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Ratio and Proportion is one of the most important chapters in mathematics, especially for competitive exams like the Sainik School Entrance Examination. Understanding these concepts helps in solving problems related to numbers, quantities, speed, distance, and comparisons.

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🔹 Ratio

• The ratio of two quantities a and b (in the same units) is expressed as:

Ratio=ab=a:b\text{Ratio} = \frac{a}{b} = a : b

• In the ratio a : b:

  • a = First term (Antecedent)

  • b = Second term (Consequent)

👉 Example:
The ratio 5 : 9 means antecedent = 5, consequent = 9.

Rule of Ratios

Multiplying or dividing each term of a ratio by the same non-zero number does not change the ratio.

👉 Examples:

• 4:5=8:10=12:154 : 5 = 8 : 10 = 12 : 15

• 4:6=2:34 : 6 = 2 : 3

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🔹 Proportion

• When two ratios are equal, they are said to be in proportion.

a:b=c:d⇒a:b::c:da : b = c : d \quad \Rightarrow \quad a : b :: c : d

• Here:

  • a & d = Extremes

  • b & c = Means

✅ Rule:

a:b::c:d⇒(a×d)=(b×c)a : b :: c : d \quad \Rightarrow \quad (a \times d) = (b \times c)

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🔹 Special Proportionals

1. Fourth Proportional

   • If a:b=c:da : b = c : d, then d is the fourth proportional to a, b, c.

2. Third Proportional

   • If a:b=b:ca : b = b : c, then c is the third proportional to a and b.

3. Mean Proportional

   • Mean proportional between a and b is:

   a×b\sqrt{a \times b}

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🔹 Comparison of Ratios

We say:

ab>cd⇒(a:b)>(c:d)\frac{a}{b} > \frac{c}{d} \quad \Rightarrow \quad (a : b) > (c : d)

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🔹 Compound and Multiple Ratios

1. Compound Ratio

   • The compound ratio of (a : b), (c : d), (e : f) is:

   ace:bdface : bdf

2. Duplicate & Sub-Duplicate Ratios

   • Duplicate ratio of (a : b) = a2:b2a^2 : b^2

   • Sub-duplicate ratio of (a : b) = a:b\sqrt{a} : \sqrt{b}

3. Triplicate & Sub-Triplicate Ratios

   • Triplicate ratio of (a : b) = a3:b3a^3 : b^3

   • Sub-triplicate ratio of (a : b) = a3:b3\sqrt[3]{a} : \sqrt[3]{b}

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🔹 Variations

1. Direct Proportion

   • If xx increases/decreases with yy:

   x∝y⇒x=kyx \propto y \quad \Rightarrow \quad x = ky

2. Inverse Proportion

   • If one quantity increases while the other decreases:

   x∝1y⇒xy=kx \propto \frac{1}{y} \quad \Rightarrow \quad xy = k

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🎯 Why Important for Sainik School Exams?

• Questions on Ratio & Proportion appear frequently in Mathematics sections.

• Helps in solving problems on:

  • Speed, Time, and Distance

  • Age Problems

  • Work and Wages

  • Mixture and Alligation

  • Profit and Loss

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