A glass contains 500 ml of milk, and a cup contains 500 ml of water. From the glass, 150 ml of milk is transferred to the cup and mixed thoroughly. Next, 150 ml of this mixture is... A glass contains 500 ml of milk, and a cup contains 500 ml of water. From the glass, 150 ml of milk is transferred to the cup and mixed thoroughly. Next, 150 ml of this mixture is transferred from the cup back to the glass. What is the ratio of the amount of water in the glass to the amount of milk in the cup? Read more

Steps to Solve

1. Initial Setup

Assume we start with 500 mL of milk in the glass and 500 mL of water in the cup.

2. First Transfer (Glass to Cup)

Transfer 100 mL of milk from the glass to the cup.

After this step:

• Glass: 400 mL milk
• Cup: 500 mL water + 100 mL milk = 600 mL mixture

3. Calculate the Ratio of Milk in the Cup

The cup now has a total of 600 mL (500 mL water + 100 mL milk).

To find the amount of milk in the cup:

• The fraction of milk in the cup is $\frac{100}{600} = \frac{1}{6}$.

4. Second Transfer (Cup Back to Glass)

Next, we transfer 100 mL of this mixture back to the glass.

In this 100 mL, the amount of milk is calculated as follows:

• Milk in the 100 mL = $\frac{1}{6} \times 100 \text{ mL} = \frac{100}{6} \text{ mL} \approx 16.67 \text{ mL}$

Thus, the glass now contains:

• Glass: $400 \text{ mL milk} + 16.67 \text{ mL milk} \approx 416.67 \text{ mL milk}$
• Water in the glass remains unchanged at 0 mL.

5. Cup after Second Transfer

After transferring back to the glass, the cup now contains:

• Remaining volume = $600 \text{ mL} - 100 \text{ mL} = 500 \text{ mL}$
• Amount of milk in cup after transfer = $100 \text{ mL} - 16.67 \text{ mL} \approx 83.33 \text{ mL}$

6. Final Ratio Calculation

The final step is to find the ratio of water in the glass to milk in the cup:

• Water in glass = 0 mL
• Milk in cup = 83.33 mL

Thus, the ratio is: $$ \text{Ratio} = \frac{0}{83.33} = 0 $$