Fractions and Decimals

Overview

Fractions and decimals form the backbone of arithmetic reasoning at the primary level and appear consistently in JKTET Paper I Mathematics. This topic tests your ability to perform operations (addition, subtraction, multiplication, division) on both fractions and decimals, convert between the two forms, and apply these skills to word problems.

For the TET exam, you must demonstrate not only computational accuracy but also conceptual clarity—understanding *why* procedures work, since pedagogy questions often probe whether you can explain these concepts to young learners. Expect 3–5 direct questions on this topic, often mixed with word problems involving money, measurement, or ratio contexts relevant to daily life in J&K.

Mastery here builds the foundation for percentage, ratio-proportion, and profit-loss topics that follow in the syllabus. A student confident with fractions and decimals will find the rest of commercial mathematics straightforward.

Key Concepts

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

• **Decimal place value**: In 45.378, the digits after the decimal point represent tenths (3), hundredths (7), and thousandths (8). Each place is 1/10 of the previous.

• **Fraction-decimal relationship**: Every fraction can be written as a decimal (divide numerator by denominator). Terminating decimals occur when the denominator has only 2 and 5 as prime factors.

• **Like and unlike fractions**: Like fractions share the same denominator; unlike fractions require a common denominator before adding or subtracting.

• **Reciprocal**: The reciprocal of a/b is b/a. Used in division of fractions ("invert and multiply").

Formulas / Key Facts

| Operation | Fractions | Decimals | |-----------|-----------|----------| | Addition/Subtraction | Convert to like fractions (same denominator), then add/subtract numerators | Align decimal points, then add/subtract as whole numbers | | Multiplication | (a/b) × (c/d) = ac/bd | Multiply as whole numbers; count total decimal places in both factors and place decimal in product | | Division | (a/b) ÷ (c/d) = (a/b) × (d/c) | Move decimal in divisor to make it whole; move decimal in dividend equally; then divide |

**Conversion formulas**:

• Fraction to decimal: Divide numerator by denominator (3/4 = 0.75)
• Decimal to fraction: Write decimal over appropriate power of 10, then simplify (0.45 = 45/100 = 9/20)

**LCM for unlike fractions**: To add 2/3 + 1/4, find LCM of 3 and 4 = 12, convert to 8/12 + 3/12 = 11/12.

**Key decimal equivalents to memorise**:

• 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75
• 1/5 = 0.2, 2/5 = 0.4, 1/8 = 0.125
• 1/3 = 0.333..., 2/3 = 0.666...

Worked Examples

**Example 1: Adding unlike fractions**

Find: 3/5 + 2/3

Step 1: Find LCM of 5 and 3 = 15

Step 2: Convert each fraction

• 3/5 = (3 × 3)/(5 × 3) = 9/15
• 2/3 = (2 × 5)/(3 × 5) = 10/15

Step 3: Add numerators: 9/15 + 10/15 = 19/15 = 1 4/15

**Example 2: Multiplying decimals**

Find: 2.5 × 0.04

Step 1: Ignore decimals, multiply as whole numbers: 25 × 4 = 100

Step 2: Count decimal places in original numbers: 1 (in 2.5) + 2 (in 0.04) = 3

Step 3: Place decimal 3 places from right in 100: 0.100 = 0.1

Answer: 0.1

**Example 3: Dividing fractions (word problem)**

A rope is 7/8 metre long. How many pieces of 1/4 metre each can be cut from it?

Step 1: Divide 7/8 by 1/4

Step 2: Invert and multiply: 7/8 × 4/1 = 28/8 = 7/2 = 3½

Answer: 3 complete pieces (with ½ piece remaining)

**Example 4: Decimal division**

Find: 4.56 ÷ 0.8

Step 1: Make divisor a whole number by multiplying both by 10

• 4.56 becomes 45.6
• 0.8 becomes 8

Step 2: Divide: 45.6 ÷ 8 = 5.7

Answer: 5.7

Common Mistakes

• **Adding fractions by adding numerators and denominators separately** → Wrong: 1/2 + 1/3 ≠ 2/5. Correct fix: Always find a common denominator first, then add only the numerators.

• **Forgetting to simplify the final answer** → 6/8 should be written as 3/4. Examiners may mark incomplete if not in lowest terms.

• **Misplacing the decimal in multiplication** → Students count decimal places incorrectly. Fix: Count total decimal places in *both* numbers being multiplied, not just one.

• **Inverting the wrong fraction in division** → In (a/b) ÷ (c/d), students sometimes invert a/b. Correct: Always invert the second fraction (the divisor), never the first.

• **Ignoring decimal alignment in addition/subtraction** → Adding 3.5 + 0.75 as 3.5 + 75 = 78.5 is wrong. Fix: Write 3.50 + 0.75 = 4.25, aligning decimal points vertically.

• **Converting recurring decimals incorrectly** → 0.333 is not exactly 1/3; it is 333/1000. For exact conversion of recurring decimals, use algebraic method or state the recurring notation (0.3̄).

Quick Reference

• **Unlike fractions**: Find LCM, convert, then add/subtract numerators only.

• **Multiply fractions**: Multiply across (numerator × numerator, denominator × denominator), then simplify.

• **Divide fractions**: Keep first, flip second, multiply.

• **Decimal multiplication**: Total decimal places = sum of decimal places in both factors.

• **Decimal division**: Shift decimal in divisor to make it whole; shift equally in dividend.

• **Always simplify**: Reduce fractions to lowest terms; remove trailing zeros in decimals.