Joely\'s Tea Shop, a store that specializes in tea blends, has available 45 poun

Question

Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?

Explanation / Answer

Grade A = 45 lbs
Grade B = 70 lbs
Total weight = A+B = 45+70 = 115 lbs
Blend Br = 1/3A + 2/3B
Blend Af = 1/2A + 1/2B

Profit = 1.50*lbs Blend Br + 2.00*lbs Blend Af
Profit = 1.50*(1/3A + 2/3B)+2,00(1/2A + 1/2B) =

45 pounds of A and 70 pounds of B yields max 24A + 72B = 98 Pounds Max Br
" " " " max 38A + 76B = 114 Pounds Max Af

The rate of profit for A is .50 per pound in BR and 1.00 per Pound in AF
For B, rate of profit for B is 1.00 for Br and 1.00 for Af.

Let X = # pounds in A
Let Y = # pounds in B

115 = A + B
A = 115-B

Let q = percent of Br
Let r = percent of Af

Let s = pounds of A in Br then 45-s = # pounds A in Af
Let t = pounds of B in Br and 70-t= # pounds in Af

.5s + 1.00*(45-s) + 1.00(t) + 1.00(70-t) = p
1.5s + 2.00t = p
If s = 45 pounds then 70 = 70/45 = 14/9s pounds
.5s + 1(45-s) + 1(14/9s) + 1(70-14/9s) = p
.5*45s + (45(1 - s)) + 14/9*45s + (45(1 - 14/9s) =p
22.5s + 45 - 45s + 70s + 35 - 70s = p
22.5s +70 - 115s = p
p = 137.5s + 70
p' = 137.5
22.5 A and 45 B for Br and 23.75 A and 23.75 B for Af.
Max profit is $137.5

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