Reading a Production Table With a Missing Cell
A factory table records units produced and defect rates by plant for one week. Plant A: 4,000 units, 2.5 percent defective. Plant B: 6,000 units, 1.5 percent defective. Plant C: units unknown, 4 percent defective. The combined defect rate across all three plants is stated as 2.5 percent. A manager wants to know Plant C's output, which plant has the worst defect performance in absolute defective units, and whether closing Plant C would lower the overall defect rate. All figures are for the same week and units are comparable across plants.
1. Solve for Plant C's unit count.
Read the worked answer
Let C = units at Plant C. Defectives: A = 4,000 × 0.025 = 100; B = 6,000 × 0.015 = 90; C = 0.04C.
Combined rate equation: (100 + 90 + 0.04C) / (4,000 + 6,000 + C) = 0.025.
190 + 0.04C = 0.025(10,000 + C) = 250 + 0.025C.
0.015C = 60, so C = 4,000 units.
Check: total 14,000, defectives 100+90+160 = 350, and 350/14,000 = 2.5 percent. Correct.
2. Which plant produces the most defective units, and does that match the highest defect rate?
Read the worked answer
Defectives: A = 100, B = 90, C = 160.
Plant C produces the most defective units (160).
It also happens to hold the highest defect rate (4 percent), so here the two rankings agree. But note the trap: Plant A has a higher rate than B (2.5 vs 1.5 percent) yet only slightly more defectives (100 vs 90) because B's volume is larger. Rate and absolute count are different questions and must be reported separately.
3. If Plant C were closed, what would the remaining combined defect rate be, and is closing it a sound way to lower the rate?
Read the worked answer
Remaining: A and B only. Defectives = 100 + 90 = 190; units = 10,000.
Rate = 190 / 10,000 = 1.9 percent, down from 2.5 percent.
Closing C does lower the blended rate, because C's 4 percent exceeds the 2.5 percent average. However, the rate falls only because the worst plant is removed from the denominator, not because anything improved. Judgement: closing a plant to flatter a metric is a reporting trick, not a quality fix; the sound action is to fix C's 4 percent rate while keeping its 4,000 units of output.