Time and Work Question 2 - CAT 2023

Step-by-Step Explanation

Let the daily work outputs of Rahul, Rakshita, and Gurmeet be R, K, and G, respectively, and let the total workload be W.

• Analyze the actual completed work:

They all work together for 6 days, and then Rakshita works alone for 3 days to finish the job:

W = 6(R + K + G) + 3K

⟹ W = 6(R + G) + 9K --- (Equation 1)

• Apply the inequality constraints:

1. Together, they take more than 7 days:

W/(R + K + G) > 7

⟹ W > 7(R + G) + 7K

Substitute Equation 1:

6(R + G) + 9K > 7(R + G) + 7K

⟹ 2K > R + G --- (Inequality 1)

2. Rahul and Gurmeet together take less than 15 days:

W/(R + G) < 15

⟹ W < 15(R + G)

Substitute Equation 1:

6(R + G) + 9K < 15(R + G)

⟹ 9K < 9(R + G)

⟹ K < R + G --- (Inequality 2)

• Combine the inequalities to bracket the total work:

From Inequality 1 and 2, we establish:

K < R + G < 2K

Now substitute this range back into the total work equation:

W = 6(R + G) + 9K

Lower Limit:

If R + G > K,

W > 6(K) + 9K

⟹ W > 15K

Upper Limit:

If R + G < 2K,

W < 6(2K) + 9K

⟹ W < 21K

This means the total work W is bounded by:

15K < W < 21K

Let TK be the number of days Rakshita takes alone:

TK = W/K

Therefore:

15 < TK < 21

Since the timeframe must be strictly less than 21 days, the time taken cannot be 21 days.

Correct Option: D

💡 FundaMakers Tip:

When a problem includes words like "more than" or "less than," translate them into inequalities. Treat compound terms like (R + G) as a single entity. Bracketing this compound term lets you define clear boundaries for the target variable K, which helps you identify the incorrect option quickly.