In what ratio wheat costing Rs. 17/kg be mixed with another wheat costing Rs. 12/kg to get a mixture costing Rs. 13.5/ kg?

Options

A. `1:2`
B. `2:5`
C. `3:7`
D. `4:9`

Answer

To solve the problem of mixing two types of wheat at different costs to achieve a desired average cost, we can use the method of alligation. Here’s a step-by-step solution:

### Step 1: Identify the Costs
We have:
- Cost of Wheat A (higher price) = Rs. 17/kg
- Cost of Wheat B (lower price) = Rs. 12/kg
- Cost of the mixture (average price) = Rs. 13.5/kg

### Step 2: Apply the Alligation Formula
The alligation formula states that the ratio of the quantities of two ingredients can be found using the formula:

\[ \text{Ratio} = \frac{\text{Cost of Higher Price} - \text{Average Price}}{\text{Average Price} - \text{Cost of Lower Price}} \]

### Step 3: Substitute the Values
Substituting the values into the formula:

- Higher Price = 17
- Lower Price = 12
- Average Price = 13.5

Now, we can calculate:

\[ \text{Ratio} = \frac{17 - 13.5}{13.5 - 12} \]

### Step 4: Calculate the Differences
Calculate the differences:

- \(17 - 13.5 = 3.5\)
- \(13.5 - 12 = 1.5\)

### Step 5: Form the Ratio
Now, we can form the ratio:

\[ \text{Ratio} = \frac{3.5}{1.5} \]

### Step 6: Simplify the Ratio
To simplify the ratio, we can multiply both the numerator and the denominator by 10 to eliminate the decimal:

\[ \text{Ratio} = \frac{35}{15} \]

Now, we can simplify this further:

\[ \text{Ratio} = \frac{7}{3} \]

### Step 7: Final Ratio
Thus, the required ratio in which the two types of wheat should be mixed is:

\[ \text{Ratio} = 7:3 \]

### Conclusion
The wheat costing Rs. 17/kg should be mixed with the wheat costing Rs. 12/kg in the ratio of 7:3 to achieve a mixture costing Rs. 13.5/kg.

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