A hardware store makes a mixed bag of 42 items using screws, bolts, and washers. The cost of screws are $3.00 each, bolts cost $2.00 each, and washers are $1.50 each. The mixture calls for four times as many screws than bolts. The total cost of the mixture is $102.00. How much of each item did the store use?
24 screws 6 bolts and 12 washers
To solve this problem, we need to set up a system of equations.
Let's use the variables:
s = number of screws
b = number of bolts
w = number of washers
We are given the following information:
1) There are 42 total items in the mixed bag:
s + b + w = 42
2) The cost of the screws, bolts, and washers combined is $102.00:
3s + 2b + 1.5w = 102
3) The mixture requires four times as many screws as bolts:
s = 4b
Now, we can solve the system of equations to find the values of s, b, and w.
First, substitute the value of s from equation 3 into equations 1 and 2:
4b + b + w = 42
3(4b) + 2b + 1.5w = 102
Simplifying equation 1:
5b + w = 42
Simplifying equation 2:
12b + 2b + 1.5w = 102
14b + 1.5w = 102
Now, we have a system of two equations and two variables:
5b + w = 42
14b + 1.5w = 102
We can solve this system of equations using any method, such as substitution or elimination.
Let's use the elimination method to solve for b and w:
Multiply the first equation by -1.5, then add the two equations together:
-7.5b - 1.5w + 14b + 1.5w = -63 + 102
6.5b = 39
b = 39 / 6.5
b = 6
Substitute the value of b back into equation 1 to solve for w:
5(6) + w = 42
30 + w = 42
w = 42 - 30
w = 12
Now that we know the value of b and w, we can substitute them back into equation 3 to solve for s:
s = 4b
s = 4(6)
s = 24
Therefore, the store used 24 screws, 6 bolts, and 12 washers to make the mixed bag of 42 items.