https://file.aiquickdraw.com/mathbot/file/1726305468030nub1lqhe.jpg

Let's break down the problem based on the image.

**Given:**
- Farm P:
  - Hens: 65
  - Ducks: 91
  - Goats: 169
- Farm Q:
  - The total number of hens, ducks, and goats is 416.
  - The ratio of hens:ducks:goats in farm Q is 5:14:13.
  
All the hens, ducks, and goats from farm Q are transferred to farm P.

**Objective:**
Determine the new ratio of hens:ducks:goats in farm P after the transfer.

### Step 1: Calculate the number of hens, ducks, and goats in farm Q.
The ratio of hens:ducks:goats in farm Q is 5:14:13, and the total number of animals is 416. So, let the number of hens in farm Q be \( 5x \), ducks be \( 14x \), and goats be \( 13x \).

\[
5x + 14x + 13x = 416
\]
\[
32x = 416
\]
\[
x = \frac{416}{32} = 13
\]

Now, we can calculate the number of hens, ducks, and goats in farm Q:
- Hens in farm Q: \( 5x = 5 \times 13 = 65 \)
- Ducks in farm Q: \( 14x = 14 \times 13 = 182 \)
- Goats in farm Q: \( 13x = 13 \times 13 = 169 \)

### Step 2: Add the animals from farm Q to farm P.
Now, add the numbers of hens, ducks, and goats from farm Q to those in farm P:

- New number of hens in farm P: \( 65 + 65 = 130 \)
- New number of ducks in farm P: \( 91 + 182 = 273 \)
- New number of goats in farm P: \( 169 + 169 = 338 \)

### Step 3: Calculate the new ratio of hens:ducks:goats in farm P.
We need to simplify the ratio \( 130:273:338 \).

Find the greatest common divisor (GCD) of 130, 273, and 338 to simplify this ratio. 

I'll calculate the GCD and simplify the ratio.

The new ratio of hens:ducks:goats in farm P is \( 10:21:26 \).

Thus, the correct answer is option **(C)**.

Do you want more details or have any questions?

Here are some related questions for practice:
1. What would the new ratio be if the total number of animals in farm Q was 500 instead of 416?
2. How would the answer change if the ratio in farm Q was \( 6:12:15 \)?
3. If farm Q had 10% more ducks, what would the new ratio in farm P be?
4. How can you determine if a ratio can be simplified further?
5. What would happen to the ratio if no hens were transferred from farm Q?

**Tip:** When working with ratios, always look for the greatest common divisor (GCD) to simplify them to their lowest terms.

The number of hens, ducks, and goats in farm P are 65, 91, and 169, respectively. The total number of hens, ducks, and goats in a nearby farm Q is 416. The ratio of hens:ducks:goats in farm Q is 5:14:13. All the hens, ducks, and goats are sent from farm Q to farm P. The new ratio of hens:ducks:goats in farm P is?

Learn how to calculate the new ratio of hens, ducks, and goats in farm P after transferring animals from farm Q. The original numbers in farm P were 65 hens, 91 ducks, and 169 goats. The ratio of animals in farm Q was 5:14:13, with a total of 416 animals. Detailed step-by-step solution included for simplifying the new ratio.

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