Time and Work Question 1 - CAT 2023
Pipes A and C are fill pipes while Pipe B is a drain pipe of a tank. Pipe B empties the full tank in one hour less than the time taken by Pipe A to fill the empty tank. When pipes A, B and C are turned on together, the empty tank is filled in two hours. If pipes B and C are turned on together when the tank is empty and Pipe B is turned off after one hour, then Pipe C takes another one hour and 15 minutes to fill the remaining tank. If Pipe A can fill the empty tank in less than five hours, then the time taken, in minutes, by Pipe C to fill the empty tank is:
Step-by-Step Explanation
Let the time taken by Pipe A to fill the tank be tA hours.
Time taken by Pipe B to empty the tank = tA − 1 hours.
Let the time taken by Pipe C to fill the tank be tC hours.
- Formulate Equation 1 (All three working together):
Pipes A, B, and C combined can fill the tank in 2 hours. Since B drains the tank, its work rate is negative:
1/tA − 1/(tA − 1) + 1/tC = 1/2 --- (Equation 1)
- Formulate Equation 2 (B and C working sequentially):
Pipe B runs for 1 hour, and Pipe C runs for 1 hour + 1 hour 15 minutes = 2.25 = 9/4 hours to fill the tank:
−1/(tA − 1) + 9/(4tC) = 1
⟹ 1/tC = 4/9(1 + 1/(tA − 1))
= 4tA/[9(tA − 1)]
- Substitute and Solve for tA:
Substitute the expression for 1/tC into Equation 1:
1/tA − 1/(tA − 1) + 4tA/[9(tA − 1)] = 1/2
After simplification:
8tA² − 18 = 9tA² − 9tA
⟹ tA² − 9tA + 18 = 0
⟹ (tA − 3)(tA − 6) = 0
We are given the constraint that Pipe A takes less than 5 hours (tA < 5), so we choose:
tA = 3 hours
- Calculate C's Time:
Substitute tA = 3 back into the expression for 1/tC:
1/tC = 4(3)/[9(3 − 1)]
= 12/18
= 2/3
⟹ tC = 1.5 hours
Converting to minutes:
1.5 × 60 = 90 minutes
Correct Option: A
💡 FundaMakers Tip:
When you have a system of fractional equations with multiple variables, try to express one variable completely in terms of another (like writing 1/tC in terms of tA). This simplifies the expression and lets you solve the problem using a standard quadratic equation. Always check the given boundaries (tA < 5) to choose the correct root immediately.