A and B together can complete one-ninth of a job in a day means that A & B together does one whole work in 9 days.
1/A +1/B = 1/9
The minimum value A can take is 10 days (Anything less than 10days is not possible because in other case, no of days taken by B will become negative, which is not possible practically)
When A takes least time (10Days) then B would take the Max time (90 days) to complete the work alone.

In order to find the Max time taken by A, find out the min time taken by B as we did earlier.
The minimum value B can take is 10 days (Anything less than 10days is not possible because in other case, no of days taken by A will become negative, which is not possible practically)
When B takes least time (10Days) then A would take the Max time (90 days) to complete the work alone.

So the required difference = Max Time taken-Min Time taken
= 90 - 10
=80 Days
Option C

Good conceptual question.
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A pure algebraic way can be this:

\(rate*time=work\)

\((A+B)*1=\frac{1}{9}\) and \(A*t=\)1 and \(B*k=1\) where t and k are the days A, B would take to finish the job and are integers.

From those we get: \(\frac{1}{t}+\frac{1}{k}=\frac{1}{9}\) or \(t+k=\frac{tk}{9}\). Lets study t (the time taken by A alone) \(t(1-\frac{k}{9})=-k\) or \(t=\frac{-9k}{9-k}\). How can we read this? Since k is a number the represents a number of days, must be positive=> -9k will be negative so because t must be positive, the denominator must be negative.
\(+ve=\frac{-ve}{-ve}\).
\(9-k<0\) so \(k>9\), but since k is an integer => THE LEAST number of days B can take to finish the job is 10.

So if this is THE LEAST for B, it must be the MOST for A => \(t=\frac{-9*10}{9-10}=90\).

You can repeat the whole analysis just by switching k and t and you find \(90-10=80\).

Hope I've explained myself well (and hope you enjoy algebra )
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solution:

1/A + 1/B = 1/9
when B max then A min. when B min A max. For minimum positive value(real), put B = 10 and consequently A=90. and vice versa. so difference= 90 - 10 = 80 (Answer)
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Asif vai.....

Given, A & B together can complete 1/9th of the job in one day.

Let the job consist of 90 units. Therefore in one day A & B together can produce 10 units.

A & B together work at a rate of 10 units/day

So the combinations for the no. of units per day produced by A & B can be (1,9), (2,8), (3,7), (4,6), (5,5), (6,4), (7,3), (8,2), (9,1)

So Maximum units produced per day by A = 9 units/ day at its highest efficiency.

Hence to complete 90 units, A will take (90/9) = 10 days, this is the fastest for A at its highest efficiency.

So Minimum Days to finish the job = 10 days

Now, Similarly, minimum units produced per day by A = 1 unit/ day at its lowest efficiency.

Hence to complete 90 units, A will take (90/1) = 90 days, this is the slowest for a its lowest efficiency.

So Maximum Days to finish the job = 90 days

Required is Difference of Max Days & Min Days = 90 - 10 = 80

Answer C
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