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Proportion Basics

Introduction

A proportion is a statement that two ratios are equal. It is usually written as a : b = c : d. This means that the ratio of a to b is the same as the ratio of c to d.

Proportion problems are very common in aptitude exams. They help in solving questions from topics like mixtures, geometry, speed-time-distance, and work. Once you know the rule of cross multiplication, these become easy to solve.

Pattern: Proportion Basics

Pattern

The key property of proportions:

If a : b = c : d, then a/b = c/d.

By cross multiplication: a × d = b × c.

This rule is used to find missing values or check if two ratios form a proportion.

Step-by-Step Example

Question

If 4 : 6 = 10 : x, find the value of x.

Solution

  1. Step 1: Write the proportion in fraction form.

    4 : 6 = 10 : x → 4/6 = 10/x

  2. Step 2: Apply cross multiplication.

    4 × x = 6 × 10

  3. Step 3: Simplify the equation.

    4x = 60 → x = 60 ÷ 4 = 15

  4. Step 4: Final Answer.

    The value of x is 15.

  5. Step 5: Quick Check.

    Left ratio: 4 : 6 = 2 : 3. Right ratio: 10 : 15 = 2 : 3. Both match ✅, so the answer is correct.

Question

Find the fourth proportional to 2, 5, and 8.

Solution

  1. Step 1: Understand the meaning.

    Fourth proportional means: if 2 : 5 = 8 : d, find d.

  2. Step 2: Apply the proportion property.

    2/5 = 8/d → 2 × d = 5 × 8

  3. Step 3: Simplify.

    2d = 40 → d = 40 ÷ 2 = 20

  4. Step 4: Final Answer.

    The fourth proportional is 20.

  5. Step 5: Quick Check.

    Ratios: 2 : 5 = 8 : 20 → both equal 0.4 ✅

Quick Variations

Third proportional: If a : b = b : c, then c is called the third proportional. Example: 2 : 4 = 4 : c → c = (4 × 4)/2 = 8.

Fourth proportional: If a : b = c : d, then d is the fourth proportional. Example: already shown above (2 : 5 = 8 : 20).

Trick to Always Use

  • Step 1: Write ratios as fractions.
  • Step 2: Apply cross multiplication (a × d = b × c).
  • Step 3: Solve for the missing value.
  • Step 4: For third/fourth proportionals, directly use the formula: - Third proportional = b² / a - Fourth proportional = (b × c) / a
  • Step 5: Always verify by simplifying both ratios.

Summary

Summary

In a proportion, a : b = c : d means a/b = c/d.

  • Key Rule: Cross multiplication → a × d = b × c.
  • Use: Solve for missing terms in proportion questions.
  • Specials: Third proportional = b²/a; Fourth proportional = (b × c)/a.
  • Always Check: Simplify both ratios and confirm equality.

With this rule, proportion problems become quick and reliable to solve in exams.

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.

Mock Test

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