The ratios of acid and water in the solutions in vessels A and B are 4 : 5 and 5 : 1, respectively. A new solution is obtained by mixing 5 litres and 4 litres of the solutions from A and B , respectively. What is the ratio of acid and water in the new solution ?

Options

A. `9 : 4`
B. `50 : 31`
C. `25 : 16`
D. `15 : 8`

Answer

To solve the problem step by step, we need to find the ratio of acid and water in the new solution obtained by mixing the solutions from vessels A and B.

### Step 1: Determine the composition of the solution in vessel A
The ratio of acid to water in vessel A is 4:5. This means that for every 9 parts of the solution (4 parts acid + 5 parts water):
- Acid = \( \frac{4}{9} \) of the solution
- Water = \( \frac{5}{9} \) of the solution

### Step 2: Calculate the amount of acid and water in 5 liters from vessel A
Since we are taking 5 liters from vessel A:
- Amount of acid from A = \( 5 \times \frac{4}{9} = \frac{20}{9} \) liters
- Amount of water from A = \( 5 \times \frac{5}{9} = \frac{25}{9} \) liters

### Step 3: Determine the composition of the solution in vessel B
The ratio of acid to water in vessel B is 5:1. This means that for every 6 parts of the solution (5 parts acid + 1 part water):
- Acid = \( \frac{5}{6} \) of the solution
- Water = \( \frac{1}{6} \) of the solution

### Step 4: Calculate the amount of acid and water in 4 liters from vessel B
Since we are taking 4 liters from vessel B:
- Amount of acid from B = \( 4 \times \frac{5}{6} = \frac{20}{6} = \frac{10}{3} \) liters
- Amount of water from B = \( 4 \times \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \) liters

### Step 5: Combine the amounts of acid and water from both vessels
Now, we add the amounts of acid and water from both vessels:
- Total acid = \( \frac{20}{9} + \frac{10}{3} \)
- To add these fractions, we need a common denominator. The least common multiple of 9 and 3 is 9.
- Convert \( \frac{10}{3} \) to ninths: \( \frac{10}{3} = \frac{30}{9} \)
- So, Total acid = \( \frac{20}{9} + \frac{30}{9} = \frac{50}{9} \) liters

- Total water = \( \frac{25}{9} + \frac{2}{3} \)
- Convert \( \frac{2}{3} \) to ninths: \( \frac{2}{3} = \frac{6}{9} \)
- So, Total water = \( \frac{25}{9} + \frac{6}{9} = \frac{31}{9} \) liters

### Step 6: Find the ratio of acid to water in the new solution
The ratio of acid to water in the new solution is:
\[ \text{Ratio} = \frac{\text{Total Acid}}{\text{Total Water}} = \frac{\frac{50}{9}}{\frac{31}{9}} = \frac{50}{31} \]

Thus, the ratio of acid to water in the new solution is \( 50:31 \).

### Final Answer
The ratio of acid and water in the new solution is **50:31**.