Arithmetic

Mixture & Alligation

Easy

A glass is filled with milk. Two-thirds of its content is poured out and replaced with water. If this process of pouring out two-thirds the content and replacing with water is repeated three more times, then the final ratio of milk to water in the glass, is

1: 26

1:80

1:27

1:81

✅ Correct Option: 2

For any dilution problem where you remove fraction p and repeat n times:

Remaining original substance = $(1-p)^n$

Here: $p = \dfrac{2}{3}$ , $n = 4$ , so remaining milk = $\left(1-\frac{2}{3}\right)^4 = \left(\frac{1}{3}\right)^4 = \dfrac{1}{81}$

Answer - $1 : 80$

Longer Explaination:

Initially: Glass is 100% milk

Each time: Remove 2/3 of current content → Add water to fill the glass

The key insight: When we remove 2/3 of the mixture, we keep 1/3 of whatever was there before.

Let us trace through each step:

After 1st operation:

Remove 2/3 of pure milk

Keep 1/3 of original milk = $\dfrac{1}{3}$ milk

Add water to fill = $\dfrac{2}{3}$ water

After 2nd operation:

Remove 2/3 of current mixture (which has $\dfrac{1}{3}$ milk and $\dfrac{2}{3}$ water)

Keep 1/3 of current mixture

Milk remaining = $\dfrac{1}{3} \times \dfrac{1}{3} = \dfrac{1}{9}$ of original milk

After 3rd operation:

Milk remaining = $\dfrac{1}{3} \times \dfrac{1}{9} = \dfrac{1}{27}$ of original milk

After 4th operation:

Milk remaining = $\dfrac{1}{3} \times \dfrac{1}{27} = \dfrac{1}{81}$ of original milk

After each operation, the milk fraction gets multiplied by $\dfrac{1}{3}$ .

After 4 operations: Milk fraction = $\left(\frac{1}{3}\right)^4 = \dfrac{1}{81}$

Why this works: Each time we remove 2/3 and keep 1/3, we're essentially multiplying the milk concentration by $\dfrac{1}{3}$ .

Since the glass is always full:

Milk fraction = $\dfrac{1}{81}$

Water fraction = $1 - \dfrac{1}{81} = \dfrac{80}{81}$

Final ratio = Milk : Water = $\dfrac{1}{81} : \dfrac{80}{81}$

Simplifying by multiplying both parts by 81:

Milk : Water = 1 : 80

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A glass is filled with milk. Two-thirds of its content is poured out and replaced with water. If this process of pouring out two-thirds the content and replacing with water is repeated three more times, then the final ratio of milk to water in the glass, is

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