Introduction / Context:
This is a typical chain rule and work problem. It tests the idea that the amount of work done is directly proportional to the number of workers, the time for which they work and, in this problem, the length of wall constructed. Such questions are common in quantitative aptitude, and a clear understanding of proportional reasoning makes them straightforward to solve.
Given Data / Assumptions:
- Twenty men build a 56 metre wall in 6 days.
- The work rate of each man is assumed to be constant throughout.
- The wall built by 35 men in 3 days is of the same type, so work is directly proportional to the length.
- We must find the length of wall built by 35 men in 3 days.
Concept / Approach:Work done is proportional to the product of men, days and length. We can write:
men * days * rate is proportional to length.Another neat method is to write that the length is directly proportional to the product men * days when other conditions are identical. Then we can set up a proportion between the known and unknown situations and solve for the missing length.
Step-by-Step Solution:Step 1: Compute the effective work input in the first case, which is 20 men working for 6 days.Step 2: In the second case, we have 35 men working for 3 days.Step 3: Since length is directly proportional to men * days, we write: 56 / L = (20 * 6) / (35 * 3).Step 4: Simplify the right hand side: (20 * 6) = 120 and (35 * 3) = 105, so the ratio is 120 / 105 = 8 / 7.Step 5: Thus 56 / L = 8 / 7, which implies L = 56 * 7 / 8 = 7 * 7 = 49 metres.Verification / Alternative check:We can check reasonableness. The product of men and days initially is 20 * 6 = 120. In the new case it is 35 * 3 = 105, which is slightly less. So the new length should be slightly less than 56 metres. However, notice that the proportionality relation was inverted in our ratio, so we ensure the algebra is correct. Directly, we can write L2 = L1 * (men2 * days2) / (men1 * days1) = 56 * (35 * 3) / (20 * 6) = 56 * 105 / 120 = 49. This gives the same result and confirms the calculation.
Why Other Options Are Wrong:46 metres would correspond to a different men days ratio and does not satisfy the proportional relation between work inputs.47 metres also fails when substituted back into the men days proportion and is simply a distractor near the correct value.48 metres is closer but still does not satisfy the exact ratio that comes from the product of men and days.Common Pitfalls:A frequent mistake is to invert the proportion incorrectly or to confuse directly proportional with inversely proportional relationships. Another common error is to mix up which product of men and days belongs in the numerator and which in the denominator. Writing the relation clearly as new length = old length * (new men * new days) / (old men * old days) avoids confusion and leads quickly to the correct answer.
Final Answer:The length of the wall built by 35 men in 3 days is 49 metres.
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