How long will it take for the water levels in the two tanks to become equal?

Here's how to solve this problem:

Step 1: Calculate the initial water level in each tank. The total height of Tank M is $50 \text{ cm}$ and it is $\frac{3}{5}$ full. The total height of Tank N is $40 \text{ cm}$ and it is $\frac{1}{2}$ full. $Initial water level in Tank M = \frac{3}{5} \times 50 cm = 30 cm$ $Initial water level in Tank N = \frac{1}{2} \times 40 cm = 20 cm$

Step 2: Calculate the base area of each tank. The base of Tank M is $60 cm \times 20 \text{ cm}$. The base of Tank N is $40 cm \times 20 \text{ cm}$. $Base area of Tank M = 60 cm \times 20 cm = 1200 cm^2$ $Base area of Tank N = 40 cm \times 20 cm = 800 cm^2$

Step 3: Convert the draining rates from litres per minute to cubic centimetres per minute. Remember that $1 L = 1000 cm^3$. $Draining rate of Tank M = 2.4 L/min \times 1000 cm^3/L = 2400 cm^3/min$ $Draining rate of Tank N = 0.8 L/min \times 1000 cm^3/L = 800 cm^3/min$

Step 4: Calculate how much the water level drops per minute for each tank. To find the drop in height, divide the volume draining rate by the base area. $Drop in water level for Tank M = \frac{2400 cm^3/min}{1200 cm^2} = 2 cm/min$ $Drop in water level for Tank N = \frac{800 cm^3/min}{800 cm^2} = 1 cm/min$

Step 5: Determine the initial difference in water levels. $Initial difference = Initial level of Tank M - Initial level of Tank N$ $Initial difference = 30 cm - 20 cm = 10 cm$

Step 6: Determine how quickly the difference in water levels changes. Tank M's level drops by $2 \text{ cm/min}$. Tank N's level drops by $1 \text{ cm/min}$. Since Tank M's level is dropping faster, the difference between the two levels is closing. $Rate at which difference closes = Drop rate of Tank M - Drop rate of Tank N$ $Rate at which difference closes = 2 cm/min - 1 cm/min = 1 cm/min$

Step 7: Calculate the time it takes for the water levels to become equal. Divide the initial difference in levels by the rate at which the difference closes. $Time = \frac{Initial difference}{Rate at which difference closes} = \frac{10 cm}{1 cm/min} = 10 minutes$

The water levels will become equal after 10 minutes.

The final answer is $\boxed{10 minutes}$.