Factory A produces x cars per hour. Factory B produces y cars every 2 hours. How many cars do both factories together produce in 8 hours?
Correct Answer: 4(2x + y)
Introduction / Context:Translate textual rates into per hour rates and aggregate outputs across the specified time window. Then sum outputs from both factories.
Given Data / Assumptions:
- A rate = x cars per hour.
- B rate = y cars every 2 hours = y/2 cars per hour.
- Time window = 8 hours.
Concept / Approach:Total cars = (rate of A + rate of B) * time. Convert B to per hour first, then multiply by 8 and simplify algebraically.
Step-by-Step Solution:
A output in 8 h = 8xB output in 8 h = 8 * (y/2) = 4yTotal = 8x + 4y = 4(2x + y)Verification / Alternative check:Unit check: both terms have units cars. Factorization 8x + 4y = 4(2x + y) is algebraically correct.
Why Other Options Are Wrong:8(x + y) incorrectly treats B as y per hour; (8x + y)/2 and 16(x + y) are algebraic or rate conversion errors; 2(4x + y) equals 8x + 2y, which undercounts B by half.
Common Pitfalls:Forgetting to convert B’s rate to per hour; distributing the 8 incorrectly; misreading parentheses in options.
Final Answer:4(2x + y)