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# How Componendo Dividendo Works?

TopComponendo and dividendo are properties that are generally used for solving Math problems of Ratios and Proportions. Let us see how componendo dividendo works?

Suppose we have a fractional equation as a /b = c /d. When we apply componendo to it we get:

(a + b) /b = (c + d) /d,

Similarly, if we apply dividendo to the above equation we get:

a / (a – b) = c / (c – d),

If we apply both componendo and dividendo together, we get the following expression:

(a + b) / (a – b) = (c + d) / (c – d),

It must always be applied on both sides of the equation to get the correct answer. Let us consider some examples of such situation to know where we should use componendo and dividendo.

Example 1: Suppose we have a fraction (x – 8) / (x + 8) = 5 /6. Find the value of x?

Solution: (x – 8) / (x + 8) = 5 /6,

Applying componendo and dividendo to the above equality we get:

((x – 8) + (x + 8)) / ((x - 8) – (x + 8)) = 5 + 6 / 5 – 6,

Or 2 x / -16 = 11 / -1,

Or 2 x / 16 = 11,

Thus we are left with a very simplified version of the original fractional equation. Thus, value of 'x' will be x = (16 * 11) /2 = 88.

Example 2: Solve (x2 - 4 – 2 x) / x2 = 1/2 for x?

Solution: Applying componendo and dividendo we get:

((x2 - 4 – 2 x) + x2) / ((x2 - 4 – 2 x) – x2) = -3,

Or 2 x2 - 4 – 2 x / (- 4 – 2x) = -3,

Or x2 - 2 – x = + 6 + 3x,

Or x2 - 4x – 8 = 0,

Solving the Quadratic Equation we can get value of x.