 The school has decided to increase the books in the library by adding 1/2 of the current amount of books to the library. However, they also have to sell 1/3 of the current books in the library. What amount will they have now? 

To determine the new amount of books in the library after the school adds and sells books, we need to calculate the net change in the number of books.

Let's break down the problem step-by-step:

1. **Fraction of books added:**  
   The school adds \(\frac{1}{2}\) of the current number of books.

2. **Fraction of books sold:**  
   The school sells \(\frac{1}{3}\) of the current number of books.

3. **Net change in the number of books:**  
   The net change in the number of books is the difference between the books added and the books sold:

   \[
   \text{Net change} = \frac{1}{2} - \frac{1}{3}
   \]

4. **Finding a common denominator:**  
   To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 2 and 3 is 6.

   Converting each fraction to have a denominator of 6:
   \[
   \frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}
   \]
   \[
   \frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}
   \]

5. **Subtracting the fractions:**  
   Now, subtract the fractions:
   \[
   \frac{3}{6} - \frac{2}{6} = \frac{1}{6}
   \]

   This means the net addition is \(\frac{1}{6}\) of the current number of books.

6. **Calculating the new total of books:**  
   Since \(\frac{1}{6}\) of the current number of books is added to the existing books, the new total number of books in the library is:
   \[
   1 + \frac{1}{6} = \frac{6}{6} + \frac{1}{6} = \frac{7}{6}
   \]

   So, the library will now have \(\frac{7}{6}\) of the original number of books.

In conclusion, after the increase and decrease, the library will have **\(\frac{7}{6}\) of the original amount of books**.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. If the library originally had 600 books, how many books will there be after the increase and decrease?
2. What percentage of the original amount does \(\frac{7}{6}\) represent?
3. If the library wants to maintain at least 500 books after any changes, how many books should they have started with?
4. What fraction of books would need to be sold to have exactly the same number of books after adding \(\frac{1}{2}\) of the current books?
5. If instead of \(\frac{1}{3}\), the library sells \(\frac{1}{4}\) of the current books, what fraction of the original amount will the library have?

**Tip:** When dealing with fractions, always ensure to find a common denominator when adding or subtracting.

Learn how to calculate changes in library books when adding 1/2 and selling 1/3 of the current amount. This problem involves fractions and arithmetic operations, explaining step-by-step the addition and subtraction of fractions. Suitable for students in Grades 5-6.

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