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?
If we let $S$ be the number of shirts, and $T$ be the number of ties, can you now state the amount spent on shirts and on ties, and what the sum of these two quantities must be?
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...
Yes, that is correct, but here is what I had in mind:
We know $(S,T)=(0,27)$ is one possible solution. As $\text{lcm}(30,70)=210$, we know then that we may add 3 shirts and subtract 7 ties. Let $n$ be the number of times we do this, and so we may state:
$(S,T)=(3n,27-7n)$
Now, minimizing $T$, we see that $n=3$ is the largest value of $n$ that allows $T$ to be non-negative, and so the desired solution is: