Inferring efficiency ratio from a scaled comparison The work done by 8 men and 4 boys in 1 day is 7 times the work done by 1 man and 1 boy in 1 day. What is the ratio of the daily work of 1 man to that of 1 boy?
Correct Answer: 3 : 1
Introduction / Context:Scaled comparisons let us deduce relative efficiencies. Here, a large crew’s one-day output equals seven times a small crew’s one-day output, which yields a linear equation in the man and boy daily rates.
Given Data / Assumptions:
- (8 men + 4 boys) in 1 day = 7 × (1 man + 1 boy) in 1 day.
- Let daily rate(man) = M and daily rate(boy) = B.
Concept / Approach:Translate the sentence into 8M + 4B = 7(M + B). Solve for M in terms of B to get the ratio M:B.
Step-by-Step Solution:
8M + 4B = 7M + 7B8M − 7M = 7B − 4B ⇒ M = 3BTherefore, M : B = 3 : 1Verification / Alternative check:Substitute M = 3B: LHS = 8(3B) + 4B = 24B + 4B = 28B; RHS = 7(3B + B) = 7(4B) = 28B. Balanced.
Why Other Options Are Wrong:2 : 1, 1 : 1, 1 : 2 do not satisfy the equality after substitution.
Common Pitfalls:Dropping coefficients or misplacing terms when moving 7(M + B) across the equality.
Final Answer:3 : 1