Arithmetic

Time & Work

Medium

Two pipes $A$ and $B$ are attached to an empty water tank. Pipe $A$ fills the tank while pipe $B$ drains it. If pipe $A$ is opened at $2$ pm and pipe $B$ is opened at $3$ pm , then the tank becomes full at $10 \mathrm{pm}$ . Instead, if pipe $A$ is opened at $2 \mathrm{pm}$ and pipe $B$ is opened at $4 \mathrm{pm}$ , then the tank becomes full at $6 \mathrm{pm}$ . If pipe $B$ is not opened at all, then the time, in minutes, taken to fill the tank is

$264$

$144$

$140$

$120$

✅ Correct Option: 2

We have two pipes working on a water tank:

Pipe A: Fills the tank (positive work)

Pipe B: Drains the tank (negative work)

The key insight is that when both pipes work together, the net filling rate = Rate of A - Rate of B

Case 1: A opens at 2pm, B opens at 3pm, tank full at 10pm

Pipe A works: 2pm to 10pm = 8 hours

Pipe B works: 3pm to 10pm = 7 hours

Result: Tank becomes full

Case 2: A opens at 2pm, B opens at 4pm, tank full at 6pm

Pipe A works: 2pm to 6pm = 4 hours

Pipe B works: 4pm to 6pm = 2 hours

Result: Tank becomes full

Instead of setting up complex equations, let's compare what changed between the two cases:

What changed from Case 1 to Case 2?

Pipe B worked 5 hours less (from 7 hours to 2 hours)

Pipe A worked 4 hours less (from 8 hours to 4 hours)

Key Insight: When pipe B works 5 hours less, pipe A needs to work 4 hours less to achieve the same result (full tank).

This tells us the relationship between their rates:

5 hours less of B's draining = 4 hours less of A's filling needed

Ratio discovered: 1 hour less of B's work = $\dfrac{4}{5}$ hours less of A's work

Now, what if B doesn't work at all?

In Case 1, B worked for 7 hours

If B doesn't work at all, that's 7 hours less of B's work

Time saved for A = $\dfrac{4}{5} \times 7 = 5.6$ hours

Therefore: A alone would take $8 - 5.6 = 2.4$ hours

Converting to minutes: $2.4 \times 60 = 144$ minutes

This approach works because we're using the principle of equivalent work. When we reduce one pipe's work time, we can calculate how much the other pipe's work time changes to maintain the same net result.

This is much faster than solving simultaneous equations, and it directly gives us the relationship between the two pipes' efficiencies.

Answer: 144 minutes

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