Let’s set up a system of equations to represent this situation.
We know:
total number of coins = 66
possible types of coins = quarter or nickel
value of each coin = $0.25 quarter and $0.05 nickel
total amount of money in the jar = $7.90
Let’s say number of quarters = Q and number of nickels = N
The first equation will only talk about the number of coins.
Some number of quarters and some number of nickels add up to 66 total coins:
Q + N = 66
The second equation will only use numbers related to the amount of money in the jar.
Some number of quarters at $0.25 each and some number of nickels at $0.05 each add up to $7.90 total money
0.25Q + 0.05N = 7.90
Now we have a system of equations that we can use to solve for Q and N. We can use either substitution or elimination. In this example I will use elimination.
Q + N = 66
0.25Q + 0.05N = 7.90
For this elimination, I am going to make the quarters disappear. I am going to take the entire top equation and multiply each term by 0.25 so that the quarters on the top matches the quarters on the bottom.
0.25Q = 0.25N = 66x0.25 —> 0.25Q + 0.25N = 16.50
Now the Q’s match on top and bottom and we can eliminate them.
0.25Q + 0.25N = 16.50
0.25Q + 0.05N = 7.90
To get the Q terms to cancel out, we would subtract them. This means we will also subtract the other matching pairs:
0.25Q - 0.25Q = 0 —> Q cancels out
0.25N - 0.05N = 0.20N
16.50-7.90 = 8.60
When we subtract the equations, we are left with
0.20N = 8.60 divide by 0.20 on both sides to get N by itself
N = 43 nickels
Now we can find the number of quarters by looking back at our total number of coins equation:
Q + N = 66 plug in 43 where N is
Q + 43 = 66 subtract 43 on both sides
Q = 23 quarters
The jar has 43 nickels and 23 quarters. :)
