Two containers, X and Y, hold mixtures of juice and water. In container X, the ratio of juice to water is 5:3, while in container Y, the ratio is 8:5. When the contents of both containers are combined into a single vat, the total volume of juice is 80 liters and the total volume of water is 49 liters. Determine the total volume of the mixture that was originally in container Y.

Correct Answer: 65 liters

Step-by-step Explanation:

Step 1: Determine the fractional components of juice and water in each container

Let the total volume of mixture in container X be $V_X$ liters and in container Y be $V_Y$ liters.

In container X, the ratio of juice to water is 5:3.
Total ratio parts for container X = 5 + 3 = 8.
- Fraction of juice in X = $\frac{5}{8}$
- Fraction of water in X = $\frac{3}{8}$

In container Y, the ratio of juice to water is 8:5.
Total ratio parts for container Y = 8 + 5 = 13.
- Fraction of juice in Y = $\frac{8}{13}$
- Fraction of water in Y = $\frac{5}{13}$

Step 2: Formulate linear equations based on the combined volumes

When the mixtures are combined in a single vat, we are given:
- Total volume of juice = 80 liters
- Total volume of water = 49 liters

Setting up the equation for total juice:

$\frac{5}{8}{V}_X+\frac{8}{13}{V}_Y=80$    --- (Equation 1)

Setting up the equation for total water:

$\frac{3}{8}{V}_X+\frac{5}{13}{V}_Y=49$    --- (Equation 2)

Step 3: Solve the equations to find $V_Y$

To eliminate $V_X$, multiply Equation 1 by 3 and Equation 2 by 5:

Multiplying Equation 1 by 3 gives:

$\frac{15}{8}{V}_X+\frac{24}{13}{V}_Y=240$    --- (Equation 3)

Multiplying Equation 2 by 5 gives:

$\frac{15}{8}{V}_X+\frac{25}{13}{V}_Y=245$    --- (Equation 4)

Subtracting Equation 3 from Equation 4:

$\left(\frac{15}{8}{V}_X+\frac{25}{13}{V}_Y\right)-\left(\frac{15}{8}{V}_X+\frac{24}{13}{V}_Y\right)=245-240$

$\frac{1}{13}{V}_Y=5$

Multiplying both sides by 13:

$V_Y=5\times 13=65$

Conclusion:
The total volume of the mixture originally in container Y is 65 liters.