Fractions and Decimals

Overview

Fractions and decimals form the backbone of numerical reasoning in primary mathematics and appear consistently in GTET Paper-I and Paper-II. This topic tests your ability to perform arithmetic operations, convert between representations, and apply these concepts to word problems involving money, measurement, and daily-life situations.

For GTET, expect questions that combine multiple operations (e.g., adding fractions then converting to decimals), comparison problems, and application-based questions. Mastery here also supports topics like percentage, ratio-proportion, and mensuration. Focus on speed and accuracy—most errors come from careless mistakes in finding common denominators or misplacing decimal points.

Key Concepts

• **Fraction fundamentals**: A fraction a/b represents 'a' parts out of 'b' equal parts. The numerator (top) counts parts; the denominator (bottom) names the size of each part.

• **Types of fractions**: Proper fractions (numerator < denominator, e.g., 3/5), improper fractions (numerator ≥ denominator, e.g., 7/4), and mixed numbers (whole + fraction, e.g., 1¾).

• **Equivalent fractions**: Fractions that represent the same value (e.g., 2/4 = 1/2 = 3/6). Multiply or divide both numerator and denominator by the same non-zero number.

• **Lowest terms**: A fraction is in lowest terms when HCF of numerator and denominator is 1. Always simplify final answers.

• **Decimal place value**: Each position after the decimal point represents tenths, hundredths, thousandths, etc. In 0.375: 3 tenths + 7 hundredths + 5 thousandths.

• **Terminating vs recurring decimals**: Fractions with denominators having only 2 and 5 as prime factors give terminating decimals (e.g., 1/8 = 0.125). Others give recurring decimals (e.g., 1/3 = 0.333...).

• **Like and unlike fractions**: Like fractions share the same denominator; unlike fractions have different denominators and require conversion before addition/subtraction.

Formulas / Key Facts

| Operation | Formula/Method | |-----------|----------------| | Adding like fractions | a/c + b/c = (a+b)/c | | Subtracting like fractions | a/c − b/c = (a−b)/c | | Adding unlike fractions | Find LCM of denominators, convert, then add numerators | | Multiplying fractions | (a/b) × (c/d) = (a×c)/(b×d) | | Dividing fractions | (a/b) ÷ (c/d) = (a/b) × (d/c) — multiply by reciprocal | | Fraction to decimal | Divide numerator by denominator | | Decimal to fraction | Write decimal over appropriate power of 10, then simplify | | Mixed to improper | a b/c = (a×c + b)/c | | Improper to mixed | Divide numerator by denominator; quotient = whole part, remainder = new numerator |