This problem involves calculating the new ratio of animals in a farm after a transfer from another farm. We need to determine the initial animal counts, calculate the number of animals transferred, and then find the combined totals to establish the final ratio.

Farm P: Initial Animal Counts

Initially, Farm P has the following number of animals:

• Hens: 65
• Ducks: 91
• Goats: 169

Farm Q: Determining Animal Numbers

Farm Q has a total of 416 animals (Hens, Ducks, and Goats) with their ratio given as 5 : 14 : 13. To find the exact number of each type of animal in Farm Q, we first sum the parts of the given ratio:

\(\text{Sum of ratio parts} = 5 + 14 + 13 = 32\)

Now, we divide the total number of animals in Farm Q by the sum of the ratio parts to find the value of one ratio unit:

\(\text{Value of one ratio unit} = \frac{\text{Total animals in Farm Q}}{\text{Sum of ratio parts}} = \frac{416}{32} = 13\)

Using this value, we can calculate the number of each type of animal in Farm Q:

• Hens in Farm Q: \(5 \times 13 = 65\)
• Ducks in Farm Q: \(14 \times 13 = 182\)
• Goats in Farm Q: \(13 \times 13 = 169\)

Let's verify the total for Farm Q:

\(65 (\text{Hens}) + 182 (\text{Ducks}) + 169 (\text{Goats}) = 416\)

This matches the total given for Farm Q.

Transferring Animals: Farm Q to Farm P

All the hens, ducks, and goats from Farm Q are sent to Farm P. This means we need to add the number of animals from Farm Q to the initial number of animals in Farm P for each category.

Farm P: New Total Animal Counts

Now, let's calculate the new total number of Hens, Ducks, and Goats in Farm P after the transfer from Farm Q. We add the animals transferred from Farm Q to the initial count in Farm P for each category.

So, the new number of animals in Farm P is: Hens = 130, Ducks = 273, Goats = 338.

Farm P: Simplifying the New Ratio

The new ratio of Hens : Ducks : Goats in Farm P is \(130 : 273 : 338\). To simplify this ratio, we need to find the greatest common divisor (GCD) of these three numbers.

Let's look for common factors. We know that the animals in Farm Q were multiples of 13. Let's check if 13 is a common factor for the new totals in Farm P:

• For Hens: \(130 \div 13 = 10\)
• For Ducks: \(273 \div 13 = 21\)
• For Goats: \(338 \div 13 = 26\)

Since all three numbers are perfectly divisible by 13, the greatest common divisor (GCD) is 13.

Dividing each part of the ratio by 13, we get the simplified new ratio:

\(130 \div 13 : 273 \div 13 : 338 \div 13\)

\(10 : 21 : 26\)

Therefore, the new ratio of hens : ducks : goats in Farm P is 10 : 21 : 26.