Componendo and Dividendo Rule

Overview:

 Topic Included: Formulas, Definitions & Exmaples. Main Topic: Quantitative Aptitude. Sub-topic: Componendo and Dividendo Rule. Questions for practice: 10 Questions.

What is Componendo and Dividendo Rule?

The Componendo and Dividendo Rule is used to ease the long calculation and solve the equations in a quick way, when dealing with equations of fractions and rational functions.

Componendo Rule:

If $$\frac{A}{B} = \frac{C}{D}$$ then, $$\frac{A + B}{B} = \frac{C + D}{D}$$

Componendo Proof:

If $$\frac{A}{B} = \frac{C}{D}$$

Then by adding 1 to both side we get $$\left(\frac{A}{B} + 1\right) = \left(\frac{A}{B} + 1\right)$$ $$\bbox[5px,border:2px solid #800000] {\frac{A + B}{B} = \frac{C + D}{D}} \qquad (1)$$

Dividendo Rule:

If $$\frac{A}{B} = \frac{C}{D}$$ then, $$\frac{A - B}{B} = \frac{C - D}{D}$$

Dividendo Proof:

If $$\frac{A}{B} = \frac{C}{D}$$

Then by subtracting 1 from both side we get $$\left(\frac{A}{B} - 1\right) = \left(\frac{A}{B} - 1\right)$$ $$\bbox[5px,border:2px solid #800000] {\frac{A - B}{B} = \frac{C - D}{D}} \qquad (2)$$

Componendo and Dividendo Rule:

If $$\frac{A}{B} = \frac{C}{D}$$ then, $$\frac{A + B}{A - B} = \frac{C + D}{C - D}$$

Componendo and Dividendo Proof:

If $$\frac{A}{B} = \frac{C}{D}$$

Then by dividing equation $$(1)$$ from equation $$(2)$$ we get $$\left[\frac{\frac{A + B}{B}}{\frac{A - B}{B}} = \frac{\frac{C + D}{D}}{\frac{C - D}{D}}\right]$$ $$\bbox[5px,border:2px solid #800000] {\frac{A + B}{A - B} = \frac{C + D}{C - D}} \qquad (3)$$ The final equation is called componendo and dividendo.

Example: If $$\frac{a}{b} = \frac{4}{3}$$, then find the value of $$\frac{a + b}{a - b}$$?

Solution: Given values, $$a = 4$$ and $$b = 3$$, we know, according to componendo dividendo $$\frac{a}{b} = \frac{c}{d}$$, then from the equation $$(3)$$, $$\frac{A + B}{A - B} = \frac{C + D}{C - D}$$ $$\frac{a + b}{a - b} = \frac{4 + 3}{4 - 3} = \frac{7}{1} = 7$$ Here we can see that the componendo and dividendo rule makes it easy to solve rational functions.