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The question involves mixing two alloys to form a new mixture. Let's break it down and solve it step by step.

### Problem breakdown:
- Alloy \( X \) (contains some percentage of copper and zinc) is mixed with 5 kg of another alloy that is 90% copper.
- The result is a mixture with 80% copper.
- A second scenario: Alloy \( X \) is mixed with 3 kg of pure copper to get a mixture that is 81.25% copper.
- We need to find how many kilograms of alloy \( X \) were used.

### Step 1: Define variables
Let \( x \) be the mass of alloy \( X \) in kilograms, and let \( p \% \) be the percentage of copper in alloy \( X \).

### Step 2: First scenario (mix with 5 kg of 90% copper)
- The total mass of the mixture is \( x + 5 \) kg.
- The amount of copper in alloy \( X \) is \( p \times x \) kg.
- The amount of copper in the 5 kg of the other alloy is \( 0.9 \times 5 = 4.5 \) kg.
- The total amount of copper in the mixture is \( p \times x + 4.5 \).
- The total percentage of copper in the mixture is 80%, so:
  \[
  \frac{p \times x + 4.5}{x + 5} = 0.80
  \]

### Step 3: Second scenario (mix with 3 kg of pure copper)
- The total mass of the mixture is \( x + 3 \) kg.
- The amount of copper in the mixture is \( p \times x + 3 \) (since pure copper is 100% copper).
- The total percentage of copper in this mixture is 81.25%, so:
  \[
  \frac{p \times x + 3}{x + 3} = 0.8125
  \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( \frac{p \times x + 4.5}{x + 5} = 0.80 \)
2. \( \frac{p \times x + 3}{x + 3} = 0.8125 \)

We can solve these two equations to find the values of \( x \) and \( p \).

---

Let me calculate the values for you.The solution to the system of equations gives us:

- \( x = 5 \) kg (the amount of alloy \( X \) used),
- \( p = 0.70 \), meaning alloy \( X \) contains 70% copper.

So, the amount of alloy \( X \) available is **5 kg**.

Would you like more details or have any questions? Here are some related questions to explore further:

1. How would the solution change if alloy \( X \) had a different percentage of copper?
2. Can you generalize the process for any two alloys mixed together?
3. What happens if alloy \( X \) is mixed with a different material instead of pure copper?
4. How would you solve if the given percentages were slightly different, say 85% instead of 90% copper?
5. How can you verify the solution using a different method?

**Tip**: In such questions, setting up equations based on weighted averages is key to solving them correctly.

An alloy X of copper and zinc is mixed with 5 kg of another alloy having 90% copper in it to get an alloy with 80% copper. If the same amount of alloy X is mixed with 3 kg of pure copper, we get an alloy with 81.25% copper. What was the amount of alloy X available (in kg)?

Learn how to solve a mixture problem involving alloys of copper and zinc. Given two different mixtures of copper, this step-by-step solution teaches how to find the amount of alloy X using weighted averages. This problem is suitable for Grades 9-12 students.

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