Question

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

• A. 5:7:13
• B. 5:14:13
• C. 10:21:26
• D. 21:10:26

Hint:

When combining ratios, ensure that the total number is correctly divided by the greatest common divisor (GCD) to simplify the ratio.

Answer 1

The Correct Option is C

We are given that:
- 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 farm Q is 416.
- The ratio of hens:ducks:goats in farm Q is 5:14:13.
Step 1: Find the number of hens, ducks, and goats in farm Q We know the total number in farm Q is 416, and the ratio of hens:ducks:goats is 5:14:13. Let the number of hens, ducks, and goats in farm Q be represented by:
- Hens in Q: \( 5x \)
- Ducks in Q: \( 14x \)
- Goats in Q: \( 13x \)
Thus, the total is: \[ 5x + 14x + 13x = 416 \] \[ 32x = 416 \] \[ x = \frac{416}{32} = 13 \] So, the number of hens, ducks, and goats in farm Q are:
- Hens in Q: \( 5 \times 13 = 65 \)
- Ducks in Q: \( 14 \times 13 = 182 \)
- Goats in Q: \( 13 \times 13 = 169 \)
Step 2: Add these to the numbers in farm P
Now, we transfer all the animals from farm Q to farm P:
- New hens in P: \( 65 + 65 = 130 \) - New ducks in P: \( 91 + 182 = 273 \) - New goats in P: \( 169 + 169 = 338 \) Step 3: Find the new ratio The new ratio of hens:ducks:goats in farm P is: \[ 130 : 273 : 338 \] Simplifying this ratio by dividing each term by their greatest common divisor, which is 13: \[ \frac{130}{13} : \frac{273}{13} : \frac{338}{13} = 10 : 21 : 26 \] Thus, the new ratio of hens:ducks:goats in farm P is 10:21:26, corresponding to Option (C).
Final Answer: (C) 10:21:26