AI Fraction Solver

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.