To solve this problem, we can set up a system of equations based on the given information.
Let's assume:
x = the number of 2 cent stamps
y = the number of 8 cent stamps
z = the number of 14 cent stamps
Based on the given information, we can create the following equations:
Equation 1: The number of 2 cent stamps is five more than twice the number of 8 cent stamps.
x = 2y + 5
Equation 2: The number of 14 cent stamps is three times the number of 8 cent stamps.
z = 3y
Equation 3: The total value of the stamps is $2.26.
2x + 8y + 14z = 226
Now we have a system of three equations. Let's solve them step-by-step.
Step 1: Substituting Equation 2 and Equation 3 into Equation 1
x = 2y + 5
z = 3y
Substituting z into Equation 1:
x = 2y + 5
14z = 42y
Dividing Equation 2 by 3:
z = 3y
Substituting z into the equation z = 42y:
14(3y) = 42y
42y = 42y
Therefore, we can conclude that Equation 1 is satisfied by the given information.
Step 2: Substituting Equation 1 and Equation 2 into Equation 3
Substituting x into the equation 2x + 8y + 14z = 226:
2(2y + 5) + 8y + 14(3y) = 226
4y + 10 + 8y + 42y = 226
54y + 10 = 226
54y = 216
y = 4
Now we know that the number of 8 cent stamps (y) is 4.
Step 3: Finding the value of x and z
Using Equation 1:
x = 2y + 5
x = 2(4) + 5
x = 8 + 5
x = 13
Using Equation 2:
z = 3y
z = 3(4)
z = 12
So, the number of stamps of each type in the collection is:
2 cent stamps: 13
8 cent stamps: 4
14 cent stamps: 12