proper and improper fractions along with explanation

Let's dive into some examples of proper and improper fractions along with explanations.

### Proper Fractions

A proper fraction is a fraction where the numerator (the top number) is less than the denominator (the bottom number). Proper fractions represent a part of a whole.

**Example 1:**

\[ \frac{3}{4} \]

- **Explanation**: The numerator (3) is less than the denominator (4). This fraction represents three parts out of four equal parts, which is less than one whole.

**Example 2:**

\[ \frac{7}{10} \]

- **Explanation**: The numerator (7) is less than the denominator (10). This fraction represents seven parts out of ten equal parts, which is also less than one whole.

### Improper Fractions

An improper fraction is a fraction where the numerator is greater than or equal to the denominator. Improper fractions represent one whole or more.

**Example 1:**

\[ \frac{9}{8} \]

- **Explanation**: The numerator (9) is greater than the denominator (8). This fraction represents more than one whole. In mixed number form, it can be expressed as \(1 \frac{1}{8}\).

**Example 2:**

\[ \frac{12}{5} \]

- **Explanation**: The numerator (12) is greater than the denominator (5). This fraction represents more than one whole. In mixed number form, it can be expressed as \(2 \frac{2}{5}\).

### Converting Improper Fractions to Mixed Numbers

To convert an improper fraction to a mixed number, divide the numerator by the denominator to get the whole number part and use the remainder as the new numerator.

**Example 1:**

Convert \(\frac{9}{4}\) to a mixed number.

- **Explanation**: Divide 9 by 4 to get 2 with a remainder of 1. The mixed number is \(2 \frac{1}{4}\).

**Example 2:**

Convert \(\frac{15}{6}\) to a mixed number.

- **Explanation**: Divide 15 by 6 to get 2 with a remainder of 3. The mixed number is \(2 \frac{3}{6}\), which simplifies to \(2 \frac{1}{2}\).

### Converting Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and add the numerator. Place this result over the original denominator.

**Example 1:**

Convert \(3 \frac{2}{5}\) to an improper fraction.

- **Explanation**: Multiply 3 (the whole number part) by 5 (the denominator) to get 15. Add 2 (the numerator) to get 17. The improper fraction is \(\frac{17}{5}\).

**Example 2:**

Convert \(4 \frac{3}{7}\) to an improper fraction.

- **Explanation**: Multiply 4 (the whole number part) by 7 (the denominator) to get 28. Add 3 (the numerator) to get 31. The improper fraction is \(\frac{31}{7}\).

### Summary

- **Proper fractions** have numerators less than denominators (e.g., \(\frac{3}{4}\), \(\frac{7}{10}\)).

- **Improper fractions** have numerators greater than or equal to denominators (e.g., \(\frac{9}{8}\), \(\frac{12}{5}\)).

- Converting improper fractions to mixed numbers involves division and finding the remainder.

- Converting mixed numbers to improper fractions involves multiplication and addition.

Here are some exercises involving proper and improper fractions, along with solutions:

### Exercise 1: Converting Improper Fractions to Mixed Numbers

Convert the following improper fractions to mixed numbers:

1. \( \frac{11}{4} \)

2. \( \frac{19}{5} \)

3. \( \frac{22}{7} \)

4. \( \frac{15}{3} \)

5. \( \frac{35}{6} \)

**Solutions:**

1. \( \frac{11}{4} \) = 2 \(\frac{3}{4}\)

2. \( \frac{19}{5} \) = 3 \(\frac{4}{5}\)

3. \( \frac{22}{7} \) = 3 \(\frac{1}{7}\)

4. \( \frac{15}{3} \) = 5

5. \( \frac{35}{6} \) = 5 \(\frac{5}{6}\)

### Exercise 2: Converting Mixed Numbers to Improper Fractions

Convert the following mixed numbers to improper fractions:

1. 2 \(\frac{3}{4}\)

2. 4 \(\frac{2}{3}\)

3. 1 \(\frac{5}{6}\)

4. 3 \(\frac{7}{8}\)

5. 5 \(\frac{1}{2}\)

**Solutions:**

1. 2 \(\frac{3}{4}\) = \( \frac{11}{4} \)

2. 4 \(\frac{2}{3}\) = \( \frac{14}{3} \)

3. 1 \(\frac{5}{6}\) = \( \frac{11}{6} \)

4. 3 \(\frac{7}{8}\) = \( \frac{31}{8} \)

5. 5 \(\frac{1}{2}\) = \( \frac{11}{2} \)

### Exercise 3: Adding Proper Fractions

Add the following fractions and simplify if possible:

1. \( \frac{2}{5} + \frac{3}{5} \)

2. \( \frac{1}{4} + \frac{2}{4} \)

3. \( \frac{3}{8} + \frac{1}{8} \)

4. \( \frac{5}{12} + \frac{7}{12} \)

5. \( \frac{1}{6} + \frac{5}{6} \)

**Solutions:**

1. \( \frac{2}{5} + \frac{3}{5} = \frac{5}{5} = 1 \)

2. \( \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \)

3. \( \frac{3}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \)

4. \( \frac{5}{12} + \frac{7}{12} = \frac{12}{12} = 1 \)

5. \( \frac{1}{6} + \frac{5}{6} = \frac{6}{6} = 1 \)

### Exercise 4: Subtracting Improper Fractions

Subtract the following fractions and simplify if possible:

1. \( \frac{9}{4} - \frac{5}{4} \)

2. \( \frac{11}{6} - \frac{7}{6} \)

3. \( \frac{10}{8} - \frac{3}{8} \)

4. \( \frac{14}{5} - \frac{9}{5} \)

5. \( \frac{13}{9} - \frac{4}{9} \)

**Solutions:**

1. \( \frac{9}{4} - \frac{5}{4} = \frac{4}{4} = 1 \)

2. \( \frac{11}{6} - \frac{7}{6} = \frac{4}{6} = \frac{2}{3} \)

3. \( \frac{10}{8} - \frac{3}{8} = \frac{7}{8} \)

4. \( \frac{14}{5} - \frac{9}{5} = \frac{5}{5} = 1 \)

5. \( \frac{13}{9} - \frac{4}{9} = \frac{9}{9} = 1 \)

These exercises should help you practice converting between improper fractions and mixed numbers, as well as adding and subtracting fractions.