At first glance, this word problem seems not to contain enough information to solve, but once you start thinking about it and making a few simple assumptions, the answer becomes clear. The first thing to consider is what types of coins you're likely to find: Besides pennies, nickels, dimes, and quarters, will you find half-dollars and dollar coins?
If I had 550 coins in a machine worth $456.25, what would be the denomination of each coin?
First of all, can you reach the desired $456.25 without half-dollars or dollars? If you assume that all the coins in the machine are quarters, you get $.25 x 550 = $137.50. Since that's way short of your goal, you know that a larger denomination of coin is required. If all 550 coins are half-dollars, you end up with $.50 x 550 = $275 — still not enough! So you know that machine has some shiny Sacagawea dollars in it.
At this point, consider your options. Half-dollars are relatively uncommon, and few (if any) vending machines take them. So what happens if you assume that all your coins are quarters and dollars? With this assumption, you can create two equations.
If q is the number of quarters and d is the number of dollars, then you know that
q + d = 550
because there are 550 coins total. You can also set up an equation for the total dollar value of the coins:
(.25)q + (1)d = 456.25
Now you can solve the first equation for q and substitute it into the second equation:
q + d = 550 (subtract q from both sides)
d = 550 – q (substitute (550 – q) for d in the second equation)
(.25)q + 550 – q = 456.25 (now solve for q)
–.75q = –93.75
q = 125
That means that the machine has 125 quarters. How many dollars does it have? Just substitute in an earlier equation:
d = 550 – q
d = 550 – 125
d = 425
So 125 quarters and 425 dollars gives you 550 coins that total $456.25.
This is a correct answer, but this isn't the only answer! If you assume that a certain number of a particular coin is in the machine, you can use the same basic equations to figure it out. Not every assumption will work (you aren't likely to find half a quarter in a vending machine, and if you did, it wouldn't be worth anything), but if you try a few common numbers, you might hit on another answer.
For example, assume that the machine contains 5 dimes, but the remaining coins are all quarters and dollars. Those dimes would be represented mathematically as 5 times .10, so you know that
(.10)5 + (.25)q + (1)d = 456.25
And because five of the coins are accounted for (as dimes), the remaining 545 coins are all quarters and dollars, which means
q + d = 545
Go through the same process as above and you'll discover another, equally correct answer!