Question 1202720: 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 $5 per pound with some Orange Pekoe tea that sells for $2 per pound to get 300 pounds of the new blend. The selling price of the new blend is to be $2.50 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?
Found 3 solutions by ikleyn, josgarithmetic, greenestamps:
Answer by ikleyn(52781)   (Show Source):
You can put this solution on YOUR website! .
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 $5 per pound with some
Orange Pekoe tea that sells for $2 per pound
to get 300 pounds of the new blend.
The selling price of the new blend is to be $2.50 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?
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It can be solved using one equation in one single unknown,
or using a system of equations in two unknowns.
I will show you the solution using one equation,
and will give the links to my lessons at this site showing how to work with systems of equations.
Let x be the pounds of the Earl Grey tea;
then the pounds of the Orange Pekoe tea is (300-x) pounds.
x pounds per $5 per pound contribute 5x dollars to the cost of the mixture.
(300-x) pounds per $2 per pound contribute 2(300-x) dollars to the cost of the mixtire.
For total cost, we obtain this equation
5x + 2*(300-x) = 2.50*300 (1)
The right side is the cost of 300 pounds of the mixture at given cost $2.50 per pound.
Simplify this equation and find x
5x + 600 - 2x = 750
5x - 2x = 750 - 600
3x = 150
x = 150/3 = 50 pounds.
ANSWER. 50 pounds of the Earl Grey tea and 300-50 = 250 pounds of the Orange Pekoe tea.
CHECK. 5*50 + 2*250 = 750 dollars, the cost of the ingredients.
2.50*300 = 750 dollars, the cost of the mixture.
The cost is the same for the ingredients and the mixture;
hence, the solution is CORRECT.
Solved.
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To see on how to solve similar problems using two equations and two unknowns, look into the lessons
- Word problems on mixtures for dry substances like coffee beans, nuts, cashew and peanuts
- Word problems on mixtures for dry substances like candies, dried fruits
- Word problems on mixtures for dry substances like soil and sand
in this site.
Answer by josgarithmetic(39617)  (Show Source):
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
For a standard formal algebraic solution, look at the response from tutor @ikleyn.
If you like doing mathematics using "magical" formulas, thereby gaining absolutely no understanding of HOW the problem is actually solved, then look at the response from tutor @josgarthmetic.
If formal algebra is not required, and you want a fast and easy way to solve the problem informally, then below are two closely related methods for solving any 2-part mixture problem like this.
First method....
(1) Look at the prices per pound of the two kinds of tea and the mixture -- on a number line, if it helps: $2.00, $2.50, and $5.00.
(2) Observe/calculate that $2.50 is 1/6 of the way from $2.00 to $5.00. ($2.00 to $5.00 is a difference of $3.00; $2.00 to $2.50 is a difference of $0.50; $0.50/$3.00 = 1/6.)
(3) That means 1/6 of the mixture is the more expensive Earl Grey tea.
1/6 of 300 pounds is 50 pounds.
ANSWER: 50 pounds of Earl Grey tea and 300-50 = 250 pounds of Orange Pekoe tea
Second method....
(1) Again look at the prices per pound of the two kinds of tea and the mixture.
(2) The difference between the prices of the Orange Pekoe tea and the mixture is $0.50; the difference between the prices of the Earl Grey tea and the mixture is $2.50. The ratio of those two differences is $0.50:$2.50 = 1:5.
(3) Interpret that to mean that the $2.50 cost per pound of the mixture is "5 times as close" to $2.00 as it is to $5.00.
(4) That means the mixture must contain 5 times as much Orange Pekoe tea as Earl Grey tea.
Use mental arithmetic (or formal algebra, if needed) to find that 300 pounds mixed in the ratio 5:1 means 250 pounds of Orange Pekoe tea and 50 pounds of Earl Grey tea.
ANSWER (again, of course): 50 pounds of Earl Grey tea and 300-50 = 250 pounds of Orange Pekoe tea
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