Question

gabby's piggy bank contains dimes and nickels worth $6.40. If she has 84 coins in all, how many of each does she have?

Answer 1

If we let Q be the number of quarters and N be number of nickels, this is the formula we would use:
[1] 25Q + 5N = 635 (using cents instead of dollars)


We also know something about Q and N.

[2] Q + N = 51


So multiply both sides of the second equation by 5. The reason we do this is because we're going to need to cancel out one of the variables (either Q or N) to solve for the other one. If we multiply times 5, we'll have 5N in both equations.
5Q + 5N = 255

Now we have two equations, and the second one can be subtracted from the first.

25Q + 5N = 635
-(5Q + 5N) = -255
---------------------
20Q = 380
so
Q = 19

There are 19 quarters.

And since there are 51 coins in all, 51-19 = 32 nickels.