Please explain how to solve this problem

The manager of a store that specializes in selling tea decides to experiment with a new blend. She will mix some Earl Grey tea that sells for $6 per pound with some Orange Pekoe tea that sells for $4 per pound to get 400 pounds of the new blend. The selling price of the new blend is to be $4.50 per pound, and there is to be no difference in revenue from selling the new blend versus selling the other types. How many pounds of the Earl Grey tea and Orange Pekoe tea are required?

The blend should have _ pounds of Earl Grey and _ pounds of Orange Pekoe.

Question

nick1272 nick1272

01-09-2021
Mathematics

contestada

Please explain how to solve this problem

The manager of a store that specializes in selling tea decides to experiment with a new blend. She will mix some Earl Grey tea that sells for $6 per pound with some Orange Pekoe tea that sells for $4 per pound to get 400 pounds of the new blend. The selling price of the new blend is to be $4.50 per pound, and there is to be no difference in revenue from selling the new blend versus selling the other types. How many pounds of the Earl Grey tea and Orange Pekoe tea are required?

The blend should have _ pounds of Earl Grey and _ pounds of Orange Pekoe.

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Answer:

100 pounds Earl Grey
300 pounds Orange Pekoe

Step-by-step explanation:

Let x represent the number of pounds of Earl Gray tea in the mix. Then the total cost of the mix is ...

6x +4(400-x)=4.50(400)

2x +1600 = 1800 . . . . . . . simplify

2x = 200 . . . . . . . . . . . subtract 1600

x = 100 . . . . . . . . . . . divide by 2. Pounds of Earl Grey

(400 -x) = 300 . . . . . Pounds of Orange Pekoe

The blend should have 100 pounds of Earl Grey and 300 pounds of Orange Pekoe.

_____

Additional comment

It is easy to show that the relationship between the quantities of high-cost (H) and low-cost (L) items in the mix (M) is ...

fraction of mix that is H is (M -L)/(H -L)

Here, we have H=6, L=4, M=4.5, so the fraction of the mix that is Earl Grey is ...

(4.5 -4)/(6 -4) = 0.5/2 = 0.25

There are 400 pounds of mix, so 0.25(400) = 100 pounds is Earl Grey.

The solution above is essentially 2-step, so using this relation doesn't necessarily save any steps. It just makes it possible to solve the problem in your head.