To solve this problem, we need to calculate the amount of syrup to be taken from the initial mixture and determine how much water to add so that the final mixture has a water-to-syrup ratio of 2:1.

Step-by-Step Solution

1. Understand the initial mixture:

   • The initial mixture has a ratio of syrup to water of 3:1.
   • This means for every 4 parts of the mixture, 3 parts are syrup, and 1 part is water.

2. Determine the final ratio:

   • We want the final mixture to have a ratio of water to syrup of 2:1.
   • This means for every 3 parts of the final mixture, 2 parts are water, and 1 part is syrup.

3. Let’s assume we take $x$ parts of the initial mixture:

   • In $x$ parts of the mixture, the syrup will be $\frac{3}{4}x$, and the water will be $\frac{1}{4}x$.

4. Add water to achieve the desired ratio:

   • We need to add enough water to make the water-to-syrup ratio 2:1.
   • Let’s add $y$ parts of water to the $x$ parts of the mixture.

5. Set up the ratio equation:

   • After adding $y$ parts of water, the total water content becomes $\frac{1}{4}x + y$.
   • The syrup content remains $\frac{3}{4}x$.
   • We want the water-to-syrup ratio to be 2:1, so we set up the equation: $\frac{\frac{1}{4}x + y}{\frac{3}{4}x} = 2$

6. Solve for $y$: $\frac{1}{4}x + y = 2 \times \frac{3}{4}x$ $\frac{1}{4}x + y = \frac{3}{2}x$ $y = \frac{3}{2}x - \frac{1}{4}x$ $y = \frac{6}{4}x - \frac{1}{4}x$ $y = \frac{5}{4}x$

7. Interpret the result:

   • To achieve a water-to-syrup ratio of 2:1, you need to take $x$ parts of the initial mixture and add $\frac{5}{4}x$ parts of water.

Summary

• Take $x$ parts of the initial syrup mixture (with a 3:1 syrup-to-water ratio).
• Add $\frac{5}{4}x$ parts of water to achieve the desired final water-to-syrup ratio of 2:1.

Would you like more details on this calculation, or do you have any further questions?

Related Questions

1. How would the calculation change if the initial ratio of syrup to water was different?
2. What if you wanted a final water-to-syrup ratio of 3:1 instead?
3. How can we calculate the amount of water needed if we start with a specific volume of the syrup mixture?
4. What if additional ingredients, like sugar or lemon juice, were added to the mixture?
5. How would you calculate the total volume of the final mixture after adding water?

Tip: When working with ratios, it's essential to understand which part of the mixture represents which component and how changes in one component affect the overall ratio.