Tea worth Rs 126 per kg and Rs 135 per kg are mixed with a third variety in the ratio ` 1: 1: 2`. If the mixture is worth Rs 153 per kg, the prices of the third variety per kg will be

Options

A. `Rs 169.5`
B. `Rs 170.0`
C. `Rs 175.5`
D. `Rs 180.0`

Answer

To solve the problem, we need to find the price of the third variety of tea when mixed with two other varieties in a specific ratio. Here’s the step-by-step solution:

### Step 1: Understand the given data
We have:
- Price of first variety (P1) = Rs 126 per kg
- Price of second variety (P2) = Rs 135 per kg
- Price of the mixture (Pm) = Rs 153 per kg
- Ratio of the varieties = 1:1:2

### Step 2: Assign variables
Let the price of the third variety be Rs x per kg.

### Step 3: Set up the equation for the average price
The average price of the mixture can be expressed as:
\[ Pm = \frac{(Quantity_1 \times Price_1) + (Quantity_2 \times Price_2) + (Quantity_3 \times Price_3)}{Total\ Quantity} \]

### Step 4: Substitute the values into the equation
From the ratio 1:1:2, we can assign:
- Quantity of first variety = 1 unit
- Quantity of second variety = 1 unit
- Quantity of third variety = 2 units

Now substituting into the equation:
\[ 153 = \frac{(1 \times 126) + (1 \times 135) + (2 \times x)}{1 + 1 + 2} \]

### Step 5: Simplify the equation
The total quantity is:
\[ 1 + 1 + 2 = 4 \]
So, substituting this into the equation gives:
\[ 153 = \frac{126 + 135 + 2x}{4} \]

### Step 6: Multiply both sides by 4 to eliminate the denominator
\[ 153 \times 4 = 126 + 135 + 2x \]
Calculating the left side:
\[ 612 = 126 + 135 + 2x \]

### Step 7: Combine the constants on the right side
Calculating the sum of 126 and 135:
\[ 126 + 135 = 261 \]
So, we have:
\[ 612 = 261 + 2x \]

### Step 8: Isolate the term with x
Subtract 261 from both sides:
\[ 612 - 261 = 2x \]
Calculating the left side:
\[ 351 = 2x \]

### Step 9: Solve for x
Divide both sides by 2:
\[ x = \frac{351}{2} = 175.5 \]

### Final Answer
The price of the third variety of tea is Rs 175.5 per kg.

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