2.1 Fractions
What is Simplifying?
Simplifying fractions is the process of reducing a fraction to its simplest form. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1.
A mixed number has a whole number and a fraction part (e.g., $2\frac{1}{2}$).
An improper fraction has a numerator larger than its denominator (e.g., $\frac{5}{2}$).
Simplifying fractions is the process of reducing a fraction to its simplest form. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1.
Using Highest Common Factor (HCF)
To simplify a fraction, divide both the numerator and denominator by their HCF.
To simplify a fraction, divide both the numerator and denominator by their HCF.
Example: Simplify $\frac{6}{8}$
Step 1: Find the HCF of 6 and 8
Factors of 6: 1, 2, 3, 6
Factors of 8: 1, 2, 4, 8
HCF = 2
Step 2: Divide both by the HCF
$$\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}$$
Step 1: Find the HCF of 6 and 8
Factors of 6: 1, 2, 3, 6
Factors of 8: 1, 2, 4, 8
HCF = 2
Step 2: Divide both by the HCF
$$\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}$$
More Examples:
Mixed Numbers and Improper Fractions
• $\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$ (HCF = 6)
• $\frac{15}{25} = \frac{15 \div 5}{25 \div 5} = \frac{3}{5}$ (HCF = 5)
• $\frac{15}{25} = \frac{15 \div 5}{25 \div 5} = \frac{3}{5}$ (HCF = 5)
A mixed number has a whole number and a fraction part (e.g., $2\frac{1}{2}$).
An improper fraction has a numerator larger than its denominator (e.g., $\frac{5}{2}$).
Converting Mixed Number → Improper Fraction
Example: Convert $2\frac{1}{2}$ to an improper fraction
Step 1: Multiply whole number by denominator: $2 \times 2 = 4$
Step 2: Add the numerator: $4 + 1 = 5$
Step 3: Keep the same denominator
$$2\frac{1}{2} = \frac{5}{2}$$
Step 1: Multiply whole number by denominator: $2 \times 2 = 4$
Step 2: Add the numerator: $4 + 1 = 5$
Step 3: Keep the same denominator
$$2\frac{1}{2} = \frac{5}{2}$$
Converting Improper Fraction → Mixed Number
Example: Convert $\frac{11}{4}$ to a mixed number
Step 1: Divide numerator by denominator: $11 \div 4 = 2$ remainder $3$
Step 2: The quotient is the whole number, remainder is the new numerator
$$\frac{11}{4} = 2\frac{3}{4}$$
Step 1: Divide numerator by denominator: $11 \div 4 = 2$ remainder $3$
Step 2: The quotient is the whole number, remainder is the new numerator
$$\frac{11}{4} = 2\frac{3}{4}$$
🧮 Fraction Simplifier
Numerator:
Denominator:
Numerator:
Denominator:
🧮 Fraction Converter
Mixed → Improper:
/
/
Improper → Mixed:
/
/
Real Life Uses:
• Cooking: Simplifying recipe measurements (e.g., $\frac{4}{8}$ cup = $\frac{1}{2}$ cup)
• Shopping: Understanding discounts and sale prices
• DIY: Measuring materials in simplest terms for easier cutting
• Time: Expressing durations ($\frac{30}{60}$ hour = $\frac{1}{2}$ hour)
• Cooking: Simplifying recipe measurements (e.g., $\frac{4}{8}$ cup = $\frac{1}{2}$ cup)
• Shopping: Understanding discounts and sale prices
• DIY: Measuring materials in simplest terms for easier cutting
• Time: Expressing durations ($\frac{30}{60}$ hour = $\frac{1}{2}$ hour)
The Golden Rule
When fractions have the same denominator, simply add or subtract the numerators and keep the denominator the same.
When fractions have different denominators, find the Least Common Multiple (LCM) of the denominators first.
⚠️ FRACTIONS MUST HAVE THE SAME DENOMINATOR TO BE ADDED OR SUBTRACTED!
Same DenominatorsWhen fractions have the same denominator, simply add or subtract the numerators and keep the denominator the same.
Examples with Same Denominators:
Different Denominators
$$\frac{3}{8} + \frac{2}{8} = \frac{3 + 2}{8} = \frac{5}{8}$$
$$\frac{7}{10} - \frac{3}{10} = \frac{7 - 3}{10} = \frac{4}{10} = \frac{2}{5}$$
When fractions have different denominators, find the Least Common Multiple (LCM) of the denominators first.
Example: Calculate $\frac{1}{4} + \frac{1}{6}$
Step 1: Find the LCM of 4 and 6
Multiples of 4: 4, 8, 12, 16, 20...
Multiples of 6: 6, 12, 18, 24...
LCM = 12
Step 2: Convert both fractions to have denominator 12
$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
$\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}$
Step 3: Add the fractions
$$\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$$
Multiples of 4: 4, 8, 12, 16, 20...
Multiples of 6: 6, 12, 18, 24...
LCM = 12
Step 2: Convert both fractions to have denominator 12
$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
$\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}$
Step 3: Add the fractions
$$\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$$
Example: Calculate $\frac{5}{6} - \frac{1}{4}$
Step 1: Find the LCM of 6 and 4 → LCM = 12
Step 2: Convert both fractions
$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
Step 3: Subtract the fractions
$$\frac{10}{12} - \frac{3}{12} = \frac{7}{12}$$
Step 2: Convert both fractions
$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$
$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
Step 3: Subtract the fractions
$$\frac{10}{12} - \frac{3}{12} = \frac{7}{12}$$
🧮 Fraction Calculator
/
/
🎯 Find the LCM:
Real Life Uses:
• Cooking: Combining ingredient amounts ($\frac{1}{2}$ cup + $\frac{1}{4}$ cup)
• Time: Adding time periods ($\frac{1}{2}$ hour + $\frac{1}{4}$ hour = $\frac{3}{4}$ hour)
• Finance: Calculating portions of budgets or expenses
• Construction: Adding measurements for materials
• Cooking: Combining ingredient amounts ($\frac{1}{2}$ cup + $\frac{1}{4}$ cup)
• Time: Adding time periods ($\frac{1}{2}$ hour + $\frac{1}{4}$ hour = $\frac{3}{4}$ hour)
• Finance: Calculating portions of budgets or expenses
• Construction: Adding measurements for materials
Multiplying Fractions
To multiply fractions, simply multiply the numerators together and the denominators together. No need for common denominators!
To divide fractions, multiply by the reciprocal (flip) of the second fraction.
To multiply fractions, simply multiply the numerators together and the denominators together. No need for common denominators!
The Rule:
$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$
$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$
Example: Calculate $\frac{2}{3} \times \frac{4}{5}$
Step 1: Multiply the numerators: $2 \times 4 = 8$
Step 2: Multiply the denominators: $3 \times 5 = 15$
$$\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$$
Step 2: Multiply the denominators: $3 \times 5 = 15$
$$\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$$
More Examples:
Dividing Fractions
$$\frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{2 \times 4} = \frac{3}{8}$$
$$\frac{3}{5} \times \frac{2}{7} = \frac{3 \times 2}{5 \times 7} = \frac{6}{35}$$
To divide fractions, multiply by the reciprocal (flip) of the second fraction.
The Rule:
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$
Remember: Keep, Change, Flip (KCF)
Keep the first fraction, Change ÷ to ×, Flip the second fraction
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$
Remember: Keep, Change, Flip (KCF)
Keep the first fraction, Change ÷ to ×, Flip the second fraction
Example: Calculate $\frac{3}{4} \div \frac{2}{5}$
Step 1: Keep the first fraction: $\frac{3}{4}$
Step 2: Change division to multiplication
Step 3: Flip the second fraction: $\frac{2}{5} \rightarrow \frac{5}{2}$
$$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$$
Step 2: Change division to multiplication
Step 3: Flip the second fraction: $\frac{2}{5} \rightarrow \frac{5}{2}$
$$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$$
More Examples:
$$\frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4} = 1\frac{1}{4}$$
$$\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2$$
🧮 Fraction Calculator V2
/
/
🎯 Find the Reciprocal:
/
/
Real Life Uses:
• Cooking: Scaling recipes (halving or doubling ingredients)
• Construction: Calculating areas ($\frac{3}{4}$ m × $\frac{2}{3}$ m)
• Finance: Finding fractions of amounts (e.g., $\frac{1}{3}$ of a price)
• Sewing: Dividing fabric into equal parts
• Science: Diluting solutions and calculating concentrations
• Cooking: Scaling recipes (halving or doubling ingredients)
• Construction: Calculating areas ($\frac{3}{4}$ m × $\frac{2}{3}$ m)
• Finance: Finding fractions of amounts (e.g., $\frac{1}{3}$ of a price)
• Sewing: Dividing fabric into equal parts
• Science: Diluting solutions and calculating concentrations