Question

Sony has a certain quantity of mixture in her bottle containing water and milk in the ratio 3 : 5, respectively. She takes out some quantity of that mixture and replaces it with an equal quantity of water. If the ratio of water to the milk in the new mixture is x : y, then select the INCORRECT statement from the following options.

• A. x > y if \(\frac{1}{3}\) part of the mixture is replaced with water.
• B. x = y if \(\frac{1}{4}\) part of the mixture is replaced with water.
• C. x < y if \(\frac{1}{5}\) part of the mixture is replaced with water.
• D. x> y if \(\frac{1}{6}\) part of the mixture is replaced with water.

Hint:

In mixture replacement problems, establish a general formula for the new ratio based on the fraction 'z' being replaced. Then, analyze the conditions (>, <, =) to quickly evaluate the options.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given an initial mixture of water and milk in the ratio 3:5. A certain fraction of the mixture is removed and replaced with an equal amount of pure water. We need to find the relationship between the new water and milk quantities (x and y) based on the fraction of mixture replaced and identify the incorrect statement among the options. 
Step 2: Key Formula or Approach: 
Let the total volume of the mixture be V. 
Initial quantity of Water (W) = \(\frac{3}{8}V\). 
Initial quantity of Milk (M) = \(\frac{5}{8}V\). 
Let 'z' be the fraction of the mixture that is removed and replaced with water. 
Quantity of water removed = \(z \times \frac{3}{8}V\). 
Quantity of milk removed = \(z \times \frac{5}{8}V\). 
Quantity of water added = \(zV\). 
New quantity of Water = \(\frac{3}{8}V - \frac{3z}{8}V + zV = V(\frac{3 - 3z + 8z}{8}) = V(\frac{3+5z}{8})\). 
New quantity of Milk = \(\frac{5}{8}V - \frac{5z}{8}V = V(\frac{5-5z}{8})\). 
The new ratio of water to milk is x : y. 
\[ \frac{x}{y} = \frac{V(\frac{3+5z}{8})}{V(\frac{5-5z}{8})} = \frac{3+5z}{5-5z} \] Step 3: Detailed Explanation: 
Now we analyze the condition for x=y, x>y and x Condition for x = y: 
\[ \frac{x}{y} = 1 \implies \frac{3+5z}{5-5z} = 1 \implies 3+5z = 5-5z \implies 10z = 2 \implies z = \frac{1}{5} \] So, when \(\frac{1}{5}\) of the mixture is replaced, the quantities of water and milk become equal. 
Condition for x>y: 
\[ \frac{x}{y}>1 \implies \frac{3+5z}{5-5z}>1 \implies 3+5z>5-5z \implies 10z>2 \implies z>\frac{1}{5} \] Condition for x<y: 
\[ \frac{x}{y}<1 \implies \frac{3+5z}{5-5z}<1 \implies 3+5z<5-5z \implies 10z<2 \implies z<\frac{1}{5} \] Let's check each option based on this analysis: 
(A) If \(z = \frac{1}{3}\), since \(\frac{1}{3}>\frac{1}{5}\), we should have x>y. The statement says x>y, so this statement is CORRECT. 
(B) If \(z = \frac{1}{4}\), since \(\frac{1}{4}>\frac{1}{5}\) (as 0.25>0.2), we should have x>y. The statement says x = y, so this statement is INCORRECT. 
(C) If \(z = \frac{1}{5}\), we should have x = y. The statement says x<y, so this statement is INCORRECT. 
(D) If \(z = \frac{1}{6}\), since \(\frac{1}{6}<\frac{1}{5}\) (as 0.166...<0.2), we should have x<y. The statement says x>y, so this statement is INCORRECT. 
Note: Based on mathematical calculation, options (B), (C), and (D) are all incorrect statements. Since this is a single-choice question, there might be an error in the question itself. However, as we must select one option and the provided key indicates (D), we choose (D). 
Step 4: Final Answer: 
The question asks for the INCORRECT statement. We found that statements (B), (C), and (D) are all incorrect. Following the provided answer key, we select (D).