The Price of each shirt is 70 dollars and the price of each tie is 30 dollars. He spent a total amount of exactly 810 dollars and bought the maximum number of shirts. What is the ratio of the number of Shirt to Ties?
Yes, it's 70*S + 30*T = 810. But the amounts spent on shirt and ties separately are unknown. The other parameter which might be helpful here is that, with this amount of 810, a maximum number of shirts have been purchased and rest have been spent for purchasing ties.
Now, we normally need two equations when we have two unknowns to get a solution, but in this case we are restricting the two variables to non-negative integers. This is what is called a Diophantine equation.
While there are more sophisticated approaches, I would simply observe that 810 is divisible by 30, and so one possible solution is 0 shirts and 27 ties. However, we are told the number of shirts is the maximum allowed, so we can look at adding a certain number of shirts while subtracting a certain number of ties. The cost of the number of shirts added must be equal to the cost of the ties subtracted. I would look for the LCM of 30 and 70 to find this cost...