Grades 9-10 Video Solutions 2025-2025_9-10

Video Transcription

Problem nine, Luca has some dogs, some rabbits and some cats, eight of his pets are not dogs,  five of his pets are not rabbits, seven of his pets are not cats. How many pets does Luca have?  So to start out, let's assign variables so that the number of rabbits be are the number of cats  be C and the number of dogs be D. Now the problem gives us some information and so we can write  equations based on the information we're given. So we know that eight of his pets are not dogs,  now the pets that aren't dogs must be the rabbits and the cats, so we know that the number of  rabbits plus the number of cats is eight. Now five of his pets are not rabbits, the pets that  aren't rabbits are the dogs and the cats, so D plus C is five, and then the seven pets that  aren't cats are dogs and rabbits, so D plus R is seven. Now we can observe that these equations,  across all of them, we have each variable twice. So if we add them, then we're going to get two  times R plus C plus D is 20. Now this is good because what we're looking for is the number of  pets, which is R plus C plus D. The total number of pets is the total number of rabbits plus the  total number of cats plus the total number of dogs. That's why adding the equations and giving  us the same number of R's, C's, and D's is going to allow us to get this two times R plus C plus D  is 20, and then if we divide by two, we can get that R plus C plus D is 10, and so we can find  out that the answer is A without having to figure out the specific numbers of dogs, rabbits, and cats.

Video Summary

Luca has a total of 10 pets. The problem provides equations based on the number of pets that are not dogs, rabbits, or cats, respectively. By setting up equations for each situation and adding them together, the sum of all pet types (dogs, rabbits, and cats) equals twice the number of total pets. Dividing by two reveals the total number of pets, without specifying how many of each type, is 10.