Question

Given below are two statements:
Statement I: Ram and Shyam can finish a task by working together in 6 days. If Shyam can finish the task by working alone in 8 days, then Ram alone will take 24 days to finish it.
Statement II: If 6 persons working 8 hours a day earn 8,400 per week, then 9 person working 6 hours a day will eam 9,450 per week.
In the light of the above statements, choose the correct answer from the options given below:

• A. Both Statement I and Statement II are true
• B. Both Statement I and Statement II are false
• C. Statement I is true but Statement II is false
• D. Statement I is false but Statement II is true

Answer 1

The Correct Option is A

To determine the truthfulness of the given statements, we need to analyze them step-by-step.

Statement I:

"Ram and Shyam can finish a task by working together in 6 days. If Shyam can finish the task by working alone in 8 days, then Ram alone will take 24 days to finish it." 

• The key to solving this is understanding the concept of work equivalence. If Shyam can complete the work alone in 8 days, we can say Shyam's work rate is \(\frac{1}{8}\) of the work per day.
• When Ram and Shyam work together, they complete the work in 6 days, which implies their combined work rate is \(\frac{1}{6}\) of the work per day.
• Let Ram's work rate be \(\frac{1}{R}\) work per day.

According to the given statements:

• \(\frac{1}{R} + \frac{1}{8} = \frac{1}{6}\)

Solving for \(\frac{1}{R}\):

• \(\frac{1}{R} = \frac{1}{6} - \frac{1}{8} = \frac{8-6}{48} = \frac{1}{24}\)
• This means Ram alone takes 24 days to finish the work, confirming that Statement I is true.

Statement II:

"If 6 persons working 8 hours a day earn 8,400 per week, then 9 persons working 6 hours a day will earn 9,450 per week."

• First, calculate the total hours worked by 6 persons in a week: 
  \(6 \times 8 \times 7 = 336\) hours
• Given earnings are 8,400 for 336 hours, the hourly wage is \(\frac{8400}{336} = 25\) per hour
• Now, calculate the total weekly hours for 9 persons working 6 hours per day: \(9 \times 6 \times 7 = 378\) hours
• Therefore, the earnings for 378 hours at 25 per hour is: \(378 \times 25 = 9,450\)
• This confirms that Statement II is true.

Conclusion: Both Statement I and Statement II are true, confirming that the provided answer option "Both Statement I and Statement II are true" is correct.