Fractions and Decimals

Overview

Fractions and decimals form the backbone of numerical reasoning in primary and upper-primary mathematics. For UPTET, this topic appears consistently across both Paper I (Classes 1–5) and Paper II (Classes 6–8), testing your ability to perform operations, convert between forms, and solve word problems. Questions typically range from straightforward computation to application-based problems involving money, measurement, and comparison.

Mastery here is non-negotiable because fractions and decimals underpin later topics—ratio and proportion, percentage, profit-loss, and mensuration. A teacher must understand not just the procedures but also the conceptual models (part-whole, division interpretation) to explain these ideas effectively to children. Expect 3–5 direct questions in the Mathematics section, plus indirect use in other problems.

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Key Concepts

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

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

• **Improper fraction**: Numerator ≥ Denominator (e.g., 9/4). Value is 1 or greater.

• **Mixed fraction**: Combination of a whole number and a proper fraction (e.g., 2¼). Every improper fraction can be written as a mixed fraction and vice versa.

• **Equivalent fractions**: Fractions representing the same value (e.g., 1/2 = 2/4 = 3/6). Obtained by multiplying or dividing numerator and denominator by the same non-zero number.

• **Decimal as a fraction with denominator 10, 100, 1000, etc.**: 0.3 = 3/10; 0.47 = 47/100; 2.135 = 2135/1000.

• **Place value in decimals**: Tenths (1/10), hundredths (1/100), thousandths (1/1000) moving right from the decimal point.

• **Like and unlike fractions**: Like fractions have the same denominator; unlike fractions have different denominators. Converting to like fractions is essential before adding or subtracting.

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Formulas / Key Facts

| Operation | Rule | |-----------|------| | **Converting improper to mixed** | Divide numerator by denominator → Quotient = whole part, Remainder = numerator of fractional part. E.g., 17/5 = 3 and 2/5. | | **Converting mixed to improper** | (Whole × Denominator) + Numerator, keep same denominator. E.g., 4 and 3/7 = (4×7+3)/7 = 31/7. | | **Addition/Subtraction of fractions** | Make denominators equal (LCM), then add/subtract numerators. | | **Multiplication of fractions** | (a/b) × (c/d) = ac/bd. Simplify before or after multiplying. | | **Division of fractions** | (a/b) ÷ (c/d) = (a/b) × (d/c). Multiply by the reciprocal. | | **Decimal to fraction** | Write digits after point over 10, 100, etc., then simplify. 0.75 = 75/100 = 3/4. | | **Fraction to decimal** | Divide numerator by denominator. 3/8 = 0.375. | | **Adding/Subtracting decimals** | Align decimal points, then add/subtract column-wise. | | **Multiplying decimals** | Multiply as whole numbers, count total decimal places in both factors, place decimal in product accordingly. | | **Dividing decimals** | Shift decimal in divisor to make it whole; shift same places in dividend; then divide. |