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The problem states that a coffee shop owner blends a gourmet coffee that sells for $7.00 per pound with a cheaper coffee that sells for $5.00 per pound. The goal is to create 50 pounds of coffee that sells for $5.50 per pound. We need to determine how much of each type of coffee should be mixed.

This is a classic "mixture problem," where the total value of the mixture should be equal to the value of the desired final blend.

### Let:
- $ x $ = pounds of gourmet coffee (costing $7.00 per pound)
- $ y $ = pounds of cheaper coffee (costing $5.00 per pound)

We know:
1. $ x + y = 50 $ (because the total weight must be 50 pounds).
2. The total value of the mixture should be $ 5.50 \times 50 = 275 $ dollars.
   - The value of the gourmet coffee is $ 7x $.
   - The value of the cheaper coffee is $ 5y $.
   - The equation for the total value of the mixture is $ 7x + 5y = 275 $.

Now, we have the system of equations:
1. $ x + y = 50 $
2. $ 7x + 5y = 275 $

I will solve this system of equations.The solution to the system of equations is:

- $ x = \frac{25}{2} = 12.5 $ pounds of the gourmet coffee (costing $7.00 per pound),
- $ y = \frac{75}{2} = 37.5 $ pounds of the cheaper coffee (costing $5.00 per pound).

Thus, the coffee shop owner should mix 12.5 pounds of the gourmet coffee with 37.5 pounds of the cheaper coffee to create 50 pounds of a blend worth $5.50 per pound.

Would you like further details or have any other questions?

### Related questions:
1. How is the value of a mixture calculated in such problems?
2. Can this problem be solved graphically?
3. What if the price per pound of the final blend were different?
4. How would the solution change if the weights of the two coffees were given instead?
5. What other real-life situations can be modeled with mixture problems?

### Tip:
When solving mixture problems, always verify your solution by plugging the values back into both the total weight and value equations to ensure consistency.

A coffee shop owner blends a gourmet brand of coffee with a cheaper brand. The gourmet coffee usually sells for $7.00 per pound. The cheaper brand sells for $5.00 per pound. How much of each type should be mixed in order to have 50 pounds of coffee that is worth $5.50 per pound?

Learn how to solve a mixture problem involving the blending of gourmet coffee at $7 per pound with cheaper coffee at $5 per pound to create a 50-pound mix worth $5.50 per pound. Suitable for high school algebra students.

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