In a pet shop, rats cost $5$ dollars each, guppies cost $3$ dollars each, and crickets cost $10 $ cents each. $100$ animals are sold, and the total cost is $100$. How many rats, guppies, and crickets were sold?
Our math team coach has shown us the solution already, but her solution requires $7$ different variables and takes at least $10$ minutes for me to solve. The math team contest has $6$ diophantine equations problems of increasing difficulty. I only have $40$ minutes to complete these $6$ questions and I am not allowed to use a calculator.
What is the fastest or simplest way to solve this problem without a calculator?