Question

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.

Answer 1

The answer is 24 screws, 6 bolts, and 12 washers.

s - the number of screws
b - the number of bolts
w - the number of washes

The cost of screws are $3.00 each: 3s
The cost of bolts are $2.00 each: 2b
The cost of washers are $1.0 each: 1.5w

A hardware store makes a mixed bag of 42 items using screws, bolts, and washers: s + b + w = 42
The mixture calls for four times as many screws than bolts: s = 4b
The total cost of the mixture is $102.00: 3s + 2b + 1.5w = 102

The system of three equations:
(i) s + b + w = 42
(ii) s = 4b
(iii) 3s + 2b + 1.5w = 102

Substitute s from (ii) equation into (i) equation and express it in the term of w:
4b + b + w = 42
5b + w = 42
w = 42 - 5b

Substitute s from (ii) equation and w from (i) equation into (iii) equation:
3 * 4b + 2b + 1.5(42 - 5b) = 102
12b + 2b + 63 - 7.5b = 102
6.5b + 63 = 102
6.5b = 102 - 63 
6.5b = 39
b = 39 : 6.5
b = 6

s = 4b = 4 * 6 = 24

w = 42 - 5b = 42 - 5 * 6 = 42 - 30 = 12