PROPORTION

 PROPORTION             PRACTICE SUMS👈

1. Definition of Proportion:

A proportion is an equation that states that two ratios are equal.

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

Here, a, b, c, d are numbers, and $b \neq 0, d \neq 0$

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🧠 Basic Terms:

• $a$ and $d$: Extremes

• $b$ and $c$: Means

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🔢 Important Rules of Proportion

🔹 1. Cross Multiplication Rule (Fundamental Rule):

If:

$$\frac{a}{b} = \frac{c}{d}$$

Then:

$$a \times d = b \times c$$

This is called the cross-product rule or Rule of Four Numbers.

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🔹 2. Invertendo:

If:

$$\frac{a}{b} = \frac{c}{d}$$

Then:

$$\frac{b}{a} = \frac{d}{c}$$

(Switch numerator and denominator on both sides)

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🔹 3. Alternendo:

If:

$$\frac{a}{b} = \frac{c}{d}$$

Then:

$$\frac{a}{c} = \frac{b}{d}$$

(Alternate terms of both ratios)

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🔹 4. Componendo:

If:

$$\frac{a}{b} = \frac{c}{d}$$

Then:

$$\frac{a + b}{b} = \frac{c + d}{d}$$

(Add numerator and denominator, divide by denominator)

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🔹 5. Dividendo:

If:

$$\frac{a}{b} = \frac{c}{d}$$

Then:

$$\frac{a - b}{b} = \frac{c - d}{d}$$

(Subtract denominator from numerator)

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🔹 6. Componendo and Dividendo (Very Important):

If:

$$\frac{a}{b} = \frac{c}{d}$$

Then:

$$\frac{a + b}{a - b} = \frac{c + d}{c - d}$$