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 800 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?

Let's solve the problem step by step. We need to find the number of pounds of Earl Grey tea and Orange Pekoe tea required to create the new blend.

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

### Given:
1. The total amount of tea is 800 pounds:
   \[
   x + y = 800
   \]
2. The cost of Earl Grey tea is $6 per pound.
3. The cost of Orange Pekoe tea is $4 per pound.
4. The selling price of the new blend is $4.50 per pound.
5. The revenue from selling the new blend should be equal to the revenue from selling the individual teas.

### Revenue Calculation:
The revenue from selling the original teas:
\[
\text{Revenue from original teas} = 6x + 4y
\]

The revenue from selling the new blend:
\[
\text{Revenue from new blend} = 4.5 \times 800 = 3600
\]

Since the revenues must be the same:
\[
6x + 4y = 3600
\]

### System of Equations:
We have the following system of equations:
\[
\begin{aligned}
1) & \quad x + y = 800 \\
2) & \quad 6x + 4y = 3600
\end{aligned}
\]

### Solve the System:
First, solve equation (1) for $ y $:
\[
y = 800 - x
\]

Substitute this into equation (2):
\[
6x + 4(800 - x) = 3600
\]

Distribute and simplify:
\[
6x + 3200 - 4x = 3600
\]

Combine like terms:
\[
2x + 3200 = 3600
\]

Subtract 3200 from both sides:
\[
2x = 400
\]

Divide by 2:
\[
x = 200
\]

Now, substitute $ x = 200 $ back into equation (1) to find $ y $:
\[
200 + y = 800
\]

\[
y = 600
\]

### Answer:
- **Earl Grey tea required:** 200 pounds
- **Orange Pekoe tea required:** 600 pounds

Would you like more details, or do you have any questions about the process?

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Here are 5 related questions to explore:

1. How would the quantities change if the blend was to be sold at $5 per pound?
2. What would happen if the manager wanted to make 1000 pounds of the blend?
3. How would the solution change if the Earl Grey tea cost $5 per pound instead?
4. What if the total revenue from the new blend needs to be $4000 instead of $3600?
5. How can we adjust the blend if the manager decides to use another tea priced at $7 per pound?

**Tip:** Understanding how to set up and solve systems of equations is essential in many real-world problems involving mixtures and blends!

Explore how to solve a tea blending problem involving Earl Grey tea at $6 per pound and Orange Pekoe tea at $4 per pound to create a new blend priced at $4.50 per pound. Discover step-by-step solutions and calculations for determining the required amounts of each tea. Suitable for high school students and anyone interested in mixture problems and revenue calculations.

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