If you have fifty coins which have a total value of one dollar what are the coins and how many of each do you have?

Answer 1

There are two possible solutions: You could have one quarter, two dimes, two nickels and forty-five pennies, or you could have two dimes, eight nickels and forty pennies.

An easy way of approaching this problem is to start by imagining that you have fifty pennies. You have the right number of coins, but are fifty cents short. Instead of adding other coins, you replace pennies with them: replacing a penny with a nickel gains four cents, a dime gains nine, and a quarter gains twenty-four. You can't possibly use more than two quarters, so there are few cases to consider there: If you replace two pennies with quarters, you've gained forty-eight cents, so you only need two more; but any further replacement will give you too much. If you use one quarter, you need to make up twenty-six more cents in steps of four or nine; it's easy to see that two of each works. Finally, with no quarters, you need to gain fifty cents using increments of four or nine; this yields the second solution.