An overall defect rate is a weighted average: each machine’s rate counts in proportion to its share of the pins. If only one machine improves, connect the change in the overall rate to that machine’s share, then solve for its rate drop with a Desmos regression. The tempting trap is treating the overall drop as the same-sized drop for Machine A.
Hints
- Hint 1
In a weighted average, a group’s change affects the overall rate in proportion to that group’s share. Since only Machine A changes, what fraction of the pins can contribute to the overall decrease?
- Hint 2
A percentage point is the direct difference between two percent rates, not a percent of the original rate. Express the overall decrease as Machine A’s share times the decrease in A’s own defect rate.
Step-by-step
Track the change from Machine A
Step 1Find how A’s change affects the overall rate
Let be the decrease in Machine A’s defect rate, measured in percentage points, or the direct difference between its old and new rates. A makes of the pins, so a drop of points in A’s rate drops the overall rate by points. Only the changing machine’s share scales the overall change. The existing -point gap between the machines doesn’t affect that change because B’s rate stays the same.
- Step 2
Write the new overall rate
The old overall rate minus the drop caused by A equals the new overall rate. Using percentage-point units throughout gives . Don’t subtract directly from : A produces only part of the pins.
- Step 3
Solve for A’s decrease
Type in Desmos. The tells Desmos to find the value of that fits the equation; under PARAMETERS, it shows . So Machine A’s defect rate must decrease by percentage points. The choices write this amount with a percent sign, but the question asks for the difference between two percent rates. Choice B.