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The problem can be broken down as follows:

- In **2010**, the library had a total of **11,500 books** in two categories: fiction and non-fiction.
- By **2015**, the library contained a total of **12,760 books**.
- During this period, there was a **10% increase** in the fiction category, and a **12% increase** in the non-fiction category.

The task is to find how many fiction books there were in the library in **2015**.

### Let's define some variables:

- Let **\( F_{2010} \)** be the number of fiction books in 2010.
- Let **\( NF_{2010} \)** be the number of non-fiction books in 2010.
- The total number of books in 2010 is the sum of fiction and non-fiction books, so:
  \[
  F_{2010} + NF_{2010} = 11,500
  \]
  
- In 2015, the number of fiction books is increased by **10%**, and the number of non-fiction books is increased by **12%**. So, the total number of books in 2015 is:
  \[
  1.1 \times F_{2010} + 1.12 \times NF_{2010} = 12,760
  \]

### System of equations:
1. \( F_{2010} + NF_{2010} = 11,500 \)
2. \( 1.1 \times F_{2010} + 1.12 \times NF_{2010} = 12,760 \)

We can solve this system of equations to find \( F_{2010} \) and \( NF_{2010} \), then calculate the number of fiction books in **2015**, which would be \( 1.1 \times F_{2010} \).

Let me solve this system for you.

### Step 1: Solving for \( F_{2010} \) and \( NF_{2010} \)

From equation (1), express \( NF_{2010} \) in terms of \( F_{2010} \):
\[
NF_{2010} = 11,500 - F_{2010}
\]

### Step 2: Substitute this into equation (2):
\[
1.1 \times F_{2010} + 1.12 \times (11,500 - F_{2010}) = 12,760
\]
\[
1.1 \times F_{2010} + 1.12 \times 11,500 - 1.12 \times F_{2010} = 12,760
\]
\[
1.1 \times F_{2010} - 1.12 \times F_{2010} = 12,760 - 1.12 \times 11,500
\]
\[
-0.02 \times F_{2010} = 12,760 - 12,880
\]
\[
-0.02 \times F_{2010} = -120
\]
\[
F_{2010} = \frac{-120}{-0.02} = 6,000
\]

### Step 3: Solve for \( NF_{2010} \)
\[
NF_{2010} = 11,500 - 6,000 = 5,500
\]

### Step 4: Calculate fiction books in 2015:
The number of fiction books in 2015 is:
\[
1.1 \times 6,000 = 6,600
\]

So, there were **6,600 fiction books** in the library in 2015.

---

Would you like more details or have any other questions? Here are five follow-up questions you could explore:

1. How many non-fiction books were in the library in 2015?
2. What was the total percentage increase in the number of books from 2010 to 2015?
3. What would happen if the growth rates of fiction and non-fiction were swapped?
4. How many books would be there if both categories increased by 15% instead?
5. What is the ratio of fiction to non-fiction books in 2010 compared to 2015?

**Tip:** To solve systems of linear equations, substitution or elimination methods are often helpful!

In 2010, a library contained 11,500 books in fiction and non-fiction categories. By 2015, the total number of books increased to 12,760. During this period, fiction books increased by 10%, and non-fiction books increased by 12%. How many fiction books were in the library in 2015?

Solve a problem involving the growth of fiction and non-fiction books in a library from 2010 to 2015, with a 10% increase in fiction books and a 12% increase in non-fiction books. The total book count grew from 11,500 to 12,760. This problem involves algebra and percentage calculations.

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