Key Concepts:
Least Common Multiple (LCM)
: The smallest number that is a multiple of two or more numbers.Highest Common Factor (HCF)
: The largest number that divides two or more numbers without leaving a remainder.Factorization
: Breaking a number into its prime factors.Addition and Subtraction of Fractions:
To add or subtract fractions, first find a common denominator (usually the least common multiple of the denominators), then combine the numerators. \[ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}, \quad \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} \]Multiplication of Fractions:
Multiply the numerators together and the denominators together: \[ \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \]Division of Fractions:
Multiply the first fraction by the reciprocal of the second: \[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \]Mixed Numbers:
A mixed number consists of a whole number and a proper fraction. To perform operations, convert mixed numbers to improper fractions: \[ a\frac{b}{c} = \frac{ac + b}{c} \] Then proceed with the appropriate operation.Example 1: Addition of Fractions
Simplify: \[ \frac{3}{4} + \frac{5}{6} \]Solution:
Step 1: Find the LCM of the denominators 4 and 6: \[ \text{LCM} = 12 \] Step 2: Convert each fraction to have a denominator of 12: \[ \frac{3}{4} = \frac{9}{12}, \quad \frac{5}{6} = \frac{10}{12} \] Step 3: Add the fractions: \[ \frac{9}{12} + \frac{10}{12} = \frac{19}{12} \] Therefore, the result is: \[ \frac{19}{12} \]Example 2: Multiplication of Fractions
Simplify: \[ \frac{2}{3} \times \frac{4}{5} \]Solution:
Multiply the numerators and the denominators: \[ \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \] Therefore, the result is: \[ \frac{8}{15} \]Example 3: Division of Fractions
Simplify: \[ \frac{5}{6} \div \frac{2}{9} \]Solution:
To divide, multiply by the reciprocal: \[ \frac{5}{6} \div \frac{2}{9} = \frac{5}{6} \times \frac{9}{2} = \frac{5 \times 9}{6 \times 2} = \frac{45}{12} \] Simplify the result: \[ \frac{45}{12} = \frac{15}{4} \] Therefore, the answer is: \[ \frac{15}{4} \]Example 4: Multiplication of Decimals
Simplify: \[ 1.25 \times 0.4 \]Solution:
Step 1: Multiply as if they are whole numbers: \[ 125 \times 4 = 500 \] Step 2: Count the total number of decimal places: 1.25 has 2 decimal places, and 0.4 has 1 decimal place, so total = 3. Step 3: Adjust the product by placing the decimal point: \[ 500 \rightarrow 0.500 \] Therefore, the result is: \[ 0.5 \]Example 5: Factorization of a Number
Factorize 84 into its prime factors.Solution:
Step 1: Divide successively by prime numbers: \[ 84 \div 2 = 42, \quad 42 \div 2 = 21, \quad 21 \div 3 = 7 \] Step 2: Write the product of prime factors: \[ 84 = 2^2 \times 3 \times 7 \]Example 6: Finding the HCF
Find the HCF of 48 and 180.Solution:
Step 1: Perform prime factorization: \[ 48 = 2^4 \times 3, \quad 180 = 2^2 \times 3^2 \times 5 \] Step 2: Identify common prime factors: Common factors are \(2^2\) and \(3\) Step 3: Multiply the common factors: \[ \text{HCF} = 2^2 \times 3 = 12 \]Example 7: Simplify an Expression with Mixed Numbers
Simplify: \[ \frac{1\frac{7}{8} \times 2\frac{2}{5}}{6\frac{3}{4} \div \frac{3}{4}} \]Solution:
Step 1: Convert all mixed numbers to improper fractions: \[ 1\frac{7}{8} = \frac{15}{8}, \quad 2\frac{2}{5} = \frac{12}{5}, \quad 6\frac{3}{4} = \frac{27}{4} \] Step 2: Multiply the fractions in the numerator: \[ \frac{15}{8} \times \frac{12}{5} = \frac{180}{40} = \frac{9}{2} \] Step 3: Divide the fractions in the denominator: \[ \frac{27}{4} \div \frac{3}{4} = \frac{27}{4} \times \frac{4}{3} = \frac{108}{12} = 9 \] Step 4: Divide the numerator by the denominator: \[ \frac{\frac{9}{2}}{9} = \frac{9}{2} \times \frac{1}{9} = \frac{1}{2} \] Final answer: \[ \boxed{\frac{1}{2}} \]