Arithmetic

Mixture & Alligation

Easy

A container has $40$ liters of milk. Then, $4$ liters are removed from the container and replaced with $4$ liters of water. This process of replacing $4$ liters of the liquid in the container with an equal volume of water is continued repeatedly. The smallest number of times of doing this process, after which the volume of milk in the container becomes less than that of water, is

Entered answer:

✅ Correct Answer: 7

we start with 40 liters of pure milk. Each time we:

Remove 4 liters of the current mixture

Add 4 liters of water

we want to find when milk volume < water volume in the container.

Since the total volume stays 40 liters, milk < water means:

Milk volume < 20 liters

Milk percentage < 50%

Let's see what happens after each operation:

Initially: 40L milk (100% milk)

After 1st operation:

Remove: 4L pure milk

Add: 4L water

Result: 36L milk, 4L water

After 2nd operation:

Current mixture: 36L milk + 4L water = 40L total

Milk concentration = $\frac{36}{40} = \frac{9}{10} = 90\%$

Remove 4L: This removes $4 \times \frac{9}{10} = 3.6$ L milk and 0.4L water

Add 4L water

Result: $(36 - 3.6) = 32.4$ L milk, $(4 + 0.4 + 4) = 8.4$ L water

Each time we remove 4L from 40L, we're removing $\frac{1}{10}$ th of the mixture. This means $\frac{9}{10}$ th of the milk remains after each operation.

After n operations, the amount of milk remaining is:

$\text{Milk remaining} = 40 \times \left(\frac{9}{10}\right)^n \text{ liters}$

Each operation leaves $\frac{9}{10}$ th of the previous milk amount, so we multiply by $\left(\frac{9}{10}\right)$ each time.

For milk < water:

$40 \times \left(\frac{9}{10}\right)^n < 20$

$\left(\frac{9}{10}\right)^n < \frac{1}{2}$

Let's calculate $\left(\frac{9}{10}\right)^n$ for different values of n:

n = 6: $\left(\frac{9}{10}\right)^6 = 0.531441 > 0.5$

n = 7: $\left(\frac{9}{10}\right)^7 = 0.4782969 < 0.5$

After 7 operations:

Milk remaining = $40 \times \left(\frac{9}{10}\right)^7 = 40 \times 0.4783 = 19.13$ liters

Water = $40 - 19.13 = 20.87$ liters

Since $19.13 < 20.87$ , milk is indeed less than water.

Answer: 7

In mixture problems, when we repeatedly remove and replace, the remaining quantity follows the pattern: Original × (fraction remaining) $^n$

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