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 or n + 2d = 72
so...
n+d = 45
n + 2d = 72
subtract them to eliminate the n , then solve for d
sub d back into the first and all is easy.
Let's start by setting up the system of equations.
Let's use the variables n and d to represent the number of nickels and dimes that Benjamin has, respectively.
The first equation represents the total number of coins Benjamin has:
n + d = 45
The second equation represents the total value of the coins:
0.05n + 0.10d = 3.60
Now, let's solve the system step-by-step.
Step 1: Solve the first equation for one variable (in this case, n) in terms of the other variable.
n = 45 - d
Step 2: Substitute the expression for n from step 1 into the second equation.
0.05(45 - d) + 0.10d = 3.60
Step 3: Simplify and solve for d.
2.25 - 0.05d + 0.10d = 3.60
0.05d = 3.60 - 2.25
0.05d = 1.35
d = 1.35 / 0.05
d = 27
Step 4: Substitute the value of d into the first equation and solve for n.
n + 27 = 45
n = 45 - 27
n = 18
Therefore, Benjamin has 18 nickels and 27 dimes.
To solve this problem, we can create a system of equations to represent the given information. Let's use the variables 'n' to represent the number of nickels and 'd' to represent the number of dimes that Benjamin has.
First, we know that Benjamin has a total of 45 coins. So the first equation is:
n + d = 45
Next, we know that the total value of his coins is $3.60. The value of a nickel is $0.05, and the value of a dime is $0.10. Hence, the second equation is:
0.05n + 0.10d = 3.60
Now we have a system of equations:
n + d = 45
0.05n + 0.10d = 3.60
To solve this system, we can use the method of substitution or elimination. Let's solve it using the substitution method.
From the first equation, we can rewrite it as:
n = 45 - d
Substitute this value of 'n' into the second equation:
0.05(45 - d) + 0.10d = 3.60
Next, distribute 0.05:
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, substitute this value of 'd' into the first equation to find 'n':
n + 27 = 45
n = 45 - 27
n = 18
Therefore, Benjamin has 18 nickels and 27 dimes.