Question

At her ranch, Georgia starts an animal shelter to save cats. After the first three days, she has 34 male cats and 13 female cats. She decides to only accept cats in male-female pairs from then on. In this problem, we find the largest number of pairs of cats she can accept to her shelter and still have at least 60% of her dogs be male.

(a) Let d be the number of pairs of cats she accepts. How many male cats will she have total? Female cats? Total cats?

(b) Write an inequality that means that the percentage of cats at her shelter that are male is at least 60%.

(c)Solve the inequality and determine the largest number of pairs of animal she can accept to her shelter and still have at least 60% of her dogs be male.

Answer 1

Explanation:

I know that both sides of the cats, males and females, receive the same # of new cats.

I write an equation, (not an inequality though):

3((34+2x)/5)=2((13+2x)/5)

From here on, I solve for x. I get x = -38. I know we can't have negative pairs and we're only bringing in more cats, and not taking any away. I add 38, and try to see what percentage of cats are male, and I get fairly close, getting to about 62.5% I think. Then, I divide 38 by 2, getting 19, and see what else I can get. I try to find a value between the 38 and lesser 19 so as to find a value which will give me a male cat percentage of 60%. I get about 57.5% with 19 plugged in, which is when I see that hey, the differences are the same. What I then do is find the average of 19 + 38, which is 57.5. I round that to 58, and I get the right answer.

I see that perhaps my solution wasn't really orthodox or anything, but with a little bit of trial and error and a bit of my creative thinking, I guess I managed to solve the problem. What worries me though is this is the first problem in the "questions" section of the inequalities section of AoPS, so I'm definitely struggling to deal with the concepts.