Rice of two category i.e. Rs. 150 per kg and Rs. 200 per kg are mixed with a third category in ratio 1:2:2, if mixture of these was sold at profit on 20% for Rs. 252, find cost price of third category rice?

Options

A. Rs. 175
B. Rs. 275
C. Rs. 250
D. Rs. 225

Answer

To solve the problem step-by-step, we need to find the cost price of the third category rice given the prices of the first two categories and the selling price of the mixture.

### Step 1: Understand the given data
We have:
- Price of first category rice = Rs. 150 per kg
- Price of second category rice = Rs. 200 per kg
- Ratio of mixing = 1:2:2 (for first, second, and third category respectively)
- Selling price of the mixture = Rs. 252
- Profit = 20%

### Step 2: Calculate the cost price of the mixture
Since the selling price includes a profit of 20%, we can find the cost price of the mixture using the formula:
\[ \text{Selling Price} = \text{Cost Price} + \text{Profit} \]
Where profit is 20% of the cost price.

Let \( CP \) be the cost price of the mixture. Then:
\[ 252 = CP + 0.2 \times CP \]
\[ 252 = 1.2 \times CP \]
\[ CP = \frac{252}{1.2} \]
\[ CP = 210 \]

### Step 3: Calculate the total cost of the mixture
The total cost of the mixture can be calculated based on the ratio of the rice categories:
- Let the quantities of the rice be:
- First category = 1 kg
- Second category = 2 kg
- Third category = 2 kg

Total quantity = \( 1 + 2 + 2 = 5 \) kg

The total cost of the mixture can also be expressed as:
\[ \text{Total Cost} = (1 \times 150) + (2 \times 200) + (2 \times x) \]
Where \( x \) is the cost price of the third category rice.

### Step 4: Set up the equation
From the previous step, we have:
\[ \text{Total Cost} = 150 + 400 + 2x \]
\[ \text{Total Cost} = 550 + 2x \]

Since we calculated the cost price of the mixture to be Rs. 210, we can set up the equation:
\[ 550 + 2x = 210 \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \):
\[ 2x = 210 - 550 \]
\[ 2x = -340 \]
\[ x = \frac{-340}{2} \]
\[ x = -170 \]

### Step 6: Correct the equation
It seems there was a misunderstanding in the cost price calculation. The cost price should be set equal to the total cost divided by the total quantity:
\[ 210 = \frac{550 + 2x}{5} \]

Multiply both sides by 5:
\[ 1050 = 550 + 2x \]
Now, solve for \( x \):
\[ 2x = 1050 - 550 \]
\[ 2x = 500 \]
\[ x = 250 \]

### Conclusion
The cost price of the third category rice is Rs. 250.