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Proportion Basics

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Introduction

एक proportion वह statement है जिसमें दो ratios बराबर होते हैं। इसे आमतौर पर a : b = c : d के रूप में लिखा जाता है। इसका मतलब है कि a का b से अनुपात, c का d से अनुपात के बराबर है।

Proportion वाले प्रश्न aptitude exams में बहुत common होते हैं। ये topics जैसे mixtures, geometry, speed-time-distance, और work में मदद करते हैं। Cross multiplication का rule समझने के बाद ये questions बहुत आसान हो जाते हैं।

Pattern: Proportion Basics

Pattern: Proportion Basics

Proportion की मुख्य property:

अगर a : b = c : d, तो a/b = c/d.

Cross multiplication से: a × d = b × c.

इसी rule का उपयोग missing value निकालने या दो ratios proportion बनाते हैं या नहीं, यह check करने के लिए किया जाता है।

Step-by-Step Example

Question

अगर 4 : 6 = 10 : x, तो x का मान निकालो।

Solution

  1. Step 1: Proportion को fraction form में लिखो।

    4 : 6 = 10 : x → 4/6 = 10/x

  2. Step 2: Cross multiplication लगाओ।

    4 × x = 6 × 10

  3. Step 3: Equation simplify करो।

    4x = 60 → x = 60 ÷ 4 = 15

  4. Step 4: Final Answer.

    x का मान 15 है।

  5. Step 5: Quick Check.

    Left ratio: 4 : 6 = 2 : 3. Right ratio: 10 : 15 = 2 : 3. दोनों match करते हैं ✅, answer सही है।

Question

2, 5 और 8 का fourth proportional निकालो।

Solution

  1. Step 1: Meaning समझो।

    Fourth proportional का मतलब: अगर 2 : 5 = 8 : d, तो d निकालना है।

  2. Step 2: Proportion property लगाओ।

    2/5 = 8/d → 2 × d = 5 × 8

  3. Step 3: Simplify करो।

    2d = 40 → d = 40 ÷ 2 = 20

  4. Step 4: Final Answer.

    Fourth proportional 20 है।

  5. Step 5: Quick Check.

    Ratios: 2 : 5 = 8 : 20 → दोनों 0.4 के बराबर हैं ✅

Quick Variations

Third proportional: अगर a : b = b : c, तो c को third proportional कहते हैं। Example: 2 : 4 = 4 : c → c = (4 × 4)/2 = 8.

Fourth proportional: अगर a : b = c : d, तो d fourth proportional कहलाता है। Example: ऊपर दिखाया हुआ (2 : 5 = 8 : 20) जैसा।

Trick to Always Use

  • Step 1: Ratios को fractions में लिखो।
  • Step 2: Cross multiplication लगाओ (a × d = b × c).
  • Step 3: Missing value निकालो।
  • Step 4: Third/Fourth proportional के लिए direct formula use कर सकते हो: - Third proportional = b² / a - Fourth proportional = (b × c) / a
  • Step 5: हमेशा दोनों ratios simplify करके verify करो।

Summary

Proportion में a : b = c : d का मतलब होता है a/b = c/d.

  • Key Rule: Cross multiplication → a × d = b × c.
  • Use: Proportion वाले missing terms निकालने में।
  • Specials: Third proportional = b²/a; Fourth proportional = (b × c)/a.
  • Always Check: दोनों ratios simplify करके equality confirm करो।

इस rule से proportion वाले questions exams में जल्दी और accurate तरीके से solve हो जाते हैं।

Practice

(1/5)
1. If 4 : 6 = 10 : x, find the value of x.
easy
A. 12
B. 15
C. 20
D. 25

Solution

  1. Step 1: Write the proportion as fractions

    4/6 = 10/x.

  2. Step 2: Cross multiply to eliminate denominators

    4 × x = 6 × 10 = 60.

  3. Step 3: Solve for the unknown

    x = 60 ÷ 4 = 15.

  4. Final Answer:

    x = 15 → Option B

  5. Quick Check:

    Ratios: 4 : 6 = 2 : 3 and 10 : 15 = 2 : 3 ✅
Hint: Use cross multiplication a×d = b×c to solve quickly.
Common Mistakes: Forgetting to reduce ratios before comparing or solving.
2. If 2 : 5 = 8 : d, find d (the fourth proportional).
easy
A. 16
B. 18
C. 20
D. 22

Solution

  1. Step 1: Express the proportion

    2/5 = 8/d.

  2. Step 2: Cross multiply

    2 × d = 5 × 8 = 40.

  3. Step 3: Solve for d

    d = 40 ÷ 2 = 20.

  4. Final Answer:

    d = 20 → Option C

  5. Quick Check:

    Ratios: 2 : 5 = 0.4 and 8 : 20 = 0.4 ✅
Hint: Apply the direct formula for fourth proportional: d = (b × c)/a.
Common Mistakes: Mixing up third and fourth proportional formulae.
3. Find the third proportional to 6 and 12.
medium
A. 18
B. 20
C. 22
D. 24

Solution

  1. Step 1: Recall the third proportional formula

    If a : b = b : c then c = b² / a.

  2. Step 2: Substitute values

    Here a = 6, b = 12 → c = (12²)/6 = 144 ÷ 6 = 24.

  3. Final Answer:

    24 → Option D

  4. Quick Check:

    Check ratios → 6 : 12 = 1 : 2, 12 : 24 = 1 : 2 ✅
Hint: Use the formula c = b²/a directly to save time.
Common Mistakes: Confusing third proportional with mean proportion (√ab).
4. If 7 : x = 21 : 18, find x.
medium
A. 5
B. 6
C. 7
D. 8

Solution

  1. Step 1: Convert to fractional form

    7/x = 21/18.

  2. Step 2: Cross multiply

    7 × 18 = 21 × x → 126 = 21x.

  3. Step 3: Solve for x

    x = 126 ÷ 21 = 6.

  4. Final Answer:

    x = 6 → Option B

  5. Quick Check:

    Ratios → 7 : 6 ≈ 1.167 and 21 : 18 ≈ 1.167 ✅
Hint: Simplify fractions first before cross multiplying to avoid big numbers.
Common Mistakes: Cross multiplying incorrectly by mixing numerator and denominator.
5. If a : b = 3 : 4 and b : c = 8 : 9, find a : c.
hard
A. 1 : 1
B. 2 : 3
C. 3 : 5
D. 6 : 9

Solution

  1. Step 1: Write the given ratios

    a/b = 3/4 and b/c = 8/9.

  2. Step 2: Make the middle term (b) equal

    LCM of 4 and 8 = 8 → rewrite a/b = 6/8 and b/c = 8/9.

  3. Step 3: Form combined ratio

    a : b : c = 6 : 8 : 9 → so a : c = 6 : 9 = 2 : 3.

  4. Final Answer:

    2 : 3 → Option B

  5. Quick Check:

    a : c = 2 : 3 ratio confirmed ✅
Hint: Equalize the middle term (b) to link two proportions easily.
Common Mistakes: Not aligning the common term before combining proportions.