SOLUTION: Needs to prepare 190 pounds of coffee beans for 5.08 per pound. Plan is to blend together high quality beans cost 6.50 per pound a cheaper bean at 3.50 a pound. To the nearest poun

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Question 1134659: Needs to prepare 190 pounds of coffee beans for 5.08 per pound. Plan is to blend together high quality beans cost 6.50 per pound a cheaper bean at 3.50 a pound. To the nearest pound how much high quality coffee bean and how much cheaper coffee bean she should blend
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
M, pounds of mixture, 190 pounds
T=5.08, dollars per pound mixture
L=3.5, dollars per pound low quality coffee
H=6.5, dollars per pound high quality coffee
-
v, pounds of the high quality higher priced coffee
M-v, pounds of the lower quality lower priced coffee


highlight_green%28Hv%2BL%28M-v%29=TM%29
-
Hv%2BLM-Lv=TM
Hv-Lv%2BLM=TM
%28H-L%29v%2BLM=TM
%28H-L%29v=TM-LM
highlight%28v=%28TM-LM%29%2F%28H-L%29%29
Substitute the values into the formula....

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Formal algebra, by the standard method....

Let x = number of pounds of low value coffee beans at $3.50 per pound
Then (190-x) = number of pounds of high value coffee beans at $6.50 per pound

The mixture of x pounds at $3.50 per pound and (190-x) pounds at $6.50 per pound yields 190 pounds at $5.08 per pound:

x%283.50%29%2B%28190-x%29%286.50%29+=+190%285.08%29

A slightly ugly equation, but easily solvable using basic algebra. I'll leave it to you to finish the solution by that method.

A much easier and faster way to the answer, if you understand how to use the method....

The key to the method is to see where the price of $5.08 per pound lies between the two costs of $3.50 and $6.50 per pound. Imagine that you are starting with the low value coffee beans and are adding the high value coffee beans until the cost per pound is the desired cost of the mixture. The question is, when do you stop.

(1) Find the fraction of the way that $5.08 is from $3.50 to $6.50.

6.50-3.50 = 3.00; 5.08-3.50 = 1.58

The fraction of the way that 5.08 is from 3.50 to 6.50 is 1.58/3.00 = 158/300. (don't bother simplifying the fraction at this point).

(2) That fraction is the fraction of the total mixture that needs to be the high value coffee beans.

190%2A%28158%2F300%29%29 = almost exactly 100

ANSWER: 100 pounds of the high value coffee beans; 190-100 = 90 pounds of low value coffee beans.