Pipe K can fill a tank in 30 minutes. Pipe L can fill the same tank five times as fast as pipe K. If both pipes K and L are opened together when the tank is empty, after how many minutes will the tank overflow?

Introduction / Context:
This is a straightforward pipes and cistern question where one pipe is several times faster than another. From the individual times, we find their rates, sum them to get the combined rate, and then compute the time taken to fill the tank starting from empty. The overflow will occur at the same time as the tank first becomes full.

Given Data / Assumptions:
- Pipe K fills the tank in 30 minutes
- Pipe L fills the tank five times as fast as K
- Both pipes are opened simultaneously when the tank is empty
- No leakage or outlet is present

Concept / Approach:
If K fills the tank in 30 minutes, its rate is 1/30 tank per minute. If L is five times as fast, its rate is 5/30 tank per minute and its individual filling time is 30 / 5 = 6 minutes. The combined rate is the sum of the individual rates, and then time to fill is simply the reciprocal of the combined rate.

Step-by-Step Solution:
Step 1: Rate of K = 1/30 tank per minute. Step 2: L is five times as fast, so rate of L = 5 * 1/30 = 5/30 = 1/6 tank per minute. Step 3: Combined rate of K and L working together = 1/30 + 1/6. Step 4: Convert 1/6 to a denominator of 30: 1/6 = 5/30, so combined rate = 1/30 + 5/30 = 6/30 = 1/5 tank per minute. Step 5: Time taken to fill the tank from empty = 1 divided by 1/5 = 5 minutes.

Verification / Alternative check:
In 5 minutes, K alone would have filled 5 * 1/30 = 1/6 of the tank, and L alone would have filled 5 * 1/6 = 5/6 of the tank. Together that is 1/6 + 5/6 = 1 full tank, which matches the requirement that the tank is full in 5 minutes.

Why Other Options Are Wrong:
3, 2 and 4 minutes: These times correspond to higher combined rates that do not match the given individual times. For example, in 4 minutes the total filled fraction would be 4 * (1/5) = 4/5, which is not a full tank.

Common Pitfalls:
Students sometimes misread “five times as fast” as adding 5 minutes or dividing the time wrongly. Remember that being multiple times as fast refers to rate, not to time directly. The correct way is to multiply or divide the time inversely and then derive the rate accurately.

Final Answer:
The tank will overflow after 5 minutes when both pipes K and L are open together.