Question

A glass is initially filled entirely with milk. In each step, $\frac{2}{3}$ of the milk is replaced with water. This process is repeated 3 times. What is the final ratio of water to milk in the glass?

• A. $1 : 27$
• B. $1 : 20$
• C. $1 : 26$
• D. $26 : 1$

Answer 1

Answer: (D)

$26 : 1$

Answer 2

Step 1: Define the process
Initially, the glass is filled entirely with milk. The process involves replacing $\frac{2}{3}$ of the milk with water in each step. After each step, the fraction of milk remaining in the glass is reduced by a factor of $\frac{1}{3}$.
Step 2: Milk remaining after each replacement
Let the initial amount of milk be 1 (full glass). After each replacement:
After the 1st replacement: Milk remaining = $\left(\frac{1}{3}\right)^1 = \frac{1}{3}$.
After the 2nd replacement: Milk remaining = $\left(\frac{1}{3}\right)^2 = \frac{1}{9}$.
After the 3rd replacement: Milk remaining = $\left(\frac{1}{3}\right)^3 = \frac{1}{27}$.
Step 3: Water added
After 3 replacements, the total amount of water in the glass is:
$\text{Water} = 1 - \text{Milk remaining} = 1 - \frac{1}{27} = \frac{27}{27} - \frac{1}{27} = \frac{26}{27}.$
**Step 4: Ratio of water to milk **
The ratio of water to milk is:
$\text{Water : Milk} = \frac{\frac{26}{27}}{\frac{1}{27}} = 26 : 1.$