (meets the requirement)
- Since we need at least 21 widgets and can only operate in whole hours, we must choose 6 hours
Answer: C (6)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students calculate \(\mathrm{21 \div 4 = 5.25}\) and think "5.25 is closest to 5, so the answer is 5 hours."
They miss the crucial insight that rounding down to 5 hours only produces 20 widgets, which doesn't meet the "at least 21" requirement. This may lead them to select Choice A (4) or Choice B (5).
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret "at least 21" as "exactly 21" and try to find a precise fractional answer.
They might calculate 5.25 hours and get confused when this isn't an option, leading to confusion and guessing rather than recognizing they need the next whole number up.
The Bottom Line:
This problem tests whether students understand that real-world constraints (whole hours of operation) combined with minimum requirements ("at least") often require rounding up rather than down, even when the calculated value seems closer to a smaller integer.
4
5
6
21