You've got:
a = number of checks worth 10 bucks
b = number of checks worth 50 bucks
You also have 10a + 50b = 1500. "Cut" a zero on each side and you get:
a + 5b = 150.
Now, you are told that a number of 7 checks were cashed during the trip and the rest were lost. You are then expected to "minimize" the damage. Since the number of $10 checks cashed was one more or one less than the number of $ 50 checks cashed, you will pick the situation when there were more $50 checks cashed than $10 checks. This is because you want to get the most value out of the checks cashed, so you pick the more valuable one.
So, we get that 4 checks of $50 were cashed and 3 checks of $10 were cashed, or 4*50 + 3*10 = 200 + 30 = $230. The minimum possible value will be 1500 - 230 = 1270.

Even if we already have the answer, you can double-check to see that it is indeed correct: if 4 $10 checks and 3 $50 checks were cashed, then the guy cashed 40 + 150 = 190. This means that the poor tourist lost 1500 - 190 = 1310, which is greater than 1270.