Write a system of equations to model the problem and solve the system:
Benjamin has nickels and dimes and his toy bank he has a total of 45 coins the total value of $3.60. How many coins of each kind does he have?
N + D = 45
5N + 10D = 360 ---> N + 2D = 72
Subtract them ---> D = 27
then in your head from the first equation, N = 18
So he has 27 dimes and 18 nickels
check:
27+18 = 45 , check!
27 dimes = $2.70
18 nickels = $0.90
total = $3.60 check!
To model this problem and solve it using a system of equations, let's define the following variables:
Let "n" represent the number of nickels.
Let "d" represent the number of dimes.
Since Benjamin has a total of 45 coins, we can write the first equation:
n + d = 45 (Equation 1)
Next, let's consider the value of the coins. Nickels have a value of $0.05 and dimes have a value of $0.10. The total value of the coins is $3.60, so the second equation can be written as:
0.05n + 0.10d = 3.60 (Equation 2)
Now we have a system of equations:
Equation 1: n + d = 45
Equation 2: 0.05n + 0.10d = 3.60
To solve this system, we can use the substitution or elimination method. Let's solve it using the substitution method:
First, solve Equation 1 for n:
n = 45 - d
Now substitute this expression for n in Equation 2:
0.05(45 - d) + 0.10d = 3.60
Distribute:
2.25 - 0.05d + 0.10d = 3.60
Combine like terms:
0.05d = 3.60 - 2.25
0.05d = 1.35
Divide both sides by 0.05:
d = 1.35 / 0.05
d = 27
Now that we have the value for d (the number of dimes), we can substitute it back into Equation 1 to find n:
n + 27 = 45
n = 45 - 27
n = 18
Therefore, Benjamin has 18 nickels and 27 dimes.