Understanding the Mixture Ratio Problem

This problem involves a mixture of alcohol and water, where we need to determine how much water to add to change the existing ratio to a new desired ratio. We are given the total volume of the mixture and the initial ratio of alcohol to water.

Initial Conditions

• Total volume of the mixture: 85 litres
• Initial ratio of alcohol to water: 11 ∶ 6
• Desired final ratio of alcohol to water: 5 ∶ 3

Calculating Initial Quantities of Alcohol and Water

First, we need to find out the actual quantities of alcohol and water present in the initial 85-litre mixture.

The total number of parts in the initial ratio is the sum of the ratio components:

Total parts = \( 11 + 6 = 17 \)

Now, we can find the volume represented by one part:

Volume per part = \(\frac{\text{Total volume}}{\text{Total parts}}\) = \(\frac{85 \text{ litres}}{17 \text{ parts}}\) = 5 litres/part

Using this, we calculate the initial amounts:

Initial quantity of alcohol = \( 11 \text{ parts} \times 5 \text{ litres/part} = 55 \text{ litres} \)

Initial quantity of water = \( 6 \text{ parts} \times 5 \text{ litres/part} = 30 \text{ litres} \)

Verification: The sum of initial quantities is \( 55 + 30 = 85 \) litres, which matches the total volume given.

Setting Up the Equation for the New Ratio

We need to add water to the mixture to achieve a new ratio of 5 ∶ 3. Let the quantity of water to be added be '\(x\)' litres.

The quantity of alcohol remains unchanged:

Quantity of alcohol = 55 litres

The quantity of water increases by '\(x\)' litres:

New quantity of water = \( 30 + x \) litres

The problem states that the new ratio of alcohol to water should be 5 ∶ 3. So, we can write the equation:

\(\frac{\text{Quantity of alcohol}}{\text{New quantity of water}} = \frac{5}{3}\)

Substituting the values:

\(\frac{55}{30 + x} = \frac{5}{3}\)

Solving for the Added Water

To find the value of '\(x\)', we solve the equation:

Cross-multiply the equation:

\( 55 \times 3 = 5 \times (30 + x) \)

\( 165 = 150 + 5x \)

Isolate the term with '\(x\)' by subtracting 150 from both sides:

\( 165 - 150 = 5x \)

\( 15 = 5x \)

Divide by 5 to find '\(x\)' :

\( x = \frac{15}{5} \)

\( x = 3 \)

Conclusion

Therefore, 3 litres of water should be added to the mixture to make the ratio of alcohol to water 5 ∶ 3.

Final Check:

New quantity of water = \( 30 + 3 = 33 \) litres.

New ratio = Alcohol : Water = \( 55 : 33 \)

Simplifying the ratio by dividing both by 11: \( \frac{55}{11} : \frac{33}{11} \) = \( 5 : 3 \). This matches the desired ratio.