A 140 ml mixture contains mixture of milk and water in the ratio of 9: 5. If x ml mixture is taken out from pack and 10 ml of water is in pack so, the new ratio of milk and water become 7: 5. Then find the quantity of milk taken out from pack?

Given:
- Initial volume of the mixture = 140 ml
- Initial ratio of milk to water = 9:5

First, calculate the initial quantities of milk and water:
\[
\text{Total parts} = 9 + 5 = 14
\]
\[
\text{Volume of milk} = \frac{9}{14} \times 140 = 90 \text{ ml}
\]
\[
\text{Volume of water} = \frac{5}{14} \times 140 = 50 \text{ ml}
\]

After taking out \(x\) ml of the mixture, 10 ml of water remains, and the new ratio of milk to water becomes 7:5. Let's denote the initial volumes of milk and water as \(M_1\) and \(W_1\) respectively, and their volumes after taking out \(x\) ml as \(M_2\) and \(W_2\) respectively.

From the problem:
\[
W_2 = 10 \text{ ml}
\]
\[
\frac{M_2}{W_2} = \frac{7}{5}
\]

We can express \(M_2\) in terms of \(W_2\):
\[
M_2 = \frac{7}{5} \times 10 = 14 \text{ ml}
\]

When \(x\) ml of the mixture is taken out, the ratio of milk and water in that \(x\) ml should be the same as the initial ratio (9:5). Therefore, the volumes of milk and water in the removed mixture can be calculated using the same initial ratio.

The quantity of mixture left is:
\[
140 - x \text{ ml}
\]

The amount of milk and water removed should also follow the 9:5 ratio:
\[
\frac{\text{Volume of milk removed}}{\text{Volume of water removed}} = \frac{9}{5}
\]

If \(M_1\) and \(W_1\) are the initial volumes of milk and water, respectively, and \(M_2\) and \(W_2\) are the volumes after removing \(x\) ml of the mixture, then:
\[
M_2 = M_1 - \frac{9}{14}x
\]
\[
W_2 = W_1 - \frac{5}{14}x
\]

Given \(W_2 = 10\):
\[
50 - \frac{5}{14}x = 10
\]
\[
40 = \frac{5}{14}x
\]
\[
x = \frac{40 \times 14}{5} = 112 \text{ ml}
\]

The volume of milk removed is:
\[
\frac{9}{14} \times 112 = 72 \text{ ml}
\]

Thus, the quantity of milk taken out from the pack is \(72 \text{ ml}\).

Would you like further details or have any questions?

Here are some related questions:
1. How do you calculate the new ratio if the quantities are different?
2. How do you convert ratios into fractions?
3. What are the steps to solve a ratio problem involving mixtures?
4. How do you determine the initial quantity from a given ratio and total volume?
5. How can you set up equations to solve mixture problems?
6. What happens to the ratio if only one part of the mixture is altered?
7. How do you handle problems with more complex ratios or multiple steps?
8. Can you generalize this method for any mixture problem?

**Tip:** Always check your calculations step-by-step to avoid errors in ratio and proportion problems.

Learn how to solve a mixture problem involving the removal of milk from a given mixture. This problem covers ratio and proportion concepts, with step-by-step calculations to find the quantity of milk removed from the initial mixture. Suitable for high school students studying ratio problems and mixture calculations.

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