#5 You Be The Judge

Otis

Otis

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Seven breeds of dogs will be judged at a local dog show. A total of 121 dogs participate. There are five small-breed groups and two large-breed groups. The fewest number of dogs in any group is 15, and the largest number of dogs in any group is 20. Only the large-breed groups contain the same number of dogs. How many dogs are in each large-breed group?

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Otis said:
Seven breeds of dogs will be judged at a local dog show. A total of 121 dogs participate. There are five small-breed groups and two large-breed groups. The fewest number of dogs in any group is 15, and the largest number of dogs in any group is 20. Only the large-breed groups contain the same number of dogs. How many dogs are in each large-breed group?

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This is my dog-patch - 42
 
[MATH]s_1,\ s_2,\ s_3,\ s_4, \text { and } s_5 \text { are the numbers of dogs in each of the small-dog groups.}[/MATH]
[MATH]\text {min}(s_1 \text { through } s_5) = 15 \text { and max}(s_1 \text { through } s_5 ) = 20.[/MATH]
[MATH]1 \le k \le 5 \implies s_k \in \mathbb Z.[/MATH]
[MATH]1 \le j \le 5 \land 1 \le k \le 5 \land j \ne k \implies s_j \ne s_k.[/MATH]
[MATH]x \text { is the number of dogs in each of the 2 large-dog groups.}[/MATH]
[MATH]x \in \mathbb Z \land 15 \le x \le 20.[/MATH]
[MATH]2x + \sum_{j=1}^5 s_j = 121 \implies \sum_{j=1}^5 s_j \text { is odd.}[/MATH]
[MATH]15 + 16 + 17 + 18 + 19 = 85, \text { odd}.[/MATH]
[MATH]16 + 17 + 18 + 19 + 20 = 90, \text { even}.[/MATH]
[MATH]\therefore 2x = 121 - 85 = 36 \implies x = 18.[/MATH]
I did not need all that apparatus, but I did not know that to start.
 
Making a simple guess, suppose the numbers in the groups are 15, 16, 17, 18, 19; 20, 20. This totals 125, which is 4 too many. We need to change a 20 to 16. So the numbers must be 15, 17, 18, 19, 20; 16, 16. This fits the requirements; so the large-breed groups each have 16.
 
[MATH]2\cdot 16 + 15 + 17 + 18 + 19 + 20 = 121[/MATH]16 dogs in the large group
 
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