Your company offers gourmet coffee beans for \$14 per pound and regular coffee beans for \$6 per pound

Your company offers gourmet coffee beans for \$14 per pound and regular coffee beans for \$6 per pound. If I put 12 pounds of the gourmet coffee beans in the mixer, how much of the regular coffee beans do I add to get a mixture that costs \$8 per pound?

Solutions

Expert Solution
Here's one way of looking at this problem.
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Let's start with the understanding that when the total cost of the mixture is divided by
the total weight of the mixture, the answer must be \$8 per lb.
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What is the total cost of the mixture. Well, the problem tells you that the mixture will
contain 12 lbs of coffee that costs \$14 per lb. Multiplying the 12 lbs times \$14 per pound,
results in \$168 dollars worth of the high value coffee in the mix.
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You do not know what the weight of the cheap coffee will be. So let's call that weight "W".
At \$6 per lb for those W pounds the value of the cheap coffee in the mix must be \$6*W.
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Therefore, the value of the mix is the sum of these two dollar amounts. This is \$168 + \$6*W.
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Now let's look at the total weight of the mixture. We know that 12 lbs of the mix is
the expensive coffee. And we have said that the weight of the cheaper coffee in the mix
will be identified as W lbs. So the total weight of the mixture will be 12 + W lbs.
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So if we divide the total cost of the coffee (168 + 6W) by the total weight of the mix
(12 + W) the answer is to be \$8 per lb. In equation form this becomes:
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{{{(168 + 6W)/(12+W) = 8}}}
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Get rid of the denominator by multiplying both sides of the equation by (12 + W) to get:
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{{{((12+W)(168+6W))/(12+W) = (12+W)*8}}}
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On the left side the (12 + W) in the numerator cancels with the (12 + W) in the denominator.
And on the right side the quantity the 8 times (12 + W) multiplies out to 96 + 8W. So
you are left with the equation:
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{{{168 + 6W = 96 + 8W}}}
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Get rid of the 8W on the right side by subtracting 8W from both sides to get:
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{{{168 -2W = 96}}}
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Then get rid of the 168 on the right side by subtracting 168 from both sides. The resulting
equation becomes:
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{{{-2W = -72}}}
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You can now solve for W which is the weight of the cheaper coffee in the mixture by dividing
both sides of this equation by -2 to get:
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{{{W = (-72)/(-2) = 36}}}
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So to get a mixture worth \$8 per pound, if you start with 12 lbs of \$14 coffee, you need
to mix in 36 lbs of coffee that sells for \$6 per lb to get the desired mixture.
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Hope this helps you to understand the problem a little better and that you can track the
above approach to get the answer.
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