There are three types of milk, Parag, Amul and Nestle. The ratio of fat to the non-fat contents in milk 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. 0.42916666666667

Answer

To solve the problem of finding the ratio of fat to non-fat contents when mixing three types of milk (Parag, Amul, and Nestle) in equal quantities, we will follow these steps:

### Step 1: Identify the ratios of fat to non-fat for each type of milk.
- For Parag: Fat : Non-fat = 4 : 5
- For Amul: Fat : Non-fat = 5 : 6
- For Nestle: Fat : Non-fat = 6 : 7

### Step 2: Convert the ratios into fractions.
- For Parag: Fat fraction = 4/(4+5) = 4/9, Non-fat fraction = 5/(4+5) = 5/9
- For Amul: Fat fraction = 5/(5+6) = 5/11, Non-fat fraction = 6/(5+6) = 6/11
- For Nestle: Fat fraction = 6/(6+7) = 6/13, Non-fat fraction = 7/(6+7) = 7/13

### Step 3: Find a common denominator for the fractions.
To combine these fractions, we need to find a common denominator. The denominators are 9, 11, and 13. The least common multiple (LCM) of these numbers is 1287.

### Step 4: Convert each fraction to have the common denominator.
- For Parag:
- Fat = 4/9 = (4 * 143)/(9 * 143) = 572/1287
- Non-fat = 5/9 = (5 * 143)/(9 * 143) = 715/1287
- For Amul:
- Fat = 5/11 = (5 * 117)/(11 * 117) = 585/1287
- Non-fat = 6/11 = (6 * 117)/(11 * 117) = 702/1287
- For Nestle:
- Fat = 6/13 = (6 * 99)/(13 * 99) = 594/1287
- Non-fat = 7/13 = (7 * 99)/(13 * 99) = 693/1287

### Step 5: Sum the fat and non-fat contents.
- Total Fat = 572 + 585 + 594 = 1751
- Total Non-fat = 715 + 702 + 693 = 2110

### Step 6: Write the final ratio of fat to non-fat.
The ratio of fat to non-fat in the mixture is:
Fat : Non-fat = 1751 : 2110

### Step 7: Simplify the ratio if possible.
To simplify the ratio, we can divide both numbers by their greatest common divisor (GCD). In this case, the GCD of 1751 and 2110 is 1, so the ratio remains:
Fat : Non-fat = 1751 : 2110

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

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