Seven machines can complete a job in h hours. How long, in terms of h,

For abstract or complicated "# of workers" questions, I use a table, where "# of workers" is the first column before rate, time, work.

It's fast. For me, after having done it a couple of times, it helped me understand the concepts because I wasn't fixated on keeping values straight.

You just add one more variable to the "LHS" of the WR=T equation ... add # of workers*R*T=W

Then rearrange the equation. This question, per this method, asks first that you find rate, then, with different number of workers and different amount of work, that you find time. See table.

Case 1: Numerator is 1. Denominator is 7*h. Thus we get RATE 1/7h

Case 2: Numerator is 2. Denominator is 5*(1/7)(h)

Case 2 numerator and denominator ====> 2 ÷ (5/7)h

= 2 x 7h/5

= 14h/5

Generalise understanding

Our objective is to generalise our understanding as much as we can, so we deal with one concept that can be applied to multiple situations with a slight change in terminology. This problem is of the type x people/machines work for y hours to do certain work. That x could be typists, painters, tree planters, machines, labourers, etc.

Approach


When 6 people/machines work for 4 hours, visualise it in your mind. 6 people sewing clothes for 4 hours. 6 painters painting a house for 4 hours. 6 machines running in a factory and producing biscuits. Now what does that mean? Simplify further. 6 people sewing clothes for 4 hours means each of them has worked for 4 hours, since they all were sitting in the factory and working for the entire 4 hours. So the total number of hours of work done is 6 x 4 = 24 hours of work done. Pause for a while. Understand what this means. This means that we got 24 hours of work done. So now we can express work done not in terms of number of shirts produced but in terms of no of hours. So if we had to get 48 hours worth of work done, those 6 people would have to sit for twice the amount of time - 8 hours, which is 48/6. Generalise this in your mind. I first need to know the total amount of work that needs to be done in terms of hours, and then divide that by the number of people who are working.

In the present case, 7 machines complete a job in h hours, so the total work done is 7h worth of work. 2 such jobs would mean 14h worth of work to be done. But we now have only 5 machines, so the time taken would be 14h/5. It will take more than twice the amount of time taken earlier because one, the work has doubled and two, the number of machines have reduced, so each machine would now have to work longer. Work it out the other way. Total work done was 7h and we had 7 machines, so we took 7h/7 = h hours to finish the work. If we had merely doubled this work to 14h, we would then take 14h/7 = 2h hours - simply double the time. However, now we also have fewer machines - only 5, so 14h/5 = 2.8h hours.

Conclusion

When x people/machines work together for y hours, each of them works for y hours each, so the total work done is worth xy hours. If the no. of people/machines decrease, the same work would need more time to complete, since everyone is working with the same efficiency or rate.

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