Relationship Between Numbers — Study Notes

Overview

Relationship Between Numbers forms the foundation of Elementary Mathematics in SSC GD. This topic tests your ability to compare, order, and establish logical connections between different types of numbers. Questions appear directly (compare two numbers, arrange in order) and indirectly (as part of number series, ratio problems, or data interpretation). Mastering this topic builds number sense—the intuitive understanding of magnitude and position that speeds up calculation across all math sections.

Expect 2–4 direct questions in the exam. The difficulty ranges from straightforward comparison of decimals and fractions to multi-step ordering problems involving mixed number types. Success requires quick mental comparison techniques rather than lengthy written calculations. Students who internalize comparison shortcuts save 30–45 seconds per question, a significant advantage in the time-pressured SSC GD math section.

The key challenge is handling mixed formats—comparing a fraction like 3/7 with a decimal like 0.44 or a percentage like 42%. You must develop reflexes for converting between formats and spotting relationships without full conversion. This topic directly supports your performance in Percentage, Ratio and Proportion, and even Data Interpretation sections.

Key Concepts

• **Natural hierarchy**: Natural numbers ⊂ Whole numbers ⊂ Integers ⊂ Rational numbers ⊂ Real numbers. Every natural number is also a whole number, integer, rational, and real number, but the reverse isn't true.

• **Comparison principle**: For positive numbers, compare magnitudes directly. For negative numbers, the number closer to zero is greater (−2 > −5). When comparing opposite-sign numbers, positive is always greater than negative.

• **Decimal comparison**: Compare from left to right—whole part first, then first decimal place, then second decimal place, and so on. 0.56 > 0.499 even though 499 > 56, because the first decimal place decides (5 > 4).

• **Fraction comparison**: When denominators are equal, larger numerator means larger fraction. When numerators are equal, smaller denominator means larger fraction. For unlike fractions, cross-multiply or convert to common denominator.

• **Percentage as hundredths**: Any percentage is that number divided by 100. So 45% = 45/100 = 0.45. This makes percentage-decimal-fraction conversions straightforward.

• **Ordering strategy**: When arranging multiple numbers, convert all to the same format (usually decimals) for easiest comparison. Pick the format that requires least calculation.

• **Inequality transitivity**: If A > B and B > C, then A > C. Use this to eliminate options in multiple-choice questions without comparing every pair.

• **Proportional relationships**: If two numbers maintain a constant ratio, they're directly proportional. If their product is constant, they're inversely proportional. These relationships help predict behavior when one number changes.

Formulas / Key Facts

• **Number of integers from A to B inclusive** = B − A + 1. From 15 to 50 = 50 − 15 + 1 = 36 integers.

• **Cross-multiplication for fractions**: a/b > c/d if and only if a×d > b×c (when b, d > 0).

• **Decimal to percentage**: Multiply by 100. 0.675 = 67.5%.

• **Percentage to decimal**: Divide by 100. 82% = 0.82.

• **Fraction to decimal**: Divide numerator by denominator. 3/8 = 0.375.

• **Fraction to percentage**: (Numerator/Denominator) × 100. 5/8 = (5/8) × 100 = 62.5%.

• **Least Common Multiple (LCM)** for comparison: Convert fractions to equivalent fractions with denominator = LCM of original denominators.

• **Benchmark fractions**: Memorize 1/2 = 0.5, 1/3 ≈ 0.33, 1/4 = 0.25, 1/5 = 0.2, 1/8 = 0.125, 3/4 = 0.75 for quick mental comparison.

Worked Examples

**Example 1: Compare 5/8, 0.62, and 64%**

Step 1: Convert all to decimals.

• 5/8 = 0.625 (divide 5 by 8)
• 0.62 remains 0.62
• 64% = 64/100 = 0.64

Step 2: Compare decimals: 0.62 < 0.625 < 0.64

Step 3: Write in original form: 0.62 < 5/8 < 64%

**Example 2: Arrange 7/12, 5/9, 11/18 in ascending order**

Step 1: Find LCM of denominators 12, 9, 18. LCM = 36

Step 2: Convert to equivalent fractions:

• 7/12 = 21/36 (multiply by 3)
• 5/9 = 20/36 (multiply by 4)
• 11/18 = 22/36 (multiply by 2)

Step 3: Compare numerators: 20 < 21 < 22

Step 4: Original order: 5/9 < 7/12 < 11/18

**Example 3: Which is greater: 3/7 or 5/11?**

Step 1: Cross-multiply.

• 3 × 11 = 33
• 7 × 5 = 35

Step 2: Compare products: 35 > 33

Step 3: Therefore 5/11 > 3/7 (the fraction whose cross-product is larger wins)

Common Mistakes

**Mistake**: Comparing decimals by number of digits after decimal point—thinking 0.5 < 0.125 because 125 > 5. **Fix**: Compare place by place from left. 0.5 = 0.500, and 5 > 1 in the first decimal place, so 0.5 > 0.125.

**Mistake**: Forgetting that larger denominator means smaller fraction when numerators are equal. Thinking 1/7 > 1/5. **Fix**: Visualize: one pizza divided among 7 people gives smaller slices than among 5 people. 1/5 > 1/7.

**Mistake**: Incorrect cross-multiplication direction—comparing a/b with c/d but mixing up which products to compare. **Fix**: Write clearly: a/b vs c/d → compare (a×d) with (b×c). Left numerator with right denominator, right numerator with left denominator.

**Mistake**: Treating negative numbers like positives—thinking −8 > −3 because 8 > 3. **Fix**: On number line, right is greater. −3 is right of −8, so −3 > −8. Number closer to zero is greater among negatives.

**Mistake**: Converting fraction to decimal by dividing denominator by numerator (3/4 calculated as 4÷3). **Fix**: Fraction means numerator ÷ denominator always. 3/4 = 3 ÷ 4 = 0.75.

Quick Reference

• Cross-multiply to compare fractions: a/b vs c/d → compare a×d with b×c.
• Convert to common format (decimals easiest) when comparing mixed number types.
• For decimals, compare digit-by-digit from left starting after the decimal point.
• Negative number closer to zero is greater: −2 > −5 > −10.
• Percentage = decimal × 100; decimal = fraction by division.
• Memorize benchmark fractions: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, 1/8 = 0.125.