There are exactly three steps—Steps 1, 2, and 3, in that order—used to assemble light bulbs at a particular factory....
GMAT Two Part Analysis : (TPA) Questions
There are exactly three steps—Steps 1, 2, and 3, in that order—used to assemble light bulbs at a particular factory. Some of the bulbs fail to make it through every step in the assembly process; if a bulb fails a given step, it will not proceed to the next step. For the most recent month, the factory manager knows the number of bulbs that were started as well as each of the following: \(\mathrm{A}\) = the number of bulbs making it through Step 1 only; \(\mathrm{B}\) = the number of bulbs making it through at least Step 2; \(\mathrm{C}\) = the number of bulbs making it through all three steps.
The manager wishes to know the fraction of the bulbs that made it through Step 1 that also made it through all three steps. In the table, choose for Numerator and Denominator the expressions that will give the factory owner the fraction desired. Make only two selections, one in each column.
Phase 1: Owning the Dataset
Understanding the Flow
Let's create a flow diagram to understand how bulbs move through the assembly process:
Started → Step 1 → Step 2 → Step 3 → Finished
↓ ↓ ↓
Failed Failed Failed
Key Definitions
- \(\mathrm{A}\) = bulbs making it through Step 1 ONLY (failed at Step 2)
- \(\mathrm{B}\) = bulbs making it through AT LEAST Step 2 (includes those that passed Step 2 but failed Step 3, AND those that passed all three steps)
- \(\mathrm{C}\) = bulbs making it through ALL THREE steps
Visual Representation
Let's use concrete numbers to test our understanding. Say we start with 100 bulbs:
Started: 100 bulbs
↓
After Step 1: Some pass, some fail
- A bulbs pass Step 1 but fail Step 2
- B bulbs pass at least Step 2
↓
After Step 2: Of the B bulbs
- Some fail Step 3
- C bulbs pass Step 3
↓
After Step 3: C bulbs completed
Phase 2: Understanding the Question
Breaking Down the Question
The manager wants: fraction of bulbs that made it through Step 1 that also made it through all three steps
Let's translate this:
- Numerator: bulbs that made it through all three steps
- Denominator: all bulbs that made it through Step 1
Key Insight
Which bulbs made it through Step 1?
- Bulbs counted in \(\mathrm{A}\) (passed Step 1 only)
- Bulbs counted in \(\mathrm{B}\) (passed at least Step 2, which means they must have passed Step 1)
Total bulbs through Step 1 = \(\mathrm{A + B}\)
Phase 3: Finding the Answer
Numerator
The numerator is straightforward: bulbs that made it through all three steps = \(\mathrm{C}\)
Denominator
All bulbs that made it through Step 1 = \(\mathrm{A + B}\)
- \(\mathrm{A}\) represents bulbs that passed Step 1 (but failed Step 2)
- \(\mathrm{B}\) represents bulbs that passed Step 1 AND Step 2 (some may have failed Step 3)
- Together, \(\mathrm{A + B}\) gives us all bulbs that passed Step 1
Phase 4: Solution
The fraction is: \(\frac{\mathrm{C}}{\mathrm{A+B}}\)
Numerator: \(\mathrm{C}\)
Denominator: \(\mathrm{A+B}\)