Question

Four men and three women can do a job in 6 days. When 5 men and 6 women work on the same job, the work gets completed in 4 days. How long will 2 women and 3 men take to do the job?

• A. 18
• B. 10
• C. 8.3
• D. 12

Answer 1

The Correct Option is C

To solve the problem, let's first define the rates at which men and women work. Let the work done by one man in one day be m and the work done by one woman in one day be w.

According to the problem:

1. Four men and three women complete the job in 6 days. We can express this as:
   \(6(4m + 3w) = 1\).
2. Five men and six women complete the job in 4 days:
   \(4(5m + 6w) = 1\).

We now have two equations:

• \(24m + 18w = 1\)
• \(20m + 24w = 1\)

Let's solve these equations to find m and w.

Rearranging the first equation:

• \(24m + 18w = 1\)
• \(4m + 3w = \frac{1}{6}\)

Rearranging the second equation:

• \(20m + 24w = 1\)
• \(5m + 6w = \frac{1}{4}\)

Now multiply the first equation by 2 and the second by 1, to eliminate a variable:

• \(8m + 6w = \frac{2}{6} = \frac{1}{3}\)
• \(5m + 6w = \frac{1}{4}\)

Subtract the second equation from the first:

• \((8m + 6w) - (5m + 6w) = \frac{1}{3} - \frac{1}{4}\)
• \(3m = \frac{1}{3} - \frac{1}{4}\)
• \(3m = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}\)
• \(m = \frac{1}{36}\)

Substitute \(m = \frac{1}{36}\) back into \(4m + 3w = \frac{1}{6}\):

• \(4\left(\frac{1}{36}\right) + 3w = \frac{1}{6}\)
• \(\frac{4}{36} + 3w = \frac{1}{6}\)
• \(\frac{1}{9} + 3w = \frac{1}{6}\)
• \(3w = \frac{1}{6} - \frac{1}{9}\)
• \(3w = \frac{3}{18} - \frac{2}{18} = \frac{1}{18}\)
• \(w = \frac{1}{54}\)

With the rates, \(m = \frac{1}{36}\) and \(w = \frac{1}{54}\), calculate the time for 2 women and 3 men to complete the job:

• Work done by 2 women and 3 men in one day is:
  \(3\left(\frac{1}{36}\right) + 2\left(\frac{1}{54}\right) = \frac{1}{12} + \frac{1}{27}\)
• Finding a common denominator (108):
  \(\frac{9}{108} + \frac{4}{108} = \frac{13}{108}\)
• Time taken to finish the job is then the inverse:
  \(\frac{108}{13} \approx 8.3\) days

Time (in days)

≈ 8.3

Therefore, 2 women and 3 men will take approximately 8.3 days to complete the job.