Calculate Orange Juice Mixture Volume

This problem requires finding the quantity of an 80% orange juice solution needed to mix with a 36-litre 25% orange juice solution to achieve a final 60% concentration.

We can use the method of alligation to solve this mixture problem efficiently.

Alligation Method Steps

• Identify the concentrations of the two solutions and the desired mean concentration.
  • Solution 1 Concentration: $80\%$
  • Solution 2 Concentration: $25\%$
  • Desired Mixture Concentration: $60\%$
• Calculate the difference between the higher concentration and the mean concentration:

  $80\% - 60\% = 20\%$

• Calculate the difference between the mean concentration and the lower concentration:

  $60\% - 25\% = 35\%$

• The ratio of the quantities of the two solutions is the inverse of these differences.

  Ratio (Quantity of 80% solution : Quantity of 25% solution) = $35 : 20$.

• Simplify the ratio:

  $ \frac{35}{20} = \frac{7}{4} $

  This means for every 4 litres of the 25% solution, we need 7 litres of the 80% solution.

• Use the given volume of the 25% solution (36 litres) to find the required volume of the 80% solution. Let the required volume be $x$ litres.

  $ \frac{\text{Volume of 80\% solution}}{\text{Volume of 25\% solution}} = \frac{7}{4} $

  $ \frac{x}{36 \text{ litres}} = \frac{7}{4} $

• Solve for $x$:

  $ x = \frac{7}{4} \times 36 \text{ litres} $

  $ x = 7 \times 9 \text{ litres} $

  $ x = 63 \text{ litres} $

Therefore, 63 litres of the 80% orange juice drink must be mixed.