Interesting sec maths question

Question: A rooster cost $5 and a hen cost $3. Chicks are sold at 3 for $1. A farmer bought 100 birds of these three types for $100. How many of each type of bird did he buy?

Hint: There are three possible sets of answers.

Solution:

Let R, H, C be the qty of rooster, hen and chicks respectively

5R+3H+1/3C = 100 (Eqn 1)
R+H+C= 100 (Eqn 2)

5X (Eqn 2)- (Eqn 1), 2H + 14/3 C = 400

H= 200- 7/3 C
R= -100+4/3 C

Since R> 0
-100+ 4/3 C> 0
C>75

Since H>0
200- 7/3 C > 0
C< 85.7

Since C must be a multiple of 3
C can be 78,81,84

So C = 78, R= 4, H=18
C= 81, R=8, H=11
C=84, R=12, H=4

the question say three types of birds. if any of them can be zero, it won’t be 3 types. else, we can have C= 75, R= 0, H=25 (2 types of birds only)

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