He sold 1/4 of his chickens so that he has 3/4 of his chickens left, which ends up being 5 units. However, 5 is not divisible by 3, so we convert the ratio 5 : 32 into 15 : 96 by multiplying both 5 and 32 by 3.

Let the number of chickens at the end be 15 units and the number of ducks at the end be 96 units.

At the end,
Chicken: 15 units
Duck: 96 units

3/4 of chickens --> 15 units
1/4 of chickens --> 5 units
4/4 of chickens --> 20 units

At the start,
Chicken: 20 units
Duck: 96 units - 368 whole ducks

This counts to a total of 560 animals, so

20 units + 96 units - 368 --> 560
116 units - 368 --> 560
116 units --> 560 + 368
116 units --> 928
1 unit --> 928 / 116 = 8

Number of chickens at first
= 20 units
= 20 x 8
= 160

Check:

160 chickens and 400 ducks at the start
120 chickens and 768 ducks at the end

120 : 768

Divide both sides by 24

5 : 32

So yes, there are 160 chickens at first.

25% = ¼

Chickens he had left as a fraction of original = 1 - ¼ = ¾

¾ is 3 units out of 4.

Ratio = 5 : 32

Change the number of units of chickens for both numerator and the ratio's to be the same, since the absolute number of chickens is the same.

¾ = 15/20

5 : 32 = 15 : 96

This means he had 20 units of chickens at first. He sold 5 units leaving him with 15 units of chickens. From the ratio, we can tell he has 96 units of ducks in the end.

Now, let's say he didn't sell that 25% of chickens but still bought the ducks.He would have 20 units of chickens and 96 units of ducks, a total of 96 + 20 = 116 units.

The number of birds would increase by 368 from 560 since he's increasing ducks but not selling chickens.

368 + 560 = 928 birds

928 birds would = 116 units

1 unit = 928 birds ÷ 116 = 8 birds

Since there are 20 units of chickens at first, chickens at first
= 8 x 20
160

Thank you Eric and J for the detailed explanations