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# Solution

Let Amol take "x" hours to complete the job all on his own. Then, Babul will take "x+5" hours.

Since Amol takes "x" hours, he will complete 1/x part of the job in one hour. Likewise, Babul will complete 1/(x+5) part of the job in one hour.

Both of them together will complete [1/x + 1/(x+5)] part of the job in one hour. [1/x + 1/(x+5)] = (2x+5)/(x^{2}+5x).

Both brothers have painted the fence in 13/2 hours (6 hours and 30 minutes) but each has taken a 30-minute break in between. So, both the brothers have worked together for 5 hours and 30 minutes (11/2 hours).

In one hour, together they complete (2x+5)/(x^{2}+5x) part of the work. So, in 11/2 hours they will
complete: 11(2x+5)/2(x^{2}+5x) part of the work.

Then, Amol works alone for 30 minutes (1/2 hour). In one hour he can complete 1/x part of the job, so in 1/2 hour he completes 1/2x part of the job.

Thereafter, Babul works alone for 1/2 hour. In one hour he completes 1/(x+5) part of the work. So, in 1/2 hour he completes 1/2(x+5) part of the work.

Therefore,

11(2x+5)/2(x^{2}+5x) + 1/2x + 1/2(x+5) = 1 (because all the parts together make up 1 job).

On simplifying, we get 11(2x+5)/2(x^{2}+5x) + (2x+5)/(2x^{2}+10x) = 1

So, 11(2x+5)/2(x^{2}+5x) = 1 - (2x+5)/(2x^{2}+10x)

So, 11(2x+5)/2(x^{2}+5x) = (2x^{2}+10x-2x-5)/(2x^{2}+10x)

So, 11(2x+5)/2(x^{2}+5x) = (2x^{2}+8x-5)/(2x^{2}+10x)

So, 11(2x+5)/2(x^{2}+5x) = (2x^{2}+8x-5)/2(x^{2}+5x)

Now, 'x' being a measure of time has to be positive; the expression 2(x^{2}+5x), therefore, cannot be zero. Since it is not zero, we
can cancel out the denominators.

So, we are left with 11(2x+5) = (2x^{2}+8x-5)

That simplifies to 2x^{2}-14x-60 = 0

Therefore, x^{2}-7x-30 = 0.

Therefore, x^{2}-10x+3x-30 = 0.

Therefore, x(x-10) + 3(x-10) = 0.

Therefore, (x-10)*(x+3) = 0.

Therefore, x = 10 or x = -3.

But, in this case x cannot be negative. So x = 10.

Therefore Amol on his own would complete the work in 10 hours and Babul would take (x+5) hours, that is 15 hours.

Of course, the simpler method would have been to consider how long the brothers would have taken to paint the fence if they had not taken any break.

The brothers completed the job in 6 hours and 30 minutes during which each took a break of 30 minutes. This means the brothers worked together for five-and-half hours and, then, each worked for 30 minutes on his own. Amol worked for 30 minutes, alone. During this period he completed 1/2x of the job. Likewise, Babul worked for 30 minutes on his own during which he completed 1/2(x+5) work. Therefore, in this one hour they completed 1/2[1/x + 1/(x+5)] of the work while working separately. This much work could have been completed in 30 minutes had they worked together because the brothers can complete [1/x + 1/(x+5)] part of the work in one hour while working together. Therefore, if the brothers had worked without taking a break, they could have completed the work in 6 hours instead of six hours and 30 minutes.

Since the brothers could have completed the job in six hours if they had worked together, in one hour they would have completed 1/6 work. But, as we have already seen that in one hour they complete [1/x + 1/(x+5)] work while working together.

So, [1/x + 1/(x+5)] = 1/6.

On simplifying, we get x^{2}-7x-30 = 0. We solve this as before. Did I do it right?

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