Final Ratio of Milk to Water in a Mixture After Repeated Operations

In this article, we will explore a method to determine the final ratio of milk to water in a mixture after multiple replacement operations. Specifically, we will analyze a mixture of 30 liters containing milk and water in the ratio 7:3. We will follow a step-by-step approach to understand the changes in the mixture after each operation.

Initial Mixture Calculation

Let's start by calculating the initial amounts of milk and water in the mixture.

The mixture is 30 liters with a milk to water ratio of 7:3. This means:

The total parts are 10 parts (7 milk 3 water). The amount of milk is 21 liters: (frac{7}{10} times 30 21). The amount of water is 9 liters: (frac{3}{10} times 30 9).

First Operation

In the first operation, 10 liters of the mixture is taken out and replaced with pure milk. We need to calculate the amount of milk and water in the 10 liters taken out and the remaining amounts in the mixture.

The amount of milk taken out is 7 liters: (frac{7}{10} times 10 7). The amount of water taken out is 3 liters: (frac{3}{10} times 10 3). After removing 10 liters, the remaining amounts are: Remaining milk: (21 - 7 14) liters. Remaining water: (9 - 3 6) liters.

Next, we add 10 liters of pure milk to the mixture. The new amounts are:

New amount of milk: (14 10 24) liters. Amount of water remains the same: 6 liters.

Second Operation

In the second operation, we take out 10 liters of the new mixture and replace it with pure milk. We need to calculate the new total mixture and the amounts taken out.

The new total mixture is 30 liters containing 24 liters of milk and 6 liters of water. The ratio of milk in the new mixture is (frac{24}{30} frac{4}{5}). The ratio of water in the new mixture is (frac{6}{30} frac{1}{5}).

Calculating the amounts taken out:

Milk in 10 liters taken out: (frac{4}{5} times 10 8) liters. Water in 10 liters taken out: (frac{1}{5} times 10 2) liters.

The remaining amounts after the second removal are:

Remaining milk: (24 - 8 16) liters. Remaining water: (6 - 2 4) liters.

Adding 10 liters of pure milk to the mixture results in:

New amount of milk: (16 10 26) liters. Amount of water remains the same: 4 liters.

Final Calculation of Ratio

Now we have the final amounts of milk and water:

Milk: 26 liters. Water: 4 liters.

The final ratio of milk to water is:

(text{Ratio} frac{26}{4} frac{13}{2} 13:2).

Thus, the final ratio of milk to water in the mixture is 13:2.

Conclusion

In conclusion, we have followed the methodical steps to calculate the final ratio of milk to water in the mixture after two replacement operations. By carefully analyzing the changes in the mixture at each step, we have determined the final ratio to be 13:2.