Concept of “Time and Work” [समय और काम]-04

Question: 38 men working 6 hrs a day can do a work in 12 days. Find the no. of days in which 57 men working 8 hrs a day can do twice the work. Assume that two men of the first group do as much work in 1 hr as 3 men of the second group do in 1.5 hr. [38 व्यक्ति 6 घंटे प्रति दिन काम करते हुए, किसी काम को 12 दिनों में कर सकते हैं. बताइए कि 57 व्यक्ति 8 घंटे प्रति दिन काम करते हुए, पहले से दोगुने काम को कितने दिनों में करेंगे. माना कि प्रथम समूह का दो व्यक्ति, एक घंटे में जितना काम कर सकता है, उतना काम करने के लिए दूसरे समूह के तीन व्यक्तियों को 1.5 घंटे लगते हैं.] [Ans. (b) 27 days]                                                          

(a) 20 days    (b) 27 days   (c) 14 days    (d) 17 days   (e) None of these

Solution:

Detailed Explanation: Out of any two of “Efficiency”, “Men” (working strength), “Days” (time period) and “Hours” (rate with respect to time) are inversely proportional to each other, so they have same suffices; they are directly proportional to “Work”, so they have opposite suffices with respect to “Work”.

Efficiency: Given that two men of the first group do as much work in 1 hr as 3 men of the second group do in 1.5 hr. One man of the first group can do a work in 2×1 days = 2 days. On the other hand, one man of the second group can do the same work in 3×1.5 days = 4.5 days. Efficiency is the inverse of no. of days taken. So, E₁:E₂=3×1.5:2×1=4.5:2

Again, the no. of men of the first group required to do a certain work in one day is 2×1 men = 2 men. On the other hand, the no. of men of the second group required to do the same piece of work in one day is 3×1.5 men = 4.5 men. Efficiency can also be defined as the inverse of the no. of men required to do in a certain time (like one day, as here). So, E₁:E₂=3×1.5:2×1=4.5:2

           

First Method: Using the formula:

E₁×M₁×D₁×H₁×W₂ = E₂×M₂×D₂×H₂×W₁

Or, 4.5×38 ×12 ×6×2 = 2×57×D₂×8×1

Or, D₂= 27 days Ans. [E₁:E₂=3×1.5:2×1=4.5:2]

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Second Method: In this method, we always interchange the value of work and then cancel them. One thing must be remembered that the row containing question mark (?) should always in the denominator.\

The question is:

E   M    D    H   W

4.5 38   12    6     1

2    57    ?     8     2       

For solution, we write it as:

E   M    D    H   W

4.5× 38 ×12×6 × 2

2   × 57 ×? × 8 × 1        

= 27 days Ans.                                                               

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Third Method: This is fractional value method. In this method, we take two values, one by one, at a time out of which one must be that value, which is to be determined. In this question, “Days” (time) is to be determined, so we write question mark (?) for time = 12 days  and take one by one, other values.

The ratio of men is either 38/57, or 57/38. One of them is less than fraction (less than one) and the other is more than fraction (more than one). For more men (for 57 men instead of 38 men), the time taken will be less, so we multiply by 38/57.

For more rate (for 8 hrs/day instead of 6 hrs/day), no. of days will be less, so we multiply by 6/8.

For more work (2 times work than that of before), there will be required more time, so we multiply by 2/1.

And again, for more efficient (E₁:E₂=3×1.5:2×1=4.5:2), there will be required less time, so we multiply by 4.5/2.

Thus, no. days required is 12 days × (38/57) × (6/8) × (2/1) × (4.5/2) = 27 days, Ans.                                                                                                                          

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