Find the proportion in which three types of sugar at 12, 14 and 20 may be mixed so as to obtain a mixture worth 15 per kg?

Options

A. `15:5:6`
B. `5:15:6`
C. `3:12:16`
D. `4:12:15`

Answer

To solve the problem of finding the proportion in which three types of sugar priced at 12, 14, and 20 per kg can be mixed to obtain a mixture worth 15 per kg, we can use the method of alligation. Here’s a step-by-step solution:

### Step 1: Identify the Prices
We have three types of sugar with the following prices:
- Sugar A: 12 per kg
- Sugar B: 14 per kg
- Sugar C: 20 per kg

### Step 2: Determine the Price of the Mixture
The target price of the mixture is 15 per kg.

### Step 3: Apply Alligation
We will use the alligation method to find the proportions. We calculate the differences between the prices of each type of sugar and the price of the mixture.

1. **Difference between Sugar A and Mixture:**
\[ 15 - 12 = 3 \]

2. **Difference between Sugar B and Mixture:**
\[ 15 - 14 = 1 \]

3. **Difference between Sugar C and Mixture:**
\[ 20 - 15 = 5 \]

### Step 4: Set Up the Alligation
Now we set up the alligation:

- For Sugar A (12): 3
- For Sugar B (14): 1
- For Sugar C (20): 5

### Step 5: Write the Proportions
The proportions of the sugars can be determined from the differences:

- The proportion of Sugar A to Sugar C is given by the differences:
\[ \text{Ratio of Sugar A to Sugar C} = 5 : 3 \]

- The proportion of Sugar B to Sugar C is:
\[ \text{Ratio of Sugar B to Sugar C} = 5 : 1 \]

### Step 6: Combine the Ratios
To find the overall ratio of the three types of sugar, we combine the ratios:

- Let the proportion of Sugar A be \(x\), Sugar B be \(y\), and Sugar C be \(z\).
- From the alligation, we have:
\[ x : y : z = 3 : 1 : 5 \]

### Step 7: Final Proportions
Thus, the final proportions in which the three types of sugar should be mixed to obtain a mixture worth 15 per kg is:
\[ 3 : 1 : 5 \]