Whole Numbers, Fractions & Decimals — Study Notes

Overview

Whole Numbers, Fractions & Decimals form the absolute foundation of Elementary Arithmetic in UPSSSC PET. These topics appear in 8–12 questions directly and underpin almost every numerical reasoning problem in the exam. Mastery here means speed and accuracy—two critical advantages in a competitive exam where every second counts.

You must be fluent in four core operations (addition, subtraction, multiplication, division) across all three number types. The exam tests your ability to **convert** between fractions and decimals, **order** mixed sets of numbers, understand **place value** in decimals, and apply these skills to word problems. Questions range from straightforward computation to tricky comparison problems designed to catch careless mistakes.

Most PET arithmetic questions reward pattern recognition and mental math over lengthy written calculations. Spend time building intuition: recognize common fraction-decimal pairs (½ = 0.5, ¼ = 0.25, ⅓ = 0.333..., ⅕ = 0.2), practice ordering mixed number sets quickly, and drill place-value positioning to avoid decimal point errors.

Key Concepts

• **Whole numbers** are the set {0, 1, 2, 3, ...}. They include zero and all positive integers. No fractions, no negatives. Operations follow standard arithmetic rules: commutative (a + b = b + a), associative ((a + b) + c = a + (b + c)), and distributive (a × (b + c) = a×b + a×c).

• **Fractions** represent parts of a whole, written as numerator/denominator. Proper fractions have numerator < denominator (⅗). Improper fractions have numerator ≥ denominator (⁷⁄₅). Mixed numbers combine a whole number and a proper fraction (1⅖ = ⁷⁄₅).

• **Like fractions** share the same denominator and can be added/subtracted directly. **Unlike fractions** require finding a common denominator (usually the LCM of denominators) before operations.

• **Decimals** are another way to represent fractions using base-10 place value. The decimal point separates the whole part from the fractional part. Each position right of the decimal is 1/10 of the previous: tenths (0.1), hundredths (0.01), thousandths (0.001).

• **Terminating decimals** end after a finite number of digits (0.75 = ¾). **Non-terminating repeating decimals** have a pattern that repeats forever (⅓ = 0.333..., ¹¹⁄₉₉ = 0.111...).

• **Place value** determines a digit's worth by its position. In 3425.67: 3 is in thousands (3000), 4 in hundreds (400), 2 in tens (20), 5 in units (5), 6 in tenths (0.6), 7 in hundredths (0.07).

• **Ordering** numbers means arranging them from smallest to largest (ascending) or largest to smallest (descending). For mixed sets of fractions and decimals, convert all to the same form first.

• **Conversion trick**: Fraction to decimal—divide numerator by denominator. Decimal to fraction—write the decimal digits as numerator, place value as denominator, then simplify (0.125 = 125/1000 = ⅛).

Formulas / Key Facts

**Operations with Fractions:**

• Addition/Subtraction: a/b ± c/d = (ad ± bc)/bd. For same denominator: a/b ± c/b = (a ± c)/b.
• Multiplication: (a/b) × (c/d) = ac/bd. Cancel common factors before multiplying to simplify.
• Division: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc. "Invert and multiply."
• Mixed to improper: 2³⁄₅ = (2×5 + 3)/5 = 13/5.
• Improper to mixed: Divide numerator by denominator; quotient is whole part, remainder is new numerator.

**Operations with Decimals:**

• Addition/Subtraction: Align decimal points vertically, then operate column-wise.
• Multiplication: Ignore decimals, multiply as whole numbers, then place decimal point. Total decimal places in answer = sum of decimal places in factors.
• Division: Move decimal point in divisor to make it whole, move decimal point in dividend by the same number of places, then divide.

**Place Value:** In the number 456.789 — 4 is in hundreds place (400), 5 in tens (50), 6 in ones (6), 7 in tenths (0.7), 8 in hundredths (0.08), 9 in thousandths (0.009).

**Common Conversions:**

• ½ = 0.5, ¼ = 0.25, ¾ = 0.75, ⅕ = 0.2, ⅖ = 0.4, ⅗ = 0.6, ⅘ = 0.8, ⅓ = 0.333..., ⅔ = 0.666..., ⅙ = 0.1666..., ⅛ = 0.125, ⅜ = 0.375, ⅝ = 0.625, ⅞ = 0.875, 1/10 = 0.1, 1/100 = 0.01.

**Ordering Rule:** For fractions with same numerator, larger denominator = smaller fraction. For same denominator, larger numerator = larger fraction. For mixed sets, convert all to decimals or all to fractions with common denominator.

Worked Examples

**Example 1: Add unlike fractions** Compute ⅔ + ⅗.

*Solution:* Find LCM of denominators 3 and 5: LCM(3,5) = 15. Convert each fraction: ⅔ = 10/15, ⅗ = 9/15. Add: 10/15 + 9/15 = 19/15 = 1⁴⁄₁₅. **Answer:** 1⁴⁄₁₅ or 1.2666...

**Example 2: Multiply decimals** Calculate 2.5 × 1.3.

*Solution:* Ignore decimals: 25 × 13 = 325. Count decimal places: 2.5 has 1 place, 1.3 has 1 place → total 2 places. Place decimal: 325 → 3.25. **Answer:** 3.25

**Example 3: Convert and order** Arrange in ascending order: 0.8, ⅔, ⅗, 0.75.

*Solution:* Convert fractions to decimals: ⅔ ≈ 0.6667, ⅗ = 0.6, 0.75 = 0.75, 0.8 = 0.8. Order: 0.6 < 0.6667 < 0.75 < 0.8. **Answer:** ⅗, ⅔, 0.75, 0.8.

**Example 4: Divide fractions** Solve (⅘) ÷ (⅖).

*Solution:* Invert the second fraction and multiply: (⅘) × (⁵⁄₂) = (4×5)/(5×2) = 20/10 = 2. **Answer:** 2

Common Mistakes

**Mistake 1:** Adding/subtracting fractions without common denominator. *Wrong:* ½ + ⅓ = (1+1)/(2+3) = ⅖. This is incorrect algebra. *Fix:* Find LCM of denominators (6). Convert: ½ = ³⁄₆, ⅓ = ²⁄₆. Add: ³⁄₆ + ²⁄₆ = ⁵⁄₆.

**Mistake 2:** Misplacing the decimal point in multiplication. *Wrong:* 1.2 × 3.4 = 408 (treating as 12 × 34 = 408). *Fix:* 12 × 34 = 408, but 1.2 has 1 decimal place, 3.4 has 1 decimal place → total 2 places → 4.08.

**Mistake 3:** Confusing division of fractions with multiplication. *Wrong:* (⅗) ÷ (⅔) = (3×2)/(5×3) = ⅔. *Fix:* Invert the second fraction: (⅗) × (³⁄₂) = 9/10 = 0.9.

**Mistake 4:** Ordering fractions by numerator or denominator alone. *Wrong:* Thinking ⅔ > ¾ because 3 > 2. *Fix:* Convert to common denominator or decimals. ⅔ = 8/12, ¾ = 9/12. So ¾ > ⅔.

**Mistake 5:** Forgetting to simplify fractions. *Wrong:* Leaving the answer as 12/16 instead of ¾. *Fix:* Always reduce to lowest terms by dividing numerator and denominator by GCD (here, 4).

Quick Reference

• **Fraction → Decimal:** Divide numerator by denominator.
• **Decimal → Fraction:** Write as numerator over place-value denominator, simplify.
• **Mixed → Improper:** Multiply whole part by denominator, add numerator.
• **Improper → Mixed:** Divide numerator by denominator; quotient = whole, remainder/denominator = fraction.
• **Add/Subtract fractions:** Convert to common denominator (LCM), then operate on numerators.
• **Multiply fractions:** Multiply numerators, multiply denominators, simplify.
• **Divide fractions:** Multiply by the reciprocal (flip the second fraction).
• **Decimal operations:** Align decimal points (add/subtract); count decimal places (multiply); shift decimal points (divide).
• **Ordering tip:** Convert all to decimals for quick comparison, or find common denominator for fractions.