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

**Options:**
A. 5
B. 6
C. 7
D. 8

**Correct Answer:** C. 7

**Title:** Milk-Water Replacement Iterations

**Difficulty:** Medium

**Section:** Quantitative Aptitude

**Category:** Arithmetic

**Topic:** Mixture & Alligation

**Tags:** Replacement Process, Iterative Calculation, Mixtures

**Solution Language:**
1. **Concept Used:**
When a proportion $$ p $$ of a mixture is replaced with an equal proportion of another liquid, the concentration of the remaining mixture decreases exponentially. After $$ n $$ iterations, the concentration of milk becomes:
$$
ext{Concentration of milk} = (1 - p)^n
$$
Here, $$ p = \frac{4}{40} = 0.1 $$, and initially, the concentration of milk is $$ 1 $$ (100%).

2. **Condition for Milk Volume to Be Less Than Water Volume:**
For the volume of milk to be less than that of water:
$$
(1 - p)^n 0.5 $$
- For $$ n = 7 $$: $$ (0.9)^7 = 0.4782969

Question

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

**Options:**  
A. 5  
B. 6  
C. 7  
D. 8

**Correct Answer:** C. 7

**Title:** Milk-Water Replacement Iterations

**Difficulty:** Medium

**Section:** Quantitative Aptitude

**Category:** Arithmetic

**Topic:** Mixture & Alligation

**Tags:** Replacement Process, Iterative Calculation, Mixtures

**Solution Language:**  
1. **Concept Used:**  
   When a proportion $$ p $$ of a mixture is replaced with an equal proportion of another liquid, the concentration of the remaining mixture decreases exponentially. After $$ n $$ iterations, the concentration of milk becomes:  
   $$
   	ext{Concentration of milk} = (1 - p)^n
   $$  
   Here, $$ p = \frac{4}{40} = 0.1 $$, and initially, the concentration of milk is $$ 1 $$ (100%).

2. **Condition for Milk Volume to Be Less Than Water Volume:**  
   For the volume of milk to be less than that of water:  
   $$
   (1 - p)^n  0.5 $$  
   - For $$ n = 7 $$: $$ (0.9)^7 = 0.4782969 < 0.5 $$

Thus, the smallest $$ n $$ for which the volume of milk becomes less than that of water is **7 iterations**.

Citations:
[1] https://pplx-res.cloudinary.com/image/upload/v1743502569/user_uploads/RwicRxPNUSAPAFN/Screenshot-2025-04-01-154539.jpg
[2] https://pplx-res.cloudinary.com/image/upload/v1743502569/user_uploads/RHOhswnXVUXnSgk/Screenshot-2025-04-01-154550.jpg

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Answer from Perplexity: pplx.ai/share

Accepted Answer

1. **Concept Used:**  
   When a proportion $$ p $$ of a mixture is replaced with an equal proportion of another liquid, the concentration of the remaining mixture decreases exponentially. After $$ n $$ iterations, the concentration of milk becomes:  
   $$
   	ext{Concentration of milk} = (1 - p)^n
   $$  
   Here, $$ p = \frac{4}{40} = 0.1 $$, and initially, the concentration of milk is $$ 1 $$ (100%).

2. **Condition for Milk Volume to Be Less Than Water Volume:**  
   For the volume of milk to be less than that of water:  
   $$
 
Thus, the smallest $$ n $$ for which the volume of milk becomes less than that of water is **7 iterations**.

Answer

Answer

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