Two mixtures P & Q of concentration 4:5 and 5:7 of juice & water poured in the ratio of 3 : 4 in a vessel. If juice in resulting mixture is 144 ml, then find quantity of water in resulting mixture?

Options

A. A)172 ml
B. B)164 ml
C. C)200 ml
D. D)192 ml

Answer

To solve the problem step by step, we will break down the information given and perform the necessary calculations.

### Step 1: Understand the Concentrations
We have two mixtures:
- Mixture P has a concentration of juice to water as 4:5.
- Mixture Q has a concentration of juice to water as 5:7.

### Step 2: Calculate Total Parts in Each Mixture
For Mixture P:
- Total parts = 4 (juice) + 5 (water) = 9 parts

For Mixture Q:
- Total parts = 5 (juice) + 7 (water) = 12 parts

### Step 3: Mix the Mixtures in the Given Ratio
The mixtures are poured in the ratio of 3:4. This means:
- For every 3 parts of Mixture P, there are 4 parts of Mixture Q.

### Step 4: Calculate the Total Parts of Juice and Water in the Mixture
Now, we will calculate the total juice and water in the resulting mixture.

- Juice from Mixture P (3 parts):
- Juice from P = (3 parts of P) × (4/9) = \( \frac{12}{9} \) parts of juice

- Water from Mixture P (3 parts):
- Water from P = (3 parts of P) × (5/9) = \( \frac{15}{9} \) parts of water

- Juice from Mixture Q (4 parts):
- Juice from Q = (4 parts of Q) × (5/12) = \( \frac{20}{12} \) parts of juice

- Water from Mixture Q (4 parts):
- Water from Q = (4 parts of Q) × (7/12) = \( \frac{28}{12} \) parts of water

### Step 5: Combine the Juice and Water
Now we will sum up the juice and water from both mixtures.

Total Juice:
- Total Juice = Juice from P + Juice from Q
- Total Juice = \( \frac{12}{9} + \frac{20}{12} \)

To add these fractions, we need a common denominator:
- The least common multiple of 9 and 12 is 36.

Converting:
- \( \frac{12}{9} = \frac{48}{36} \)
- \( \frac{20}{12} = \frac{60}{36} \)

Now adding:
- Total Juice = \( \frac{48}{36} + \frac{60}{36} = \frac{108}{36} = 3 \) parts of juice

Total Water:
- Total Water = Water from P + Water from Q
- Total Water = \( \frac{15}{9} + \frac{28}{12} \)

Converting:
- \( \frac{15}{9} = \frac{60}{36} \)
- \( \frac{28}{12} = \frac{84}{36} \)

Now adding:
- Total Water = \( \frac{60}{36} + \frac{84}{36} = \frac{144}{36} = 4 \) parts of water

### Step 6: Calculate the Total Quantity of Juice in the Mixture
We know from the problem that the total juice in the resulting mixture is 144 ml.

### Step 7: Find the Value of Each Part
From the total juice:
- 3 parts = 144 ml
- 1 part = \( \frac{144}{3} = 48 \) ml

### Step 8: Calculate the Quantity of Water
Now, we need to find the quantity of water:
- Total Water = 4 parts
- Quantity of Water = 4 parts × 48 ml/part = 192 ml

### Final Answer
The quantity of water in the resulting mixture is **192 ml**.

---