SOLUTION: How much coffee costing $6.00 per pound should be mixed with 3 pounds of coffee costing $4.00 per pound to create a mixture costing $4.50 per pound? I tried every way I could t

Algebra ->  Customizable Word Problem Solvers  -> Mixtures -> SOLUTION: How much coffee costing $6.00 per pound should be mixed with 3 pounds of coffee costing $4.00 per pound to create a mixture costing $4.50 per pound? I tried every way I could t      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 574679: How much coffee costing $6.00 per pound should be mixed with 3 pounds of coffee costing $4.00 per pound to create a mixture costing $4.50 per pound?
I tried every way I could think of to find the answer but I can`t seem to figure it out. Please help and thank you in advance.
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Let x represent the number of pounds of $6 coffee to be added to form a mix of the two coffees.
.
The value added to the mixture by the $6 coffee will be $6 times the number of pounds of the $6 coffee. This value can be written as 6*x.
.
The problem tells you that there are 3 pounds of coffee costing $4 per pound. Therefore, the value of the $4 coffee in the mixture will be 3 pounds times $4 per pound. This product is $12.
.
So the total value of the coffee will be 6*x + $12
.
The number of pounds of coffee in the mixture will be the sum of the number of pounds of the two types of coffee. The problem tells you that there are 3 pounds of the cheaper coffee, and we plan to add x pounds of the more expensive coffee. So when we are done mixing the two types of coffee, the total weight of the mixture will be 3 + x pounds.
.
Then the cost per pound of the mixture will be the total value of the coffee in the mixture divided by the total weight of the mixture. This will be the value 6x + 12 divided by the total weight 3 + x. The problem tells us that this cost per pound is to equal $4.50. Therefore, we can write the equation:
.
%286%2Ax%2B12%29%2F%283%2Bx%29+=+4.50
.
To get rid of the denominator, multiply both sides by 3 + x as follows:
.
%28%283+%2B+x%29%2A%286%2Ax%2B12%29%29%2F%283%2Bx%29+=+4.50%2A%283%2Bx%29
.
On the left side the 3 + x in the numerator cancels with the denominator. On the right side the 4.50 multiplies each of the terms in the parentheses. The result is:
.
%28cross%283+%2B+x%29%29%2A%286%2Ax%2B12%29%2F%28cross%283%2Bx%29%29+=+13.50%2B4.50%2Ax
.
and this simplifies to:
.
6%2Ax+%2B+12+=+13.50+%2B+4.50%2Ax
.
Get rid of the 4.50*x on the right side by subtracting 4.50*x from both sides. Then get rid of the 12 on the left side by subtracting 12 from both sides. These two subtractions will result in:
.
%286+-4.50%29%2Ax+=+13.50+-+12
.
This simplifies to:
.
1.50%2Ax+=+1.50
.
Finally we can solve for x by dividing both sides by 1.50 and we get:
.
x+=+1
.
This tells us that we should add 1 pound of coffee costing $6 per pound to the 3 pounds of coffee costing $4 per pound and we will get a mixture that should sell for $4.50 per pound.
.
As a check, we can see that the mixture will be a total of 4 pounds. The total value of the coffee will be $6 for the 1 pound of $6 coffee and $12 for the 3 pounds of $4 coffee. Therefore, the total value of the 4 pound mixture is $6 + $12 or $18.
.
The $18 divided by the total weight of 4 pounds results in $4.50 just as the problem wanted. So the answer checks.
.
Hope that this helps you to see how this problem can be solved.
.