In this chapter, you will learn
- —Understand different types of fractions and their properties
- —Perform addition, subtraction, multiplication, and division of fractions
- —Convert between fractions and decimals
- —Perform operations with decimal numbers
Types of Fractions
A fraction represents a part of a whole and is written as a/b where a is the numerator and b is the denominator.
Key Types:
- Proper Fraction: Numerator < Denominator (e.g., 3/5, 7/9) - value is less than 1
- Improper Fraction: Numerator ≥ Denominator (e.g., 7/5, 9/9) - value is ≥ 1
- Mixed Fraction: Combination of whole number and proper fraction (e.g., 2 3/5) = 13/5
- Like Fractions: Same denominator (e.g., 3/7, 5/7)
- Unlike Fractions: Different denominators (e.g., 2/3, 5/7)
Converting Mixed to Improper: 2 3/5 = (2×5 + 3)/5 = 13/5
Exam Tip
Exam questions often mix proper, improper, and mixed fractions. Practice converting between them quickly.
Common Mistake
Confusing improper fractions with mixed fractions. 7/5 is improper, 1 2/5 is its mixed form.
Operations with Fractions
Addition and Subtraction: Convert to like fractions, then add/subtract numerators.
Example: 2/3 + 1/5 = 10/15 + 3/15 = 13/15 (LCM of 3 and 5 is 15)
Multiplication: Multiply numerators and denominators separately. Simplify by canceling common factors.
Example: 3/4 × 2/5 = 6/20 = 3/10
Division: Multiply by the reciprocal (flip the second fraction).
Example: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8
Exam Tip
For division, always flip and multiply. It's the most reliable method and reduces errors.
Common Mistake
In division, students forget to flip the second fraction: 3/4 ÷ 2/5 ≠ 3/4 ÷ 5/2.
Decimals and Decimal Operations
A decimal is a fraction expressed using base 10 with a decimal point.
Decimal Place Value: 3.245 = 3 ones + 2 tenths + 4 hundredths + 5 thousandths
0.1 (tenths), 0.01 (hundredths), 0.001 (thousandths)
Addition & Subtraction: Align decimal points and add/subtract as usual.
Example: 2.35 + 1.08 = 3.43
Multiplication: Ignore decimals, multiply, then place decimal in answer equal to sum of decimal places.
Example: 2.5 × 1.2 = 25 × 12 = 300, then 3.00 = 3
Division: Move decimal point in divisor to make it whole, then adjust dividend similarly.
Example: 1.44 ÷ 1.2 = 14.4 ÷ 12 = 1.2
Exam Tip
Decimal point placement in multiplication and division is critical. Count decimal places carefully.
Common Mistake
Misplacing decimal points in multiplication: 0.2 × 0.3 = 0.06 (not 0.6)
Fraction and Decimal Conversion
Fraction to Decimal: Divide numerator by denominator.
Example: 3/4 = 0.75 (divide 3 by 4)
1/2 = 0.5, 1/4 = 0.25, 1/5 = 0.2, 3/8 = 0.375
Decimal to Fraction: Write decimal over place value denominator, then simplify.
Example: 0.35 = 35/100 = 7/20 (simplified)
0.6 = 6/10 = 3/5, 0.25 = 25/100 = 1/4
Real-World Application: Price conversion (₹2.50 = 2 50/100), percentages, measurements.
Exam Tip
Learn common fraction-decimal pairs: 1/2=0.5, 1/4=0.25, 3/4=0.75, 1/5=0.2, etc.
Common Mistake
Not simplifying fractions after conversion: 0.5 = 5/10 is correct but 1/2 is the simplified form.
Chapter Summary
Fractions and Decimals are essential for understanding parts and proportions. Key points:
- Types: Proper, improper, mixed, like, and unlike fractions
- Addition/Subtraction: Use LCM to make denominators equal
- Multiplication: Multiply numerators and denominators
- Division: Multiply by reciprocal (flip and multiply)
- Decimals: Base-10 representation with place values
- Conversion: Divide for fraction-to-decimal, simplify for decimal-to-fraction
Exam Focus: Operations on fractions, decimal place value, conversion problems, word problems involving measurements and money.