Here's an arithmetic approach.
3/4ths of the llamas were sold. This means (at least) two things:
1/4th of the llamas remain, and the original number of llamas must be a multiple of 4:
4, 8, 12, 16, 20, 24, …
1/3rd of the alpacas were sold. This means (at least) two things:
2/3rds of the alpacas remain, and the original number of alpacas must be a multiple of 3:
3, 6, 9, 12, 15, 18, …
We can discover patterns, by writing out specific cases and comparing the resulting numbers. For example, we know that the original number of alpacas is a multiple of 3, so let's see what happens if there were originally three alpacas (i.e., the first multiple of 3).
3 alpacas means 397 llamas because there were 400 animals originally. But 397 is not a multiple of 4 because all multiples of 4 are even numbers. (Dividing by 4 is the same as dividing by 2 twice.)
Okay, next case: 6 alpacas means 394 llamas. But 394 is not a multiple of 4 because 394÷4 has a remainder of 2.
Okay, next case: 9 alpacas means 391 llamas. 391 is odd, so it's not a multiple of 4.
Okay, next case: 12 alpacas means 388 llamas. Success! 388 is the 97th multiple of 4 because 388÷4 is 97 with no remainder.
Therefore, after the sale: 1/4th of 388 llamas (which is 97 animals) and 2/3rds of 12 alpacas (which is 8 animals) remain.
8 + 97 = 105
We need the remaining animals to total 125, so we keep going.
Next case: 15 alpacas means 385 llamas. 385 is odd, so it's not a multiple of 4.
Next case: 18 alpacas means 382 llamas. But 382 is not a multiple of 4 because 382÷4 has a remainder of 2.
If you have been organized, you might notice a pattern in the results, so far.
Code:
[FONT=courier new]Alpacas Llamas After Sale
------- ------ ----------
3 397
6 394
9 391
12 388 8+97 = 105
15 385
18 382
[/FONT]
We need the original number of llamas to be even. That only happens when the number of alpacas is even. Therefore, we can skip all the cases where the multiple of 3 (i.e., original alpacas) is odd. Keep going. Be organized.
Code:
[FONT=courier new]Alpacas Llamas After Sale
------- ------ ----------
6 394
12 388 8+97 = 105
18 382
24 376 16+94 = 110
30 370
36 364 24+91 = 115
[/FONT]
Two more patterns have emerged.
1) The number of llamas is a multiple of 4 only when the number of alpacas is a multiple of 12.
2) Each new multiple of 12 alpacas increases the number of remaining animals by 5.
I hope that the patterns I have shown help you to answer the exercise.