The correct option is (d) MAN = 54 days, Woman = 96 days

The explanation is: Let 1 man TAKE x days to finish the job and 1 woman take y days to finish the same job.

Then, 1 man’s 1 day work will be \(\frac {1}{x}\) days

1 woman’s 1 day work will be \(\frac {1}{y}\) days

10 men and 6 women can finish the job in 6 days

(10 men’s 1 day work + 6 women’s 1 day work = \(\frac {1}{4}\))

\(\frac {10}{x}+\frac {6}{y}=\frac {1}{4}\)

Let, \(\frac {1}{x}\) = u, \(\frac {1}{y}\) = v

10u+6v=\(\frac {1}{4}\)(1)

5 men and 7 women can finish the job in 6 days.

(5 men’s 1 day work + 7 women’s 1 day work = \(\frac {1}{6}\))

\(\frac {5}{x}+\frac {7}{y}=\frac {1}{6}\)

Let, \(\frac {1}{x}\) = u, \(\frac {1}{y}\) = v

5u + 7v = \(\frac {1}{6}\)(2)

Multiplying equation by 2 and then subtracting both the equations we get,

10u + 14v = \(\frac {1}{6}\)

-10u + 6v = \(\frac {1}{4}\)

8v = \(\frac {1}{3}-\frac {1}{4}\)

8v = \(\frac {1}{12}\)

v = \(\frac {1}{96}\)

v = \(\frac {1}{y}=\frac {1}{96}\)

y = 96

Substituting the value of v in equation (1) we get,

10u + 6\((\frac {1}{96})=\frac {1}{4}\)

10u + \(\frac {1}{16}=\frac {1}{4}\)

10u = \(\frac {3}{16}\)

u = \(\frac {3}{160}\)

u = \(\frac {1}{x}=\frac {3}{160}\)

x = \(\frac {160}{3}\) ≈ 54

Hence, a man alone can finish the job in 54 days and a woman can finish the job in 96 days.