Question

Equal quantities of three mixtures of milk and water are mixed in the ratio of \(1: 2,2: 3\) and \(3: 4\). The ratio of water and milk in the mixture is : (a) \(193: 122\) (b) \(122: 193\) (c) \(61: 97\) (d) \(137: 178\)

Short answer

Answer: The ratio of water and milk in the final mixture is 193:97.

Step-by-step solution

Analyze the given information

The ratios of milk and water in the three mixtures are given as: 1. \(1:2\) 2. \(2:3\) 3. \(3:4\) Since equal quantities of each mixture are taken, let's assume the quantity of each mixture to be \(x\).

Calculate the quantities of milk and water in each mixture

Using the given ratios, we can calculate the amount of milk and water in each mixture. 1. Mixture 1: Milk = \(\frac{1}{1+2}*x = \frac{1}{3}*x\), Water = \(\frac{2}{1+2}*x = \frac{2}{3}*x\) 2. Mixture 2: Milk = \(\frac{2}{2+3}*x = \frac{2}{5}*x\), Water = \(\frac{3}{2+3}*x = \frac{3}{5}*x\) 3. Mixture 3: Milk = \(\frac{3}{3+4}*x = \frac{3}{7}*x\), Water = \(\frac{4}{3+4}*x = \frac{4}{7}*x\)

Find the total quantities of milk and water

We need to find the total quantities of milk and water in the final mixture after mixing all three: Total milk = milk in mixture 1 + milk in mixture 2 + milk in mixture 3 = \(\frac{1}{3}*x + \frac{2}{5}*x + \frac{3}{7}*x\) Total water = water in mixture 1 + water in mixture 2 + water in mixture 3 = \(\frac{2}{3}*x + \frac{3}{5}*x + \frac{4}{7}*x\)

Simplify the total quantities

Now we need to simplify the total quantities of milk and water to find the final ratio: Total milk = \(x(\frac{1}{3} + \frac{2}{5} + \frac{3}{7}) = x\frac{97}{105}\) Total water = \(x(\frac{2}{3} + \frac{3}{5} + \frac{4}{7}) = x\frac{193}{105}\)

Calculate the final ratio

To find the final ratio of water and milk in the mixture, we need to divide the total water by the total milk: Final ratio = \(\frac{\text{Total water}}{\text{Total milk}} = \frac{\frac{193}{105}*x}{\frac{97}{105}*x}\) Since \(x\) and \(105\) are common in both numerator and denominator, we can cancel them: Final ratio = \(\frac{193}{97}\) This can be simplified as: Final ratio = \(193:97\) Hence, the ratio of water and milk in the final mixture is \(193:97\). The correct option is (a) \(193: 122\).

Key concepts

Milk and Water Mixtures

In problems involving mixtures, it's common to deal with multiple substances combined in varying ratios. In our exercise, we examine mixtures of milk and water combined in different proportions. Understanding how each mixture contributes to the whole is central to solving such problems.

• Mixture 1 has milk to water in a ratio of \(1:2\). This means for every 1 part of milk, there are 2 parts of water.
• Mixture 2 is in a ratio of \(2:3\), suggesting for every 2 parts of milk, 3 parts are water.
• For Mixture 3, the ratio is \(3:4\), meaning 3 parts of milk go with 4 parts of water.

Each ratio helps us determine the composition of the mixture. Assuming equal quantities for each mixture (let's say \(x\)), helps standardize how we calculate the contribution of milk and water from each.

Ratio Simplification

Simplifying ratios is a critical step in solving mixture problems. Not only does it make our calculations easier, but it also helps in accurately comparing proportions across different mixtures. To simplify, we divide each part of the mixture by the sum of its parts to find individual contributions. This technique can clarify complex relationships.

Consider how we derived the proportions for each mixture:

• For mixture 1 (\(1:2\)), we divided 1 and 2 by their sum \((1+2 = 3)\). - Milk is \(\frac{1}{3}x\) and water is \(\frac{2}{3}x\).
• Mixture 2's (\(2:3\)) sum is 5. - Milk is \(\frac{2}{5}x\) and water is \(\frac{3}{5}x\).
• Finally, for mixture 3 (\(3:4\)), sum is 7. - Milk is \(\frac{3}{7}x\) and water is \(\frac{4}{7}x\).

Simplification reveals standardized units, allowing us to compare and calculate the overall mixture.

Proportions

Proportions help in understanding the relationship between parts of a mixture and their overall contribution. Solving proportion problems in the context of mixtures involves careful calculation of totals and simplification.

In the exercise, we add each component's contribution to get the total milk and water across all mixtures. Here's how we did it:

• Total milk from all mixtures = \(\frac{1}{3}x + \frac{2}{5}x + \frac{3}{7}x\).
• Total water from all mixtures = \(\frac{2}{3}x + \frac{3}{5}x + \frac{4}{7}x\).

With these totals, dividing water by milk derives the final ratio. By understanding individual contributions from each mixture, calculating the overall effect becomes simpler and allows us to find the desired final ratio effortlessly. This problem teaches the power of precise proportion analysis in determining accurate results.