Equivalent Ratios and Fractions Operations

Equivalent Ratios

• Two fractions are equivalent if they have the same value
• Equivalent fractions can be found by multiplying or dividing both numerator and denominator by the same number
• Example: 1/2 = 2/4 = 3/6
• To find an equivalent fraction, multiply or divide both numbers by the same number

Adding and Subtracting Fractions

• To add or subtract fractions, they must have the same denominator
• If the denominators are different, find the least common multiple (LCM) of the two denominators
• Convert both fractions to have the LCM as the denominator
• Add or subtract the numerators and keep the denominator the same
• Example: 1/4 + 1/6 = 3/12 + 2/12 = 5/12

Multiplying Fractions

• To multiply fractions, multiply the numerators and multiply the denominators
• Example: 1/2 × 3/4 = (1 × 3) / (2 × 4) = 3/8
• Multiplying by a whole number is the same as multiplying by an equivalent fraction
• Example: 2 × 3/4 = 2/1 × 3/4 = 6/4 = 3/2

Dividing Fractions

• To divide fractions, invert the second fraction (i.e., flip the numerator and denominator) and then multiply
• Example: 1/2 ÷ 3/4 = 1/2 × 4/3 = (1 × 4) / (2 × 3) = 4/6
• Dividing by a whole number is the same as dividing by an equivalent fraction
• Example: 3/4 ÷ 2 = 3/4 ÷ 2/1 = 3/4 × 1/2 = 3/8

Converting to Decimals

• A fraction can be converted to a decimal by dividing the numerator by the denominator
• Example: 3/4 = 3 ÷ 4 = 0.75
• Decimals can be converted back to fractions by finding the equivalent fraction
• Example: 0.5 = 1/2

Equivalent Ratios

• Equivalent fractions have the same value, and can be found by multiplying or dividing both numerator and denominator by the same number
• Example of equivalent fractions: 1/2 = 2/4 = 3/6
• Multiplying or dividing both numbers by the same number finds an equivalent fraction

Adding and Subtracting Fractions

• Fractions must have the same denominator to add or subtract
• If denominators differ, find the least common multiple (LCM) of both denominators to convert fractions
• Add or subtract the numerators and keep the denominator the same after converting
• Example: 1/4 + 1/6 = 3/12 + 2/12 = 5/12

Multiplying Fractions

• Multiply the numerators and multiply the denominators to multiply fractions
• Example: 1/2 × 3/4 = (1 × 3) / (2 × 4) = 3/8
• Multiplying by a whole number is the same as multiplying by an equivalent fraction
• Example: 2 × 3/4 = 2/1 × 3/4 = 6/4 = 3/2

Dividing Fractions

• Invert the second fraction (flip numerator and denominator) and then multiply to divide fractions
• Example: 1/2 ÷ 3/4 = 1/2 × 4/3 = (1 × 4) / (2 × 3) = 4/6
• Dividing by a whole number is the same as dividing by an equivalent fraction
• Example: 3/4 ÷ 2 = 3/4 ÷ 2/1 = 3/4 × 1/2 = 3/8

Converting to Decimals

• Divide the numerator by the denominator to convert a fraction to a decimal
• Example: 3/4 = 3 ÷ 4 = 0.75
• Convert decimals back to fractions by finding the equivalent fraction
• Example: 0.5 = 1/2