Two types of tea costing 180per kg and 280 per kg should be miced in the ratio so that the mixture obtained, sold at 320per kg to earn a profit of 20% in

Options

A. `3:13`
B. `1:13`
C. `4:13`
D. `2:13`

Answer

To solve the problem of mixing two types of tea costing ₹180 per kg and ₹280 per kg to achieve a selling price of ₹320 per kg with a profit of 20%, we will use the method of allegation. Here’s a step-by-step solution:

### Step 1: Calculate the Cost Price (CP) of the Mixture
To find the CP of the mixture, we first need to determine the CP that corresponds to a selling price (SP) of ₹320 per kg with a profit of 20%.

The formula for selling price with profit is:
\[ SP = CP + \text{Profit} \]
Given that profit is 20% of CP, we can express this as:
\[ SP = CP + 0.20 \times CP = 1.20 \times CP \]

Now, substituting the given SP:
\[ 320 = 1.20 \times CP \]

To find CP, we rearrange the equation:
\[ CP = \frac{320}{1.20} = \frac{320 \times 100}{120} = \frac{32000}{120} = \frac{800}{3} \approx 266.67 \]

### Step 2: Set Up the Allegation Method
Now we will use the allegation method to find the ratio in which the two types of tea should be mixed.

We have:
- Cost of tea A (₹180 per kg)
- Cost of tea B (₹280 per kg)
- Cost of the mixture (CP) = ₹266.67 (or \(\frac{800}{3}\))

### Step 3: Calculate the Differences
We set up the differences as follows:
- Difference between tea A and the mixture:
\[ \text{Difference A} = 266.67 - 180 = 86.67 \]
- Difference between tea B and the mixture:
\[ \text{Difference B} = 280 - 266.67 = 13.33 \]

### Step 4: Determine the Ratio
The ratio of the quantities of tea A to tea B is the inverse of the differences calculated:
\[ \text{Ratio} = \frac{\text{Difference B}}{\text{Difference A}} = \frac{13.33}{86.67} \]

To simplify this ratio, we can multiply both sides by 100 to eliminate the decimals:
\[ \text{Ratio} = \frac{13.33 \times 100}{86.67 \times 100} = \frac{1333}{8667} \approx \frac{2}{13} \]

### Final Answer
Thus, the ratio in which the two types of tea should be mixed is:
\[ \text{Ratio of tea A to tea B} = 2 : 13 \]

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