Understanding the Mixture and Ratio Problem

This question involves a mixture of two liquids, A and B, where their quantities are given in a specific ratio. We need to find out how much of liquid A must be added to change the ratio to a new value.

Step-by-Step Solution

Let's break down the problem into smaller steps to find the solution.

1. Identify the initial conditions:
   • Total volume of the mixture: 70 litres
   • Initial ratio of liquid A to liquid B (A:B): 5 ∶ 9
2. Calculate the initial amounts of liquid A and B:
   • The total parts in the initial ratio are $5 + 9 = 14$ parts.
   • The value of one part in the 70-litre mixture is $\frac{70 \text{ litres}}{14 \text{ parts}} = 5$ litres/part.
   • Initial quantity of liquid A = $5 \text{ parts} \times 5 \text{ litres/part} = 25$ litres.
   • Initial quantity of liquid B = $9 \text{ parts} \times 5 \text{ litres/part} = 45$ litres.
   • We can verify this: $25 \text{ litres} + 45 \text{ litres} = 70$ litres, which matches the total volume.
3. Define the change:
   • Liquid A is added to the mixture. Let the amount of liquid A added be $x$ litres.
   • The quantity of liquid B remains unchanged.
4. Formulate the new quantities:
   • New quantity of liquid A = (Initial quantity of A) + (Amount added) = $25 + x$ litres.
   • New quantity of liquid B = Initial quantity of B = $45$ litres.
5. Set up the equation based on the new ratio:
   • The new desired ratio of liquid A to liquid B (A:B) is 2 ∶ 3.
   • This means the ratio of the new quantity of A to the new quantity of B is $\frac{\text{New quantity of A}}{\text{New quantity of B}} = \frac{2}{3}$.
   • Substituting the expressions for the new quantities, we get the equation: $\frac{25 + x}{45} = \frac{2}{3}$.
6. Solve the equation for x:
   • To solve for $x$, we can cross-multiply: $3 \times (25 + x) = 2 \times 45$.
   • Distribute the 3 on the left side: $75 + 3x = 90$.
   • Subtract 75 from both sides: $3x = 90 - 75$.
   • Simplify: $3x = 15$.
   • Divide by 3: $x = \frac{15}{3}$.
   • Calculate $x$: $x = 5$ litres.

So, 5 litres of liquid A must be added to the mixture.

Verification

If 5 litres of liquid A are added:

• New quantity of A = $25 + 5 = 30$ litres.
• New quantity of B = $45$ litres.
• New ratio A:B = $30 : 45$.
• Simplifying the ratio by dividing both numbers by their greatest common divisor (15): $\frac{30}{15} : \frac{45}{15} = 2 : 3$.

This matches the desired new ratio, confirming our answer.

Revision Table: Mixture Ratio Problem

Reviewing the key steps for solving mixture ratio problems with additions:

• Calculate initial quantities based on the total volume and ratio.
• Identify which component is being added and by how much (use a variable).
• Write expressions for the new quantities of the components.
• Set up an equation using the new ratio of the components.
• Solve the equation for the unknown variable.
• Verify the answer using the calculated amount.

Additional Information: Ratios and Proportions

Ratios are used to compare the relative amounts of two or more quantities. A ratio like A:B = 5:9 means that for every 5 units of A, there are 9 units of B. Proportions are equations that state that two ratios are equal, like $\frac{a}{b} = \frac{c}{d}$. Solving mixture problems often involves setting up and solving proportions.

When a quantity is added to a mixture, only the amount of that specific component changes. The amount of other components remains constant unless otherwise stated. The total volume of the mixture also changes when a component is added.