A and B together can complete one-ninth of a job in a day.

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 )

Thanks Zarrolou

Regards
Vinni

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)

1/A + 1/B = 1/9

so A = 9B/(B-9)

A will be max when denominator is minimum (1) so B = 10 and A = 90
For A to be minimum these values will just be reverse so A = 10
Difference 80

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

We a = the number of days A needs to take to complete the job alone and b = the number of days B needs to take to complete the job alone. So A and B’s rates are 1/a and 1/b, respectively and we have:

1/a + 1/b = 1/9

1/a = 1/9 - 1/b

1/a = (b - 9)/(9b)

a = 9b/(b - 9)

Since a and b are positive integers, we see that the smallest positive integer for b is 10. In that case, a = 9(10)/(10 - 9) = 90 and this must be the maximum value of a since b is minimum.

Notice that in the above we’ve solved a in terms of b. If we solve b in terms of a, we should have:
b = 9a/(a - 9)

In this case we see that the minimum value of a is 10. Therefore, the difference between the maximum and minimum number of days A takes to complete the job alone is 90 - 10 = 80.

Answer: C

I did in a different way.
R(A+B) = 1/(9 )=10/90=(1+9)/90=1/90+9/90=1/90+1/10
so, their rates can be divided as 1/90 & 1/10
i.e. Time TA & TB can be divided as 90 & 10
Thus, Time(Max – MIN) = 90-10 = 80 days.
Bunuel, is this approach OK?

A very good question indeed!

A pure algebraic solution shall be as follows:

Let a be the number of days taken by A to do the job and b be the number of days taken by B to complete the job.

From the information given, we can deduce that

1/a + 1/b = 1/9

Solving further will lead to the following equation:

9a + 9b = ab

Since we are told that the number of days shall always be whole numbers, there is merit in solving the equation further. Even though we have the classic problem of having 2 variables and one equation, there is still enough information to be derived from this.

Let us rewrite the equation as follows

9a + 9b -ab =0
a(9-b) + 9b = 0
a(9-b) + 9b - 81 + 81 = 0
a(9-b) -9(9-b) +81 = 0
=> (a-9)x(b-9) = 81

The problem is now a simple number properties problem where the two terms on the left (in order to be whole) have to be factors of 81.

Case 1: Minimize a by choosing a - 9 as 1 and b - 9 as 81, leading to a value of 10 for a
Case 2: Maximize a by choosing a-9 as 81 and b-9 as 1, leading to a value of 90 for a

We can see that the difference between the maximum and minimum value of a shall come out to be 90 - 10 = 80­