Three men A, B and C working together can do a job in 6 hours less time than A alone, in 1 hour less time than B alone and in one half the time needed by C when working along. Then A and B together can do the job in

The correct option is 4/3.

To find the time taken by A and B together to complete the job, we can set up an equation representing their rates of work.

Let the time taken by A, B, and C working together to complete the job be $x$ hours.
Based on the information given in the question:
- Time taken by A alone = $x+6$ hours
- Time taken by B alone = $x+1$ hours
- Time taken by C alone = $2x$ hours

The rate of work of A, B, and C working together is the sum of their individual rates of work:

$\frac{1}{x}=\frac{1}{x+6}+\frac{1}{x+1}+\frac{1}{2x}$

Now, we solve for $x$ by subtracting $\frac{1}{2x}$ from both sides:

$\frac{1}{x}-\frac{1}{2x}=\frac{1}{x+6}+\frac{1}{x+1}$

$\frac{1}{2x}=\frac{(x+1)+(x+6)}{(x+6)(x+1)}$

$\frac{1}{2x}=\frac{2x+7}{x^2+7x+6}$

Cross-multiplying to solve the equation:

$x^2+7x+6=2x(2x+7)$

$x^2+7x+6=4x^2+14x$

$3x^2+7x-6=0$

Factoring the quadratic equation:

$3x^2+9x-2x-6=0$

$3x(x+3)-2(x+3)=0$

$(3x-2)(x+3)=0$

Since time cannot be negative, we have:

$3x-2=0\Rightarrow x=\frac{2}{3}$ hours

Now, we determine the time taken by A and B working together.
- Time taken by A alone = $x+6=\frac{2}{3}+6=\frac{20}{3}$ hours
- Rate of work of A = $\frac{3}{20}$ of the job per hour
- Time taken by B alone = $x+1=\frac{2}{3}+1=\frac{5}{3}$ hours
- Rate of work of B = $\frac{3}{5}$ of the job per hour

Combined rate of A and B working together:

$\text{Combined Rate}=\frac{3}{20}+\frac{3}{5}=\frac{3}{20}+\frac{12}{20}=\frac{15}{20}=\frac{3}{4}$ of the job per hour

Thus, the time taken by A and B together to complete the job is the reciprocal of their combined rate:

$\text{Time}=\frac{4}{3}$ hours