Ratio and Proportion – Complete Short Notes for Competitive Exams

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1. Basic Concepts

Ratio

• A ratio is a comparison of two quantities of the same kind by division.

• Represented as:

  $$\text{Ratio of a to b} = \frac{a}{b} \text{ or } a : b$$

• Example: Ratio of 8 to 4 = 8 : 4 = 2 : 1

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Proportion

• When two ratios are equal, they form a proportion.

  $$a:b=c:d\text{or}\frac{a}{b}=\frac{c}{\mathrm{d}}$$

• Example: (2 : 3 = 8 : 12) (both = 2/3)

• In (a : b = c : d), a and d are extremes, b and c are means.

  • Product of extremes = Product of means
    
    a × d = b × c
    
    

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2. Key Properties of Ratio

1. Ratio has no units (it’s a pure number).

2. If each term of a ratio is multiplied or divided by the same non-zero number, the ratio remains unchanged.

   • Example: (6 : 9 = 2 : 3) (dividing both by 3).

3. Ratios can be simplified like fractions.

4. For equal ratios, cross-products are equal:
   (a:b = c:d ⇒ a×d = b×c)

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3. Types of Ratios

1. Simple Ratio → Ratio of two quantities (e.g., 4:5).

2. Compound Ratio → Product of two or more ratios.

   • Example: (a:b) × (c:d) = (a×c):(b×d)

3. Duplicate Ratio → Square of the ratio.

   • Example: If a:b = 2:3, duplicate ratio = 4:9

4. Triplicate Ratio → Cube of the ratio.

   • Example: 2:3 → 8:27

5. Sub-duplicate Ratio → Square root of the ratio.

   • Example: 4:9 → 2:3

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4. Comparison of Ratios

To compare ratios a:b and c:d, convert both to fractions and compare:

$$\frac{a}{b} \text{ vs } \frac{c}{d}$$​
Whichever is greater, that ratio is higher.

Example: Compare 3:4 and 5:7

3/4 = 0.75, 5/7 ≈ 0.714 ⇒ 3:4 > 5:7

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5. Proportion Formulas

If (a:b = c:d), then:

a×d = b×c

Mean Proportion (Geometric Mean):

If (a, b, c) are in proportion,

$$a:b=b:c\Rightarrow b^2=a\times c\Rightarrow b=\sqrt{a\times c}$$

Third Proportion:

If (a : b = b : c),
then c is the third proportional to a and b.

$$c=\frac{b^2}{a}$$

Fourth Proportion:

If (a : b = c : d),
then d is the fourth proportional to a, b, c.

$$d = \frac{b×c}{a}$$

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6. Continued Proportion

If (a : b = b : c = c : d),
then a, b, c, d are in continued proportion.

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7. Important Results

1. If (a:b = c:d), then:

   • (a+b):(b+d) = (c+d):(d+d) — only when the ratio is maintained.

2. If a:b = c:d = e:f,
   then (a+b+e):(b+d+f) also follows the same ratio.

3. If two numbers are in ratio a:b, then:

   • First number = a × k

   • Second number = b × k
     (where k is common multiple or constant of proportionality)

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8. Application in Problems

1. Partitive Division

If a sum (S) is divided in ratio a : b

$$\text{First part} = \frac{a}{a+b} × S,\quad \text{Second part} = \frac{b}{a+b} × S$$

Example:
Divide ₹2000 in ratio 3:2
→ 1 part = 2000 ÷ (3+2) = 400
→ Shares = 3×400=1200 and 2×400=800

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2. Ratio of Ages

If present age ratio = (a:b) and difference = d,

$$\text{Each year} = \frac{d}{a-b}$$
Then actual ages = a×year difference, b×year difference

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3. Ratio in Speed, Time, and Distance

• Speed ∝ Distance / Time

• If distance is constant:

  $$\text{Speed ratio}=\frac{1}{\text{Time ratio}}$$

• If time is constant:

  $$\text{Speed ratio}=\text{Distance ratio}$$

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4. Ratio in Income and Expenditure

If income:expenditure = m:n and savings = s,

$$\text{Savings ratio}=(m-n):n$$

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9. Examples for Practice

1. Find the fourth proportional to 2, 3, 8
   (d = (3×8)/2 = 12)

2. If a:b = 2:3 and b:c = 4:5, find a:b:c
   Make b same: LCM(3,4)=12
   → a:b:c = 8:12:15

3. Divide ₹1200 in ratio 5:3
   Total = 8 parts → 1 part = 1200/8 = 150
   → Shares = 5×150=750, 3×150=450

4. Mean proportional between 9 and 16
   (b = √(9×16) = √144 = 12)

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10. One-Liner Formulas

Concept	Formula
Ratio of a to b	a:b = a/b
Proportion	a:b = c:d ⇒ a×d = b×c
Fourth Proportional	d = (b×c)/a
Third Proportional	c = (b²)/a
Mean Proportional	b = √(a×c)
Equal Ratios	a:b = c:d ⇒ ad = bc
Division in Ratio	(a/(a+b))×Sum , (b/(a+b))×Sum

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11. Tricks & Shortcuts

✅ Simplify ratios before solving.
✅ Use cross-multiplication for proportion questions.
✅ Convert ratios to fractions for clarity.
✅ Always ensure same units before comparing ratios.
✅ Memorize:

• (a:b = c:d ⇒ a×d = b×c)

• Mean proportional = √(product of extremes).

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12. Exam Day Quick Revision Checklist

☑ Ratio basics & simplification
☑ Types of proportion (mean, third, fourth)
☑ Formula for division in ratio
☑ Compound ratio tricks
☑ Word problems on ratio (ages, income, mixture, etc.)

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✅ In One Line:

“Ratio compares quantities; Proportion equates ratios.
Hints: (a:b = c:d ⇒ ad=bc), (Mean = √(ac)), (Third = b²/a), (Fourth = bc/a).”

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