## AI Fraction Solver

AI fraction solver is a MathCrave Math program that uses artificial intelligence algorithms to solve mathematical problems involving fractions. It can handle various operations such as addition, subtraction, multiplication, and division of fractions. The AI fraction solver uses machine learning techniques to understand and analyze the problem, apply the appropriate

### Lesson Notes: Understanding Fractions

#### 1. Understanding Numerator and Denominator in Fractional Expressions

#### Definition:

– Fraction: A fraction represents a part of a whole and is written as \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \neq 0 \).

– Numerator (a): The top part of the fraction, representing how many parts we have.

– Denominator (b): The bottom part of the fraction, representing how many equal parts the whole is divided into.

#### Examples:

– \( \frac{3}{4} \): 3 is the numerator, 4 is the denominator.

– \( \frac{7}{5} \): 7 is the numerator, 5 is the denominator.

#### 2. Proper and Improper Fractions and Mixed Numbers

Proper Fractions:

– The numerator is less than the denominator.

– Examples: \( \frac{3}{4}, \frac{2}{5} \)

Improper Fractions:

– The numerator is greater than or equal to the denominator.

– Examples: \( \frac{5}{3}, \frac{8}{8} \)

Mixed Numbers:

– A combination of a whole number and a proper fraction.

– Examples: \( 1\frac{2}{3}, 3\frac{1}{4} \)

Conversion between Mixed Numbers and Improper Fractions:

– Mixed to Improper: Multiply the whole number by the denominator and add the numerator, then place over the original denominator.

– Example: \( 2\frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{4} \)

– Improper to Mixed: Divide the numerator by the denominator to get the whole number, and the remainder becomes the new numerator.

– Example: \( \frac{11}{4} = 2\frac{3}{4} \)

#### 3. Add and Subtract Fractions

**Adding and Subtracting Fractions with the Same Denominator:**

– Add or subtract the numerators and keep the same denominator.

– Example: \( \frac{2}{5} + \frac{1}{5} = \frac{2+1}{5} = \frac{3}{5} \)

**Adding and Subtracting Fractions with Different Denominators:**

– Find a common denominator, convert the fractions, then add or subtract the numerators.

– Example: \( \frac{2}{3} + \frac{1}{4} \)

1. Common denominator: 12.

2. Convert: \( \frac{2}{3} = \frac{8}{12} \) and \( \frac{1}{4} = \frac{3}{12} \)

3. Add: \( \frac{8}{12} + \frac{3}{12} = \frac{11}{12} \)

**Subtracting Example:**

– Example: \( \frac{5}{6} – \frac{1}{4} \)

1. Common denominator: 12.

2. Convert: \( \frac{5}{6} = \frac{10}{12} \) and \( \frac{1}{4} = \frac{3}{12} \)

3. Subtract: \( \frac{10}{12} – \frac{3}{12} = \frac{7}{12} \)

#### 4. Multiply and Divide Fractions

**Multiplying Fractions:**

– Multiply the numerators and denominators.

– Example: \( \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \)

**Dividing Fractions:**

– Multiply by the reciprocal of the second fraction.

– Example: \( \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{2 \times 5}{3 \times 4} = \frac{10}{12} = \frac{5}{6} \)

#### 5. Order of Precedence in Expressions Involving Fractions

Order of Operations (PEMDAS/BODMAS):

– Parentheses/Brackets

– Exponents/Orders

– Multiplication and Division (from left to right)

– Addition and Subtraction (from left to right)

Example:

– \( \frac{1}{2} + 3 \times \left( \frac{4}{5} – \frac{2}{3} \right) \)

1. Calculate inside parentheses first: \( \frac{4}{5} – \frac{2}{3} \)

– Common denominator: 15.

– Convert: \( \frac{4}{5} = \frac{12}{15} \) and \( \frac{2}{3} = \frac{10}{15} \)

– Subtract: \( \frac{12}{15} – \frac{10}{15} = \frac{2}{15} \)

2. Multiply: \( 3 \times \frac{2}{15} = \frac{6}{15} = \frac{2}{5} \)

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

– Common denominator: 10.

– Convert: \( \frac{1}{2} = \frac{5}{10} \) and \( \frac{2}{5} = \frac{4}{10} \)

– Add: \( \frac{5}{10} + \frac{4}{10} = \frac{9}{10} \)

By following these steps and understanding the concepts, students will be able to work with fractions confidently in various mathematical contexts.