karla has $4.60 in nickels and quarters. she has a total of 36 coins. How many nickels and how many quarters does she have?
This is a "system of equations" problem. You have two unknown quantities (number of nickels and number of quarters) so you need to write two equations to solve for the two unknowns. One equation has to do with the number of coins, and the other one has to do with the value of the coins.
Let N be the number of nickels and Q be the number of quarters.
For the first equation: N + Q = 36 because she has a total of 36 coins.
For the other equation, multiply the number of each type of coin by the value of that type of coin, and then add those products together:
5N + 25Q = 460
Take the first equation and solve for either N or Q. I will solve for N by subtracting Q from both sides:
N + Q - Q = 36 - Q
So N = 36 - Q.
I can now substitute (36-Q) in place of N in the other equation:
5N + 25Q = 460
5(36-Q) + 25Q = 460
Distribute the 5 into the parentheses:
180 - 5Q + 25Q = 460
Combine like terms:
180 + 20Q = 460
Subtract 180 from both sides:
180 + 20Q - 180 = 460 - 180
So: 20Q = 280
Divide both sides by 20:
20Q/20 = 280/20
Q = 14: There are 14 quarters. Now substitute 14 for Q in the first equation:
N + Q = 36
N + 14 = 36
Subtract 14 from both sides:
N + 14 - 14 = 36 - 14
N = 22: There are 22 nickels.
We can check the result by verifying that 22 nickels = $1.10 and 14 quarters = $3.50, so we do get a total of $4.60.