A stamp collector bought 295 stamps for $87.75. The purchase included 5 cent, 25 cent, and 40 cent stamps. How many of each type stamp was purchased?
If there are x,y,z stamps of each price, we know that
x+y+z = 295
5x+25y+40z = 8775
There is no unique solution...
To solve this problem, we can set up a system of equations based on the information given.
Let's assume that the number of 5 cent stamps is x, the number of 25 cent stamps is y, and the number of 40 cent stamps is z.
Based on the given information, we have three equations:
1) The total number of stamps is 295: x + y + z = 295.
2) The total cost of the stamps is $87.75: 0.05x + 0.25y + 0.40z = 87.75.
3) We also know that there are no negative numbers of stamps: x, y, and z are all non-negative integers.
Now, we need to solve this system of equations.
One approach is to use trial and error or wishful thinking to find a combination of stamp quantities that satisfies the given conditions. However, this is not the most efficient method.
Instead, let's use a more systematic approach called substitution.
First, we can isolate one variable from one equation and substitute it into the other equation(s).
From the first equation, we can isolate x: x = 295 - y - z.
Now, substitute this value of x into the second equation:
0.05(295 - y - z) + 0.25y + 0.40z = 87.75.
Simplifying this equation:
14.75 - 0.05y - 0.05z + 0.25y + 0.40z = 87.75.
Combining like terms:
0.20y + 0.35z = 73.
Next, we can isolate another variable, such as y, from this equation and substitute it into the remaining equation(s).
From the equation above, we can isolate y: y = (73 - 0.35z) / 0.20.
Now, we substitute this value of y into the first equation:
(295 - (73 - 0.35z) / 0.20) + z = 295.
Simplifying this equation:
295 - (73 - 0.35z) / 0.20 + z = 295.
Removing the denominators:
295 - (73 - 0.35z) + 0.20z = 295.
Simplifying this equation:
222 + 0.15z = 295.
Subtracting 222 from both sides:
0.15z = 73.
Dividing both sides by 0.15:
z = 73 / 0.15.
Calculating z:
z = 486.67.
Since z cannot be a decimal value (as stated in the problem), this means our assumption or calculations must be incorrect.
To fix this error, review the calculations for any mistakes or re-evaluate the problem statement.