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 $4 per pound with some Orange Pekoe tea that sells for $ 2 per pound to get 200 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?

This is a mixture problem involving two types of tea, Earl Grey and Orange Pekoe, with different prices, and a blend that should have a specific price per pound. Let's set up the problem step by step.

### Step 1: Define the variables
Let:
- $ x $ be the number of pounds of Earl Grey tea.
- $ y $ be the number of pounds of Orange Pekoe tea.

From the problem:
- We know the total amount of tea in the blend is 200 pounds, so:
  \[
  x + y = 200 \quad \text{(Equation 1)}
  \]

### Step 2: Set up the revenue equation
The total revenue from selling 200 pounds of the new blend should equal the revenue from selling the individual components.

- Earl Grey tea sells for $4 per pound, so the revenue from $ x $ pounds of Earl Grey is:
  \[
  4x
  \]
  
- Orange Pekoe tea sells for $2 per pound, so the revenue from $ y $ pounds of Orange Pekoe is:
  \[
  2y
  \]

- The new blend sells for $2.50 per pound, and the total amount of the blend is 200 pounds, so the revenue from the blend is:
  \[
  2.5 \times 200 = 500 \, \text{dollars}
  \]

Thus, the revenue equation is:
\[
4x + 2y = 500 \quad \text{(Equation 2)}
\]

### Step 3: Solve the system of equations

We now have the system:
1. $ x + y = 200 $
2. $ 4x + 2y = 500 $

#### Solving for $ y $ in terms of $ x $ from Equation 1:
\[
y = 200 - x
\]

#### Substitute $ y = 200 - x $ into Equation 2:
\[
4x + 2(200 - x) = 500
\]
Simplify the equation:
\[
4x + 400 - 2x = 500
\]
\[
2x + 400 = 500
\]
\[
2x = 100
\]
\[
x = 50
\]

Now, substitute $ x = 50 $ back into Equation 1:
\[
50 + y = 200
\]
\[
y = 150
\]

### Final Answer:
- 50 pounds of Earl Grey tea are required.
- 150 pounds of Orange Pekoe tea are required.

Would you like further details or have any questions? Here are some related problems:

1. What if the total blend needed to be 300 pounds? How would that affect the mix?
2. What if the price of the blend was $3 per pound instead of $2.50?
3. How would the problem change if the price of Earl Grey increased to $5 per pound?
4. What if you needed to maintain a 60-40 ratio of the two teas instead of setting prices?
5. Can you mix other teas with different prices and still keep the blend at $2.50 per pound?

**Tip:** Always check if your system of equations accurately reflects both the quantities and values involved!

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 $4 per pound with some Orange Pekoe tea that sells for $2 per pound to get 200 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?

Learn how to solve a system of equations to determine the correct blend of Earl Grey and Orange Pekoe tea that maintains equal revenue while creating a 200-pound mixture priced at $2.50 per pound. This example covers algebraic concepts, such as solving linear equations. Suitable for Grades 8-10.

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