We are given the following:

• A 280 ml pack contains a mixture of mango juice and orange juice in the ratio of 9:5.
• Some $x$ ml of juice is taken out from the pack.
• After removing $x$ ml, 20 ml of orange juice is added, and the new ratio of mango juice to orange juice becomes 7:5.

We need to find how much mango juice was taken out.

Step 1: Initial quantity of mango and orange juice

The total volume is 280 ml, and the initial ratio of mango juice to orange juice is 9:5.

Let the initial amount of mango juice be $M_0$ ml, and the initial amount of orange juice be $O_0$ ml.

From the ratio $M_0 : O_0 = 9 : 5$, we can write: $M_0 = \frac{9}{14} \times 280 = 180 \text{ ml}$ $O_0 = \frac{5}{14} \times 280 = 100 \text{ ml}$

Step 2: Juice taken out and new ratio after addition

Let $x$ ml of juice be taken out. Since the juice is taken from the mixture, the amount of mango juice and orange juice taken out will also be in the ratio of 9:5.

Thus, the amount of mango juice taken out is $\frac{9}{14}x$ and the amount of orange juice taken out is $\frac{5}{14}x$.

Now, after removing $x$ ml, 20 ml of orange juice is added. The new amounts of mango juice and orange juice are:

• Mango juice remaining = $180 - \frac{9}{14}x$ ml
• Orange juice remaining = $100 - \frac{5}{14}x + 20$ ml

Step 3: Using the new ratio of mango juice to orange juice

After the changes, the ratio of mango juice to orange juice becomes 7:5. Thus, we can write the equation: $\frac{180 - \frac{9}{14}x}{100 - \frac{5}{14}x + 20} = \frac{7}{5}$

Simplifying the denominator: $100 - \frac{5}{14}x + 20 = 120 - \frac{5}{14}x$

Substitute into the equation: $\frac{180 - \frac{9}{14}x}{120 - \frac{5}{14}x} = \frac{7}{5}$

Step 4: Solving for $x$

Cross-multiply to eliminate the fraction: $5 \left( 180 - \frac{9}{14}x \right) = 7 \left( 120 - \frac{5}{14}x \right)$ Expanding both sides: $900 - \frac{45}{14}x = 840 - \frac{35}{14}x$ Move all $x$-terms to one side and constants to the other: $900 - 840 = \frac{45}{14}x - \frac{35}{14}x$ $60 = \frac{10}{14}x$ Multiply both sides by 14 to get rid of the denominator: $60 \times 14 = 10x$ $840 = 10x$ $x = 84 \text{ ml}$

Step 5: Finding the amount of mango juice taken out

The amount of mango juice taken out is $\frac{9}{14}x$. So: $\text{Mango juice taken out} = \frac{9}{14} \times 84 = 54 \text{ ml}$

Final Answer:

The quantity of mango juice taken out initially is 54 ml.

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Would you like further details on any part of the process?

Here are 5 related questions:

1. How would the problem change if 30 ml of orange juice were added instead of 20 ml?
2. What happens to the ratio if no juice is taken out but 20 ml of orange juice is added?
3. Can you generalize a formula for any ratio of two liquids?
4. What if the initial volume of the pack was 350 ml, with the same ratio of mango and orange juice?
5. How would the solution change if mango and orange juice were in a different ratio, such as 3:2?

Tip: In mixture problems, ratios help maintain proportionality between components, and small changes can significantly affect outcomes!