Bronze which costs $9.10/kg is made by combining copper which costs $8.90/kg with ton which costs $9.50/kg.  Find the number of kg of copper and tin required to make 15.3 kg of bronze.

To solve this problem, we can use the concept of weighted averages. Let's define the variables first:

- Let $ x $ be the mass (in kg) of copper used.
- Let $ y $ be the mass (in kg) of tin used.

Given:
- The total mass of the bronze is $ x + y = 15.3 $ kg.
- The cost per kg of copper is $8.90/kg.
- The cost per kg of tin is $9.50/kg.
- The cost per kg of bronze is $9.10/kg.

We can set up the following system of equations based on the cost:

1. **Mass Equation**:  
   \[
   x + y = 15.3
   \]

2. **Cost Equation**:  
   The total cost of the copper and tin should equal the total cost of the bronze:
   \[
   8.90x + 9.50y = 9.10 \times 15.3
   \]

Now, let's solve these equations.

### Step 1: Substitute the mass equation into the cost equation
From the mass equation:
\[
y = 15.3 - x
\]
Substitute this into the cost equation:
\[
8.90x + 9.50(15.3 - x) = 9.10 \times 15.3
\]

### Step 2: Simplify the cost equation
Expanding and simplifying:
\[
8.90x + 9.50 \times 15.3 - 9.50x = 9.10 \times 15.3
\]
\[
(8.90x - 9.50x) + 9.50 \times 15.3 = 9.10 \times 15.3
\]
\[
-0.60x + 9.50 \times 15.3 = 9.10 \times 15.3
\]

### Step 3: Solve for $ x $
\[
-0.60x = 9.10 \times 15.3 - 9.50 \times 15.3
\]
\[
-0.60x = 15.3 \times (9.10 - 9.50)
\]
\[
-0.60x = 15.3 \times (-0.40)
\]
\[
x = \frac{15.3 \times 0.40}{0.60}
\]
\[
x = \frac{6.12}{0.60} = 10.2 \text{ kg}
\]

### Step 4: Find $ y $
Using $ y = 15.3 - x $:
\[
y = 15.3 - 10.2 = 5.1 \text{ kg}
\]

### Final Answer
- Copper required: **10.2 kg**
- Tin required: **5.1 kg**

Would you like more details or have any questions about this solution?

Here are 5 related questions for further practice:
1. How would the solution change if the cost of tin increased to $10.00/kg?
2. What would the required quantities be if the total weight of the bronze needed was 20 kg instead of 15.3 kg?
3. If the price of copper decreases to $8.50/kg, what would be the new quantities of copper and tin required?
4. How much would it cost to make 30 kg of bronze using the same cost per kg for copper and tin?
5. What proportion of the total cost does copper contribute in the original scenario?

**Tip:** When dealing with problems involving mixtures or alloys, setting up the system of equations based on total mass and cost relationships is crucial for finding the correct solution.

Bronze which costs $9.10/kg is made by combining copper which costs $8.90/kg with tin which costs $9.50/kg. Find the number of kg of copper and tin required to make 15.3 kg of bronze.

Learn how to determine the quantities of copper and tin required to make 15.3 kg of bronze, given their respective costs. This problem involves algebraic concepts, weighted averages, and solving a system of linear equations. Suitable for students in Grades 10-12.

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