SOLUTION: Please help me solve this word problem? I am not sure where to start with this.
A certain brand of razor blades comes in packages of 6, 12, 24 blades, costing $2, $3, and $4 per
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Question 812011: Please help me solve this word problem? I am not sure where to start with this.
A certain brand of razor blades comes in packages of 6, 12, 24 blades, costing $2, $3, and $4 per package, respectively. A store sold 12 packages containing a total of 162 razors blades and took in $35. How many packages of each type were sold?
Found 2 solutions by richwmiller, DrBeeee:
Answer by richwmiller(17219) (Show Source): You can put this solution on YOUR website!
call the packages a b and c
costs 2a+3b+4c=35,
packages a+b+c=12,
blades 6a+12b+24c=162
a = 5, b = 3, c = 4
check
2*5+3*3+4*4=35
10+9+16=35
35=35
ok
check
5+3+4=12
12=12
ok
check
5*6+12*3+24*4=162
3+36+96=162
66+96=162
162=162
ok
Answer by DrBeeee(684) (Show Source): You can put this solution on YOUR website!
You said the magic word please. So here's your answer!
Let a = number of packages of 6 razor blades @ $2
Let b = number of packages of 12 razor blades @ $3
Let c = number of packages of 24 razor blades @ $4
This is three unknowns, so we need three independent (all different) equations to solve the problem. The problem is similar to a coin problem.
We get the three equations from the following three givens;
I) Total number of packages sold was 12 and
II) Total cost of the purchase was 35 and
III) Total number of razor blades sold was 162
From I we get
(1) a + b + c = 12
From II we get
(2) 2*a + 3*b + 4*c = 35
From III we get
(3) 6*a + 12*b + 24*c = 162
Do you understand these three equations? The first one is easy, it's the sum of each size package sold.
The second is the cost, e.g. if we bought 3 packages of 6 razor @ $2 we pay 6*2 or $12. Since "a" is the number of packages of 6 razors we pay 2*a. Likewise we pay 3*b for the packages containing 12 razors @ $3 and 4*c for the 24 razor size packages.
The third is the number of individual razor blades, "a" packages of 6 razors each is 6*a razor blades, 12*b for the 12 packages and 24*c for the 24 razor blade packages.
Now all we need to to do is solve the three equations for a, b and c.
We can do this many different ways. If you know how to solve matrices on your fancy calculator you can do that. I'm an old timer so I'll show you how to do it by "hand", by a process of elimination.
First we fix the three equation so that the coefficient of "a" is 2 for all three equation. We multiply both sides of (1) by 2 to get
(4) 2*a + 2*b + 2*c = 24
Keep (2) as is
(5) 2*a + 3*b + 4*c = 35
Divide (3) by 3 on each side to get
(6) 2*a + 4*b + 8*c = 54
Now you have each equation (4), (5) and (6) starting with 2*a. Now we start the elimination process.
Firstly subtract (4) from (6) to get
(7) 0 + 2*b + 6*c = 30
Then subtract (5) from (6) to get
(8) 0 + 1*b + 4*c = 19
Now we have eliminated the variable "a" from the problem and have two equations (7) and (8) to solve for the remaining variables b and c.
Continuing with the elimination process divide (7) by 2 to get
(9) b + 3*c = 15 and subtract it from (8) to get
(10) 0 + c = 19 - 15 or
(11) c = 4
Now use c = 4 in (9) to get
(12) b + 3*4 = 15 or
(13) b = 15 - 12 or
(14) b = 3
Now put b and c into (1) to get
(15) a + 3 + 4 = 12 or
(16) a = 12 - 7 or
(17) a = 5
ALWAYS check your answer.
We will use (1), (2) and (3) to check.
Is (5 + 3 + 4 = 12)?
Is (12 = 12)? Yes
Is (2*5 + 3*3 + 4*4 = 35)?
Is (10 + 9 + 16 = 35)?
Is (35 = 35)? Yes
Is (6*5 + 12*3 + 24*4 = 162)?
Is (30 + 36 + 96 = 162)?
Is (66 + 96 = 162)?
Is (162 = 162)? Yes
Answer: The store sold five 6-bladed packages, three of the 12-bladed packages and four of the 24-bladed packages.
PS The process of elimination is the same way linear equations are solved on a computer, not matrices.
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