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 hens, ducks and goats are sent from farm Q to farm P. The new ratio of hens: ducks: goats in farm P is _____

The correct option is 10:21:26.

Let us break down the solution step-by-step to understand how to arrive at the new ratio of hens, ducks, and goats in farm P after adding the animals from farm Q.

Step 1: Identify the initial number of animals in Farm P
Initially, Farm P has:
Number of hens in P = 65
Number of ducks in P = 91
Number of goats in P = 169

Step 2: Calculate the number of each type of animal in Farm Q
The total number of animals in Farm Q is 416, and they are distributed in the ratio of hens : ducks : goats = 5 : 14 : 13.
First, find the sum of the ratio parts:
Sum of ratio parts = 5 + 14 + 13 = 32
Now, calculate the number of each animal in Farm Q:
Number of hens in Q = $\frac{5}{32}\times 416$ = 5 × 13 = 65
Number of ducks in Q = $\frac{14}{32}\times 416$ = 14 × 13 = 182
Number of goats in Q = $\frac{13}{32}\times 416$ = 13 × 13 = 169

Step 3: Calculate the new total number of animals in Farm P
When all the animals from Farm Q are sent to Farm P, the new counts in Farm P become:
New number of hens = Hens in P + Hens in Q = 65 + 65 = 130
New number of ducks = Ducks in P + Ducks in Q = 91 + 182 = 273
New number of goats = Goats in P + Goats in Q = 169 + 169 = 338

Step 4: Find the new ratio of hens : ducks : goats in Farm P
The new ratio is:
New Ratio = 130 : 273 : 338
To simplify this ratio, we find the greatest common divisor (GCD) of 130, 273, and 338.
Let us check for divisibility by 13:
130 ÷ 13 = 10
273 ÷ 13 = 21
338 ÷ 13 = 26
Since 10, 21, and 26 have no common factors other than 1, the ratio is in its simplest form.
Therefore, the new ratio of hens : ducks : goats in Farm P is 10:21:26.