In what ratio must a grocer mix two varieties of teas at Rs. 60 a kg, and Rs. 65 a kg, so that by selling the mixture at Rs. 68.20 a kg, he may gain 10% ?

Options

A. `3:2`
B. `3:4`
C. `3:5`
D. `4:5`

Answer

To solve the problem of finding the ratio in which a grocer must mix two varieties of teas priced at Rs. 60 per kg and Rs. 65 per kg, so that by selling the mixture at Rs. 68.20 per kg, he gains a profit of 10%, we can follow these steps:

### Step 1: Calculate the Cost Price (CP) of the Mixture
To find the cost price of the mixture, we first need to determine the effective cost price that allows for a 10% profit when selling at Rs. 68.20 per kg.

The formula for profit percentage is:
\[ \text{Profit Percentage} = \frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}} \times 100 \]

Given:
- Selling Price (SP) = Rs. 68.20
- Profit Percentage = 10%

We can rearrange the formula to find the Cost Price (CP):
\[ 10 = \frac{68.20 - CP}{CP} \times 100 \]

### Step 2: Rearranging the Equation
Rearranging the equation gives:
\[ 10 \cdot CP = 68.20 - CP \]
\[ 10 \cdot CP + CP = 68.20 \]
\[ 11 \cdot CP = 68.20 \]

### Step 3: Solve for CP
Now, we can solve for CP:
\[ CP = \frac{68.20}{11} = 6.20 \]
Thus, the cost price of the mixture is Rs. 62 per kg.

### Step 4: Apply the Allegation Method
Now, we will use the allegation method to find the ratio in which the two types of teas should be mixed.

1. **Type 1 Tea (CP = Rs. 60)**
2. **Type 2 Tea (CP = Rs. 65)**
3. **Mixture CP = Rs. 62**

Using the allegation method:
- The difference between the cost price of Type 1 and the mixture:
\[ 65 - 62 = 3 \]
- The difference between the cost price of Type 2 and the mixture:
\[ 62 - 60 = 2 \]

### Step 5: Determine the Ratio
The ratio of the quantities of Type 1 and Type 2 teas is given by the differences calculated:
\[ \text{Ratio} = \frac{3}{2} \]

### Conclusion
The grocer must mix the two varieties of teas in the ratio of **3:2**.