If five men and two boys working together can do four times as much work in one hour as one man and one boy working together, what is the ratio of the work done by a man to the work done by a boy in a given time?

Introduction / Context:
This is a basic algebraic application of work rate comparisons between men and boys. We are given that a larger group of men and boys performs four times as much work per hour as a smaller group. From this information, we must determine how the individual productivity of a man compares to that of a boy.

Given Data / Assumptions:
• Five men and two boys do four times as much work per hour as one man and one boy.
• Work rate of each man is assumed identical.
• Work rate of each boy is assumed identical.
• We must find the ratio of work done by a man to work done by a boy in the same time.

Concept / Approach:
We model the work rates using variables: one for a man and one for a boy. The statement about four times as much work becomes an equation relating the combined rates of the two groups. Solving this equation gives a relationship between the man and boy rates, which directly yields the desired ratio.

Step-by-Step Solution:
Let m = work done by one man in one hour. Let b = work done by one boy in one hour. Work done by five men and two boys in one hour = 5m + 2b. Work done by one man and one boy in one hour = m + b. Given that: 5m + 2b = 4(m + b). Expand the right side: 5m + 2b = 4m + 4b. Rearrange: 5m − 4m = 4b − 2b. This gives m = 2b. Thus, the ratio of man work to boy work in the same time is m : b = 2b : b = 2 : 1.

Verification / Alternative check:
If we assume b = 1 unit per hour, then m = 2 units per hour. One man and one boy together produce 2 + 1 = 3 units per hour. Five men and two boys produce 5 × 2 + 2 × 1 = 10 + 2 = 12 units per hour. This is exactly four times 3 units per hour, confirming the calculation.

Why Other Options Are Wrong:
Ratios such as 1 : 2 or 1 : 3 imply that a boy is more productive than a man, which contradicts m = 2b. A ratio of 3 : 1 implies m = 3b, which does not satisfy the equation 5m + 2b = 4(m + b). Only 2 : 1 is consistent with the stated condition.

Common Pitfalls:
Some learners might incorrectly divide the number of men by the number of boys or directly take 5 : 2 and compare it with 1 : 1, rather than forming the correct algebraic equation. It is important to remember that the relationship involves work rates, not headcounts alone.

Final Answer:
The ratio of work done by a man to that by a boy is 2 : 1.