Fractions and Decimals

Overview

Fractions and decimals form the backbone of numerical reasoning in the AP TET Mathematics section. This topic tests your ability to perform basic operations, convert between forms, and apply these concepts to word problems—skills essential for teaching primary and upper primary students.

Expect questions on simplifying fractions, comparing unlike fractions, performing operations on mixed numbers, and converting recurring decimals to fractions. These concepts also underpin later topics like percentage, ratio-proportion, and mensuration. A strong grip here saves time across multiple question types.

Mastery requires both procedural fluency (can you compute quickly and accurately?) and conceptual clarity (can you explain *why* 0.25 equals 1/4 to a student?). Both dimensions appear in AP TET questions.

Key Concepts

• **Fraction** = Part of a whole, written as numerator/denominator (e.g., 3/5 means 3 parts out of 5 equal parts).

• **Types of fractions**: Proper (numerator < denominator), Improper (numerator ≥ denominator), Mixed (whole number + proper fraction).

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

• **Decimal**: A way of writing fractions using place value (tenths, hundredths, thousandths). Example: 0.75 = 75/100.

• **Terminating decimals**: Decimals that end (e.g., 0.25, 0.125). Occur when denominator has only 2 and/or 5 as prime factors.

• **Non-terminating recurring decimals**: Decimals that repeat infinitely (e.g., 1/3 = 0.333...). Indicated by a bar over repeating digits.

• **Like fractions** share the same denominator; **unlike fractions** have different denominators. Convert to like fractions before adding/subtracting.

• **LCM of denominators** is used to find the common denominator for unlike fractions.

Formulas / Key Facts

| Operation | Rule | |-----------|------| | Addition/Subtraction of like fractions | a/c ± b/c = (a ± b)/c | | Addition/Subtraction of unlike fractions | Find LCM, convert to equivalent fractions, then add/subtract numerators | | Multiplication of fractions | (a/b) × (c/d) = (a×c)/(b×d) | | Division of fractions | (a/b) ÷ (c/d) = (a/b) × (d/c) — multiply by reciprocal | | Fraction to decimal | Divide numerator by denominator | | Decimal to fraction | Write decimal as fraction over power of 10, then simplify | | Recurring decimal to fraction | Let x = decimal, multiply to shift repeating part, subtract, solve |

**Key conversions to memorize:**

• 1/2 = 0.5
• 1/4 = 0.25, 3/4 = 0.75
• 1/5 = 0.2, 2/5 = 0.4, 3/5 = 0.6, 4/5 = 0.8
• 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, 7/8 = 0.875
• 1/3 = 0.333..., 2/3 = 0.666...

Worked Examples

**Example 1: Add 2/5 + 3/4**

Step 1: Find LCM of 5 and 4 → LCM = 20

Step 2: Convert to equivalent fractions

• 2/5 = (2×4)/(5×4) = 8/20
• 3/4 = (3×5)/(4×5) = 15/20

Step 3: Add numerators → 8/20 + 15/20 = 23/20 = 1 3/20

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**Example 2: Multiply 2 1/3 by 1 2/5**

Step 1: Convert mixed numbers to improper fractions

• 2 1/3 = (2×3 + 1)/3 = 7/3
• 1 2/5 = (1×5 + 2)/5 = 7/5

Step 2: Multiply → (7/3) × (7/5) = 49/15

Step 3: Convert to mixed number → 49 ÷ 15 = 3 remainder 4 → **3 4/15**

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**Example 3: Convert 0.272727... to a fraction**

Step 1: Let x = 0.272727...

Step 2: Multiply by 100 (since 2 digits repeat) → 100x = 27.272727...

Step 3: Subtract → 100x − x = 27.2727... − 0.2727... → 99x = 27

Step 4: Solve → x = 27/99 = **3/11** (after simplifying by dividing by 9)

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**Example 4: Divide 4.5 by 0.15**

Step 1: Remove decimals by multiplying both by 100 → 4.5 × 100 = 450 → 0.15 × 100 = 15

Step 2: Divide → 450 ÷ 15 = **30**

Common Mistakes

• **Forgetting to find LCM before adding unlike fractions** → Students add numerators and denominators directly (2/3 + 1/4 ≠ 3/7). Fix: Always convert to like fractions first.

• **Inverting the wrong fraction in division** → When dividing a/b ÷ c/d, students flip a/b instead of c/d. Fix: Remember "Keep-Change-Flip" — keep first fraction, change ÷ to ×, flip the second.

• **Not simplifying the final answer** → Leaving 8/12 instead of 2/3 loses marks. Fix: Always check if numerator and denominator share common factors.

• **Misplacing decimal point in multiplication/division** → Count total decimal places in multiplication; in division, shift decimal in both numbers equally. Fix: Verify by estimation (4.5 ÷ 0.15 should be larger than 4.5, not smaller).

• **Confusing terminating vs recurring decimals** → 1/6 = 0.1666... (recurring), not 0.16 (terminating). Fix: Actually perform the division or check denominator's prime factors.

• **Converting mixed numbers incorrectly** → For 3 2/5, students write 32/5 instead of (3×5+2)/5 = 17/5. Fix: Use formula: whole × denominator + numerator, all over denominator.

Quick Reference

• **Add/Subtract fractions**: Same denominator → operate on numerators; different denominators → find LCM first.

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

• **Divide fractions**: Multiply by reciprocal of divisor.

• **Decimal to fraction**: Place over 10, 100, or 1000 based on decimal places, then reduce.

• **Recurring decimal to fraction**: Set up equation, multiply to shift repeating block, subtract, solve.

• **Always simplify** your final answer to lowest terms.