Fractions

Overview

Fractions form the backbone of numerical reasoning in primary mathematics and appear frequently in KAR TET Paper I. This topic tests your conceptual understanding of part-whole relationships, ability to perform operations with fractions, and skill in converting between different fraction forms. Mastery here directly supports later topics like ratio, proportion, percentage, and decimal arithmetic.

For the exam, expect questions on identifying fraction types, comparing and ordering fractions, performing basic operations (addition, subtraction, multiplication, division), and converting between mixed numbers and improper fractions. Word problems involving fractions in real-life contexts—sharing, measuring, cooking—are common. From a pedagogy standpoint, you must understand how children develop fraction sense and the typical misconceptions they carry.

Key Concepts

• **Fraction as part-whole**: A fraction a/b represents 'a' equal parts out of 'b' total equal parts. The denominator tells how many equal parts the whole is divided into; the numerator tells how many parts are taken.

• **Proper fraction**: Numerator is less than denominator (e.g., 3/7). Value is always less than 1.

• **Improper fraction**: Numerator is greater than or equal to denominator (e.g., 9/4). Value is 1 or greater.

• **Mixed fraction (mixed number)**: A whole number combined with a proper fraction (e.g., 2¼). It represents a quantity greater than 1.

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

• **Decimal fractions**: Fractions with denominators that are powers of 10 (10, 100, 1000…). These convert directly to decimal notation (e.g., 7/10 = 0.7, 25/100 = 0.25).

• **Like and unlike fractions**: Like fractions share the same denominator; unlike fractions have different denominators. Operations require converting unlike to like fractions first.

• **Unit fraction**: A fraction with numerator 1 (e.g., 1/5, 1/8). Fundamental building block for understanding all fractions.

Formulas / Key Facts

| Concept | Formula / Rule | |---------|----------------| | Mixed to improper | a b/c = (a × c + b) / c | | Improper to mixed | Divide numerator by denominator; quotient = whole part, remainder = new numerator | | Equivalent fraction | a/b = (a × k) / (b × k) for any k ≠ 0 | | Simplest form | Divide numerator and denominator by their HCF | | Addition (like) | a/c + b/c = (a + b) / c | | Addition (unlike) | Find LCM of denominators, convert, then add numerators | | Subtraction | Same as addition but subtract numerators | | Multiplication | a/b × c/d = (a × c) / (b × d) | | Division | 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 |