Fraction, Ratio & Proportion

Fraction किसी whole (पूर्ण) का एक part (भाग) होता है। यह p/q के रूप में व्यक्त किया जाता है, जहाँ p = Numerator (अंश) और q = Denominator (हर)।

Example:

• 3/4 means: 4 equal parts में से 3 parts
• 1/2 means: 2 equal parts में से 1 part
• 5/8 means: 8 equal parts में से 5 parts

Key Point: Denominator never can be zero (q ≠ 0).

• Proper Fraction — उचित भिन्न: Numerator < Denominator (e.g., 3/4, 5/8)
• Improper Fraction — अनुचित भिन्न: Numerator ≥ Denominator (e.g., 5/4, 7/3)
• Mixed Fraction — मिश्रित भिन्न: Whole number + Proper fraction (e.g., 1½, 2¾)
• Like Fractions — समान भिन्न: Same denominator (e.g., 2/7, 3/7, 5/7)
• Unlike Fractions — असमान भिन्न: Different denominators (e.g., 2/5, 3/7, 4/9)
• Equivalent Fractions — तुल्य भिन्न: Same value with different numerator/denominator (e.g., 1/2 = 2/4 = 3/6 = 4/8)

Conversion:

• Improper → Mixed: Divide numerator by denominator. Example: 7/3 = 2⅓
• Mixed → Improper: (Whole × Denominator) + Numerator / Denominator. Example: 2¾ = (2×4+3)/4 = 11/4

• Addition/Subtraction:
  • Like fractions: Add/subtract numerators, keep denominator same. (e.g., 2/7 + 3/7 = 5/7)
  • Unlike fractions: Find LCM of denominators, convert to like fractions, then add/subtract. (e.g., 1/2 + 1/3 = 3/6 + 2/6 = 5/6)
• Multiplication:
  • Multiply numerators together, multiply denominators together. (e.g., 2/3 × 4/5 = 8/15)
  • Simplify before multiplying if possible (cross-cancellation).
• Division:
  • Multiply by reciprocal of the second fraction. (e.g., 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6)

Formula:

• Like Fractions: Larger numerator = larger fraction. (e.g., 5/7 > 3/7)
• Unlike Fractions: Cross-multiply. (e.g., Compare 3/4 and 5/8 → 3×8=24, 5×4=20 → 24>20 → 3/4 > 5/8)
• Decimal Method: Convert to decimals and compare. (e.g., 3/4 = 0.75, 5/8 = 0.625 → 0.75 > 0.625)

Rule for Comparison:

Ratio दो या दो से अधिक quantities के बीच के relationship को दर्शाता है।

Example:

• Ratio of 4 to 6 is 4:6 = 2:3 (simplified)
• Ratio 2:3 means: for every 2 parts of first, there are 3 parts of second

Key Points:

• Ratio has no units (same units required)
• Ratio can be expressed in lowest terms (simplified)
• Order matters: a:b ≠ b:a (unless a=b)

• a : b = c : d → a/b = c/d
• Multiplying/Dividing by same number: a:b = ka:kb (k ≠ 0)
• Inverse Ratio: b:a (reciprocal)
• Duplicate Ratio: a² : b²
• Triplicate Ratio: a³ : b³
• Sub-duplicate Ratio: √a : √b
• Sub-triplicate Ratio: ∛a : ∛b
• Componendo: (a+b)/b = (c+d)/d
• Dividendo: (a−b)/b = (c−d)/d
• Componendo and Dividendo: (a+b)/(a−b) = (c+d)/(c−d)

Compound Ratio two or more ratios को multiply करके बनाया जाता है।

Example:

• Compound ratio of 2:3 and 4:5 = (2×4) : (3×5) = 8:15
• Compound ratio of 2:3, 4:5, and 6:7 = (2×4×6) : (3×5×7) = 48:105 = 16:35

Application: Used in problems involving multiple ratios (e.g., A:B = 2:3, B:C = 4:5 → A:B:C = 8:12:15)

Proportion दो ratios की equality है।

Example:

• 2:3 = 4:6 (because 2/3 = 4/6)
• This is read as "2 is to 3 as 4 is to 6"

Terms:

• a, d = Extremes (चरम पद)
• b, c = Means (मध्य पद)
• Product of Extremes = Product of Means

• Direct Proportion — प्रत्यक्ष समानुपात: When one quantity increases, the other also increases (or decreases together). x ∝ y → x/y = k (constant).
• Inverse Proportion — व्युत्क्रम समानुपात: When one quantity increases, the other decreases. x ∝ 1/y → x × y = k (constant).
• Continued Proportion — अनुक्रमानुपात: a, b, c are in continued proportion if a/b = b/c → b² = ac. Here b is the mean proportional.

Examples:

• Direct: More workers → More work done (in same time). Speed ∝ Distance (time constant).
• Inverse: More workers → Less time to finish work. Speed ∝ 1/Time (distance constant).

Partnership Problems:

• Profit/Loss is divided in the ratio of Capital × Time
• If capitals are invested for the same time → profit ratio = capital ratio
• If capitals are invested for different times → profit ratio = capital1×time1 : capital2×time2

Example:

• A invests ₹5000 for 6 months, B invests ₹4000 for 8 months → Profit ratio = (5000×6) : (4000×8) = 30000 : 32000 = 15:16

Mixture/Alligation Problems:

• Used to find the ratio in which two quantities should be mixed to get a desired average
• Alligation Rule: (Cheaper quantity) / (Dearer quantity) = (Mean price − Cheaper price) / (Dearer price − Mean price)

• Fraction: p/q (q≠0), Proper (p
• Ratio: a:b = a/b
• Proportion: a:b :: c:d → ad = bc
• Compound Ratio: a:b × c:d = ac:bd
• Direct Proportion: x/y = k
• Inverse Proportion: x×y = k
• Continued Proportion: a:b = b:c → b² = ac
• Partnership: Profit ratio = Capital × Time
• Alligation Rule: (C)/(D) = (M−C)/(D−M)

• Trap 1: Ratio a:b means a/b = ? → ❌ It means for every a, there are b.
• Trap 2: Direct proportion = always increasing → ❌ Both can decrease together too.
• Trap 3: Inverse proportion = x∝y → ❌ x∝1/y (not x∝y)
• Trap 4: Proportion a:b::c:d → a/d = b/c → ❌ ad = bc (not a/d = b/c)
• Trap 5: Compound ratio = sum of ratios → ❌ It is product of ratios.
• Trap 6: Partnership profit = only capital → ❌ It is capital × time.
• Trap 7: Mixed fraction to improper: 2¾ = 11/4 → ❌ Correct: (2×4+3)/4 = 11/4.

What is 2/3 + 3/4?

• A. 5/7
• B. 17/12
• C. 7/12
• D. 15/12

Explanation: LCM of 3 and 4 is 12 → (2/3 = 8/12, 3/4 = 9/12) → 8/12 + 9/12 = 17/12.

Which is greater: 5/8 or 3/5?

• A. 5/8
• B. 3/5
• C. Both equal
• D. Cannot be determined

Explanation: Cross multiply: 5×5=25, 3×8=24 → 25>24 → 5/8 > 3/5.

Simplify the ratio 12:18.

• A. 3:4
• B. 2:3
• C. 4:6
• D. 6:9

Explanation: Divide both by HCF (6) → 12÷6=2, 18÷6=3 → 2:3.

Find the value of x: 2:3 = 8:x

• A. 6
• B. 10
• C. 12
• D. 14

Explanation: 2/3 = 8/x → 2×x = 3×8 → 2x = 24 → x = 12.

Find the compound ratio of 2:3 and 4:5.

• A. 6:8
• B. 8:15
• C. 15:8
• D. 6:15

Explanation: (2×4) : (3×5) = 8:15.

A invests ₹5000 for 6 months, B invests ₹4000 for 8 months. What is the profit ratio?

• A. 15:16
• B. 16:15
• C. 5:4
• D. 4:5

Explanation: A = 5000×6 = 30000, B = 4000×8 = 32000 → Ratio = 30000:32000 = 15:16.

If 4, x, 9 are in continued proportion, find x.

• A. 5
• B. 6
• C. 7
• D. 8

Explanation: 4:x = x:9 → x² = 4×9 = 36 → x = 6.

If 10 workers can complete work in 12 days, how many workers are needed to complete it in 8 days?

• A. 10
• B. 12
• C. 15
• D. 20

Explanation: Inverse proportion: 10×12 = W×8 → W = 120/8 = 15.

Fraction Types:

• Proper (p

Ratio Properties:

• a:b = ka:kb, Inverse = b:a, Duplicate = a²:b², Triplicate = a³:b³
• Sub-duplicate = √a:√b, Sub-triplicate = ∛a:∛b

Proportion Types:

• Direct (x/y = k), Inverse (x×y = k), Continued (a:b = b:c → b² = ac)

Partnership:

• Profit ratio = Capital × Time

Exam Formula:

UGC NET में Fraction, Ratio & Proportion के questions basic arithmetic और logical application को test करते हैं।