In what ratio must a person mix three kinds of wheat costing him 1.20, 1.44 and 1.74 per kg, so that the mixture may be worth 1.41 per kg?

Options

A. `1:2:3`
B. `4:5:7`
C. `11:77:7`
D. `13:7:9`

Answer

To solve the problem of mixing three kinds of wheat costing ₹1.20, ₹1.44, and ₹1.74 per kg to achieve a mixture worth ₹1.41 per kg, we will use the method of alligation. Here is a step-by-step solution:

### Step 1: Identify the costs
Let:
- Cost of first kind of wheat (C1) = ₹1.20
- Cost of second kind of wheat (C2) = ₹1.44
- Cost of third kind of wheat (C3) = ₹1.74
- Cost of the mixture (Cm) = ₹1.41

### Step 2: Calculate the differences using alligation
Using the alligation method, we calculate the differences between the costs of the wheat and the cost of the mixture:

1. For the first and second kinds of wheat:
- Difference between C2 and Cm:
\[ D1 = C2 - Cm = 1.44 - 1.41 = 0.03 \]
- Difference between C1 and Cm:
\[ D2 = Cm - C1 = 1.41 - 1.20 = 0.21 \]

### Step 3: Determine the ratio of the first and second kinds of wheat
The ratio of the quantities of the first kind of wheat (Q1) to the second kind of wheat (Q2) is given by the inverse of the differences calculated:
\[ \frac{Q1}{Q2} = \frac{D1}{D2} = \frac{0.03}{0.21} = \frac{1}{7} \]
Thus, the ratio of the first kind to the second kind is 1:7.

### Step 4: Calculate the differences for the first and third kinds of wheat
Now, we will consider the first kind of wheat and the third kind of wheat:
1. Difference between C3 and Cm:
\[ D3 = C3 - Cm = 1.74 - 1.41 = 0.33 \]
2. Difference between C1 and Cm:
\[ D4 = Cm - C1 = 1.41 - 1.20 = 0.21 \]

### Step 5: Determine the ratio of the first and third kinds of wheat
The ratio of the quantities of the first kind of wheat (Q1) to the third kind of wheat (Q3) is:
\[ \frac{Q1}{Q3} = \frac{D3}{D4} = \frac{0.33}{0.21} = \frac{11}{7} \]
Thus, the ratio of the first kind to the third kind is 11:7.

### Step 6: Combine the ratios
Now we have two ratios:
1. \( Q1 : Q2 = 1 : 7 \)
2. \( Q1 : Q3 = 11 : 7 \)

To find a common ratio for all three kinds of wheat, we can express \( Q2 \) and \( Q3 \) in terms of \( Q1 \):
- Let \( Q1 = 1x \)
- Then \( Q2 = 7x \)
- From \( Q1 : Q3 = 11 : 7 \), we can express \( Q3 \) as:
\[ Q3 = \frac{7}{11} \cdot Q1 = \frac{7}{11} \cdot 1x = \frac{7}{11}x \]

### Step 7: Find a common multiple
To combine these ratios, we can find a common multiple:
- The least common multiple of the denominators (1, 7, and \( \frac{7}{11} \)) can be taken as 77.

Thus, we can express the ratios as:
- \( Q1 = 11 \)
- \( Q2 = 77 \)
- \( Q3 = 49 \)

### Final Ratio
The final ratio of the three kinds of wheat is:
\[ Q1 : Q2 : Q3 = 11 : 77 : 49 \]

### Conclusion
The required ratio in which the person must mix the three kinds of wheat is **11 : 7 : 7**.

---