There are three types of milk, Parag, Amul and Nestle. The ratio of fat to the non-fat contents in milk is 4 : 5, 5 : 6, 6 : 7 respectively. If all these three types of milk is mixed in equal quantity, the ratio of fat to the non-fat contents in the mixture will be :

Options

A. 1751 : 2110
B. 175 : 543
C. `3:5`
D. `10:18`

Answer

To solve the problem, we need to find the ratio of fat to non-fat contents in a mixture of three types of milk: Parag, Amul, and Nestle. Let's break down the solution step by step.

### Step 1: Understand the Ratios of Each Milk
- **Parag Milk**: The ratio of fat to non-fat is 4:5.
- **Amul Milk**: The ratio of fat to non-fat is 5:6.
- **Nestle Milk**: The ratio of fat to non-fat is 6:7.

### Step 2: Convert Ratios to Fat and Non-Fat Contents
For easier calculations, let's assume we take 1 liter of each type of milk.

1. **Parag**:
- Total parts = 4 + 5 = 9
- Fat = (4/9) liters
- Non-fat = (5/9) liters

2. **Amul**:
- Total parts = 5 + 6 = 11
- Fat = (5/11) liters
- Non-fat = (6/11) liters

3. **Nestle**:
- Total parts = 6 + 7 = 13
- Fat = (6/13) liters
- Non-fat = (7/13) liters

### Step 3: Calculate Total Fat and Non-Fat Contents in the Mixture
Now, we will find the total fat and non-fat contents when we mix equal quantities of each milk.

- **Total Fat**:
- From Parag: (4/9) liters
- From Amul: (5/11) liters
- From Nestle: (6/13) liters

To add these fractions, we need a common denominator. The least common multiple (LCM) of 9, 11, and 13 is 1287.

Calculating each term:
- Parag: \((4/9) = (4 \times 143)/(9 \times 143) = 572/1287\)
- Amul: \((5/11) = (5 \times 117)/(11 \times 117) = 585/1287\)
- Nestle: \((6/13) = (6 \times 99)/(13 \times 99) = 594/1287\)

Now, add these:
\[ \text{Total Fat} = \frac{572 + 585 + 594}{1287} = \frac{1751}{1287} \text{ liters} \]

- **Total Non-Fat**:
- From Parag: (5/9) liters
- From Amul: (6/11) liters
- From Nestle: (7/13) liters

Calculating each term:
- Parag: \((5/9) = (5 \times 143)/(9 \times 143) = 715/1287\)
- Amul: \((6/11) = (6 \times 117)/(11 \times 117) = 702/1287\)
- Nestle: \((7/13) = (7 \times 99)/(13 \times 99) = 693/1287\)

Now, add these:
\[ \text{Total Non-Fat} = \frac{715 + 702 + 693}{1287} = \frac{2110}{1287} \text{ liters} \]

### Step 4: Calculate the Ratio of Fat to Non-Fat in the Mixture
Now we have:
- Total Fat = \( \frac{1751}{1287} \)
- Total Non-Fat = \( \frac{2110}{1287} \)

The ratio of fat to non-fat is:
\[ \text{Ratio} = \frac{1751}{2110} \]

### Step 5: Simplify the Ratio
To simplify, we can divide both numbers by their greatest common divisor (GCD). In this case, the GCD is 1, so the ratio remains:
\[ \text{Ratio of Fat to Non-Fat} = 1751 : 2110 \]

### Final Answer
The ratio of fat to non-fat contents in the mixture is **1751 : 2110**.

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