In vessel A mixture, petrol and kerosene oil are in the ratio of 7:5 and in vessel B it is in the ratio of 8:5. P litre of mixture from vessel A and Q litre of mixture from vessel B are taken out and poured into vessel C. If vessel C contains total 150 litres mixture with 40% kerosene oil, then find value of `P/Q` ?

Options

A. A)`12/19`
B. B)`12/17`
C. C)`11/13`
D. D)`12/13`

Answer

To solve the problem step by step, we will follow the given information and apply the concepts of mixtures and ratios.

### Step 1: Understand the Ratios in Each Vessel
- In **Vessel A**, the ratio of petrol to kerosene is **7:5**. This means:
- Petrol = 7 parts
- Kerosene = 5 parts
- Total parts = 7 + 5 = 12 parts

- In **Vessel B**, the ratio of petrol to kerosene is **8:5**. This means:
- Petrol = 8 parts
- Kerosene = 5 parts
- Total parts = 8 + 5 = 13 parts

### Step 2: Determine the Composition of Vessel C
- Vessel C contains a total of **150 litres** of mixture with **40% kerosene**. This implies:
- Kerosene in Vessel C = 40% of 150 = 0.4 × 150 = 60 litres
- Petrol in Vessel C = 150 - 60 = 90 litres

### Step 3: Express the Ratios in Vessel C
- The ratio of petrol to kerosene in Vessel C can be calculated as follows:
- Petrol : Kerosene = 90 : 60 = 3 : 2

### Step 4: Set Up the Allegation Method
- We will use the allegation method to find the ratio of P and Q (the amounts taken from Vessel A and Vessel B).
- For Vessel A:
- Proportion of petrol = \( \frac{7}{12} \)

- For Vessel B:
- Proportion of petrol = \( \frac{8}{13} \)

- For Vessel C:
- Proportion of petrol = \( \frac{3}{5} \)

### Step 5: Apply the Allegation Formula
Using the allegation method, we can set up the following:

1. Calculate the difference between the proportion of petrol in Vessel A and Vessel C:
- \( \frac{7}{12} - \frac{3}{5} \)

2. Calculate the difference between the proportion of petrol in Vessel B and Vessel C:
- \( \frac{8}{13} - \frac{3}{5} \)

### Step 6: Find a Common Denominator
To perform the calculations, we need a common denominator for the fractions:
- The least common multiple of 12, 13, and 5 is **780**.

### Step 7: Calculate Each Part
1. For Vessel A:
- \( \frac{7}{12} \) of 780 = \( 7 \times 65 = 455 \)

2. For Vessel B:
- \( \frac{8}{13} \) of 780 = \( 8 \times 60 = 480 \)

3. For Vessel C:
- \( \frac{3}{5} \) of 780 = \( 3 \times 156 = 468 \)

### Step 8: Set Up the Allegation
Now we can set up the differences:
- From Vessel A to Vessel C: \( 455 - 468 = -13 \)
- From Vessel B to Vessel C: \( 480 - 468 = 12 \)

### Step 9: Find the Ratio P:Q
The ratio \( P:Q \) can be expressed as:
- \( P:Q = 12:13 \)

### Step 10: Calculate \( \frac{P}{Q} \)
Thus, \( \frac{P}{Q} = \frac{12}{13} \).

### Final Answer
The value of \( \frac{P}{Q} \) is **12:13**.

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