Understanding the Mixture Ratio Problem

This problem involves a mixture of fruit juice and water. We are given the initial total volume and the ratio of the two components. We need to find out how much water must be added to change the ratio to a new value.

Let's break down the problem step by step to find the amount of water needed to achieve the desired ratio.

Initial Mixture Analysis

The total volume of the initial mixture is 55 litres.

The initial ratio of fruit juice to water is $4 \ratio 1$. This means for every 4 parts of fruit juice, there is 1 part of water.

The total number of parts in the initial ratio is $4 + 1 = 5$ parts.

To find the volume represented by one part, we divide the total volume by the total number of parts:

Volume per part = $\frac{\text{Total Volume}}{\text{Total Parts}} = \frac{55 \text{ litres}}{5 \text{ parts}} = 11 \text{ litres/part}$.

Now we can calculate the initial quantities of fruit juice and water:

• Initial Fruit Juice: $4 \text{ parts} \times 11 \text{ litres/part} = 44 \text{ litres}$
• Initial Water: $1 \text{ part} \times 11 \text{ litres/part} = 11 \text{ litres}$

Let's verify the total: $44 \text{ litres} + 11 \text{ litres} = 55 \text{ litres}$. This matches the given total volume.

Adding Water to Change the Ratio

We are adding water to the mixture, but the amount of fruit juice remains unchanged.

Let 'x' be the amount of water (in litres) added to the mixture.

• Final Fruit Juice: 44 litres (remains the same)
• Final Water: Initial Water + Added Water = $(11 + x)$ litres

The new ratio of fruit juice to water is desired to be $2 \ratio 1$.

We can set up an equation based on the final ratio:

$\frac{\text{Final Fruit Juice}}{\text{Final Water}} = \frac{2}{1}$

Substitute the calculated final quantities into the ratio equation:

$\frac{44}{11 + x} = \frac{2}{1}$

Solving for the Amount of Water Added

To solve for 'x', we can cross-multiply the equation:

$44 \times 1 = 2 \times (11 + x)$

$44 = 22 + 2x$

Now, we isolate the term with 'x' by subtracting 22 from both sides of the equation:

$44 - 22 = 2x$

$22 = 2x$

Finally, divide by 2 to find the value of 'x':

$x = \frac{22}{2}$

$x = 11$

Conclusion

The amount of water that must be added to the mixture is 11 litres.

Let's check the final ratio: Fruit Juice : Water = 44 : 22. Dividing both numbers by their greatest common divisor (22), we get $44 \div 22 = 2$ and $22 \div 22 = 1$. So, the final ratio is $2 \ratio 1$, which matches the desired ratio.

Revision Table: Key Concepts for Mixture Ratio Problems

Additional Information: Ratio and Proportion in Mixtures

Ratio and proportion are fundamental concepts used frequently in problems involving mixtures. A ratio tells us the relative amounts of different substances in a mixture. When we add or remove a substance, the ratio changes unless the substance is added/removed in the same proportion as its current presence in the mixture.

In mixture problems, it's crucial to identify which component's quantity remains constant and which component's quantity changes. In this problem, adding only water means the amount of fruit juice stays constant, while the amount of water increases.

Setting up the equation based on the unchanging component (or the ratio of the components) is a common strategy to solve these types of problems. For example, if fruit juice is constant, its amount in the initial mixture equals its amount in the final mixture. If water is added, the ratio $\frac{\text{Fruit Juice}}{\text{Water}}$ changes, and we use the constant fruit juice amount to find the new water amount.