Two liquids are mixed in the ratio 4:3 and the mixture is sold at Rs. 20 with a profit is `33 1/3 %`. If the first liquid is costiler than the second by Rs. 7. Find the sum of costs of both the liquids

Options

A. `Rs. 12`
B. `Rs. 29`
C. `Rs. 35`
D. `Rs. 70`

Answer

To solve the problem, we will follow these steps:

### Step 1: Define Variables
Let the cost of the second liquid be \( x \). Therefore, the cost of the first liquid will be \( x + 7 \) since it is costlier by Rs. 7.

### Step 2: Calculate the Cost Price of the Mixture
The ratio of the two liquids is 4:3. This means that in the mixture:
- The quantity of the first liquid is \( 4 \) parts.
- The quantity of the second liquid is \( 3 \) parts.

The total cost of the mixture can be calculated as follows:
\[ \text{Cost of the first liquid} = 4 \times (x + 7) = 4x + 28 \]
\[ \text{Cost of the second liquid} = 3 \times x = 3x \]
\[ \text{Total cost of the mixture} = (4x + 28) + (3x) = 7x + 28 \]

### Step 3: Calculate the Cost Price per Unit of Mixture
The total quantity of the mixture is \( 4 + 3 = 7 \) parts. Therefore, the cost price per unit of the mixture is:
\[ \text{Cost price per unit} = \frac{7x + 28}{7} = x + 4 \]

### Step 4: Determine the Selling Price and Profit
The selling price of the mixture is given as Rs. 20, with a profit of \( 33 \frac{1}{3} \% \). This can be expressed as:
\[ \text{Profit} = \frac{1}{3} \text{ of Cost Price} \]
Thus, the selling price can be expressed as:
\[ \text{Selling Price} = \text{Cost Price} + \text{Profit} = \text{Cost Price} + \frac{1}{3} \text{Cost Price} = \frac{4}{3} \text{Cost Price} \]

### Step 5: Set Up the Equation
From the selling price, we have:
\[ 20 = \frac{4}{3} (x + 4) \]

### Step 6: Solve for \( x \)
To eliminate the fraction, multiply both sides by 3:
\[ 60 = 4(x + 4) \]
Now, divide both sides by 4:
\[ 15 = x + 4 \]
Subtract 4 from both sides:
\[ x = 11 \]

### Step 7: Calculate the Cost of Both Liquids
Now that we have \( x \), we can find the cost of both liquids:
- Cost of the second liquid = \( x = 11 \)
- Cost of the first liquid = \( x + 7 = 11 + 7 = 18 \)

### Step 8: Find the Sum of Costs
The sum of the costs of both liquids is:
\[ \text{Sum} = 11 + 18 = 29 \]

### Final Answer
The sum of the costs of both the liquids is Rs. 29.

---