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.

Answer 1

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?

How would you go about solving this problem exactly?

Answer 2

Hello

Answer 3

Hi

Answer 4

I'm sorry I thought I could answer it I am just as confuse as you are Im so sorry for the lead on please forgive me

Answer 5

It's fine. It is complicated and I have no idea where to begin. My teachers are killing me.

Answer 6

Mine as well I hate when they do questions like these like how are you supposed to know how much time he worked

Answer 7

Exactly! Lol I guess I'll attempt to try it again later.

Answer 8

Grade A = 45 lbs
Grade B = 70 lbs
Total weight = A+B = 45+70 = 115
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

This should be right.

Answer 9

Thank you. I kind of get it.

Answer 10

YOur welcome

Answer 11

Call Breakfast Blend "m" and Afternoon Blend "n" for morning and noon

Then, divide up the poundage into 1/6 pound portions. You will have:

45 x 6 = 270 portions of A
70 x 6 = 420 portions of B

Each bag of either "m" or "n" tea is a pound and you start out with:

45 + 70 = 115 pounds of tea

So, you will sell 115 bags of "m" and "n" tea in total

Here's where you can make it easy. Imagine putting your "portions" into barrels. You have to use up all your portions by putting them into either the "m" or "n" barrel. Each bag of "m" requires 2 portions of A. Each bag of "n" requires 3 portions of A. Let "x" be the number of portions of A going into the "m" barrel. Then "270 - x" portions go into the "n" barrel :

(1/2)x + (1/3)(270 - x) = 115

where again, the 115 is the total bags sold. "(1/2)x" is the number of bags of "m" and "(1/3)(270 - x)" is the number of bags of "n".

"x" solves to be 150 so (1/2)(150) = 75 bags of "m" can be made from the 270 portions of A. And (1/3)(270 - 150) = 40 bags of "n" can be made from the remainder.

Notice that you get your desired 75 + 40 = 115 bags of tea.

You can come up with a similar equation for the B tea that will produce a breakdown of 75 bags of "m" and 40 bags of "n". Let "y" be the number of portions of B going into the "m" barrel. Then "420 - y" portions go into the "n" barrel :

(1/4)y + (1/3)(420 - y) = 115

where again, the 115 is the total bags sold. "(1/4)y" is the number of bags of "m" and "(1/3)(420 - y)" is the number of bags of "n".

"y" solves to be 300 so (1/4)(300) = 75 bags of "m" can be made from the 420 portions of B. And (1/3)(420 - 300) = 40 bags of "n" can be made from the remainder.

Notice that you get your desired 75 + 40 = 115 bags of tea yet again, confirming the method.

(75)(1.50) + (40)(2.00) = 192.50 dollars in profit.

Answer 12

Any questions, or are you still reading, @Bstyles ?

Answer 13

The key to this problem is not so much maximizing profit as using up all your portions of the A and B teas that you start with. You find out how much "m" and "n" tea you can make using up all your starting ingredients. Then you find the profit for each type and add them up. You could play with taking some "m" bags, breaking them up to use to make "n" bags with some ingredients left over, but you will get less profit. For example, if you wanted to make even one more bag of the more profitable "n" blend, you would have to use 2 bags of "m" to make one bag of "n" (having ingredients leftover that are worthless). You would sacrifice $3.00 to make $2.00, thus your profit would be less overall that way.

Answer 14

TMI tcarro, use simple small words. :)

Answer 15

@MKTY do you see how the answer is 192.50 and not 137.50?

Answer 16

Sorry. I just got back to the laptop. Give me a minute to read it over and process to see if I understand it now.

Answer 17

no

Answer 18

@MKTY , you use up all of your starting ingredients evenly, with nothing left over. You will end up maximizing because the problem is a simplified maximization problem where the starting ingredients, if apportioned correctly, will automatically maximize. If you use my split of blends, you will at least see that I legitimately get $192.50 and that is of course larger than $137.50. If you can't understand my method, then you should at least be able to use my split and just get the end answer.

Answer 19

In other words, put 25 pounds of your A grade and 50 pounds of your B grade into the Breakfast Blend. You'll get 75 pounds of Breakfast Blend. Put the remaining 20 pounds of A and the remaining 20 pounds of B into the Afternoon Blend. You'll get 40 pounds of Afternoon Blend. You will get maximum profits this way. Try anything else and you won't be as profitable.

Answer 20

Is (1/4)y + (1/3)(420 - y) = 115 and (1/2)x + (1/3)(270 - x) = 115 two separate equations or is it the same thing. That threw me off

Answer 21

They are 2 totally different equations. Remember, at the beginning of the "y" variable equation, I said that that referred to using the "B" grade tea. You can (and should) go through using that "y" equation to get the same breakdown. It confirms the method.

Answer 22

Oh right. I think I'm finally understanding. Much more than I did beforehand.

Answer 23

If the problem were made to be harder, like with leftover ingredients, then we could resort to true maximization techniques. Here, we started with "using up all ingredients evenly" and we see that we can't break up bags of "m" to make bags of "n" without losing profit.

Answer 24

Thank you so much! You've really helped a bunch :)

Answer 25

uw! Good luck to you in all of your studies and thx for the recognition! @Bstyles