Problem 195

Solution 1

Since each emu gets 2 treats and each chicken gets 4 treats, each of Karl’s 30 birds gets at least 2 treats .

If Karl begins by giving his 30 birds exactly 2 treats each, then Karl will have given out $30\times2=60$ of the treats.

Since Karl has 100 treats to hand out, then he has $100-60=40$ treats left to give.

However, each emu has already received their 2 treats (since all 30 birds were given 2 treats).

So the remaining 40 treats must be given to chickens.

Each chicken is to receive 4 treats and has already received 2 treats.

Therefore, each chicken must receive 2 more treats.

Since there are 40 treats remaining, and each chicken receives 2 of these treats, then there are $40\div2=20$ chickens.

(We may check that if there are 20 chickens, then there are $30-20=10$ emus, and Karl would then give out $4\times20+2\times10=100$ treats.)

Solution 2

Using a trial and error approach, if Karl had 5 emus and $30-5=25$ chickens, then he would need to hand out $5\times2+25\times4=110$ treats.

Since Karl hands out 100 treats, we know that Karl has more emus than 5 (and fewer chickens than 25).

We show this attempt and continue with this approach in the table below.

Number of Emus
Number of Chickens
Number of Emu Treats
Number of Chicken Treats
Total Number of Treats

5
$30-5=25$
$5\times2=10$
$25\times4=100$
$10+100=110$

7
$30-7=23$
$7\times2=14$
$23\times4=92$
$14+92=106$

10
$30-10=20$
$10\times2=20$
$20\times4=80$
$20+80=100$

Therefore, Karl has 20 chickens.