In 729 litres solution of acid and water, the ratio of acid and water `7:2.` How many litres of water must be added to it to get the solution in which the ratio of acid and water is `5:3` ?

Options

A. `187.2`
B. `182.4`
C. `178.2`
D. `180.4`

Answer

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

### Step 1: Understand the initial conditions
We have a total solution of 729 liters, consisting of acid and water in the ratio of 7:2.

### Step 2: Set up the equations
Let the amount of acid be \(7x\) liters and the amount of water be \(2x\) liters. Therefore, we can write the equation:
\[ 7x + 2x = 729 \]

### Step 3: Solve for \(x\)
Combine the terms:
\[ 9x = 729 \]
Now, divide both sides by 9:
\[ x = \frac{729}{9} = 81 \]

### Step 4: Calculate the amounts of acid and water
Now, substitute \(x\) back into the expressions for acid and water:
- Amount of acid:
\[ 7x = 7 \times 81 = 567 \text{ liters} \]
- Amount of water:
\[ 2x = 2 \times 81 = 162 \text{ liters} \]

### Step 5: Determine the new ratio
We need to find out how much water (\(y\) liters) must be added to achieve a new ratio of acid to water of \(5:3\).

### Step 6: Set up the new ratio equation
After adding \(y\) liters of water, the total amount of water becomes \(162 + y\) liters. The new ratio can be expressed as:
\[ \frac{567}{162 + y} = \frac{5}{3} \]

### Step 7: Cross-multiply to solve for \(y\)
Cross-multiplying gives:
\[ 5(162 + y) = 3 \times 567 \]
Calculating the right side:
\[ 3 \times 567 = 1701 \]
Now, substituting this back into the equation:
\[ 5(162 + y) = 1701 \]

### Step 8: Expand and simplify
Expanding the left side:
\[ 810 + 5y = 1701 \]
Now, isolate \(5y\):
\[ 5y = 1701 - 810 \]
Calculating the right side:
\[ 5y = 891 \]

### Step 9: Solve for \(y\)
Now, divide both sides by 5:
\[ y = \frac{891}{5} = 178.2 \]

### Conclusion
The amount of water that must be added to achieve the desired ratio of \(5:3\) is **178.2 liters**.

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