Some important results

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Given a, b, c and d are non-zero real numbers, we can deduce other proportions by simple Algebra. These results are often referred by the names mentioned along each of the properties obtained.

(1) If then bc = ad

This property is known as INVERTENDO.

(2) If , then ad = bc

This property is known as ALTERNENDO.

(3) If we add 1 to both sides of then

This property is known as COMPONENDO.

(4) If we subtract 1 from both sides of then

This property is known as DIVIDENDO.

(5) If result (3) is divided by the result (4), then

or

This property is known as COMPONENDO & DIVIDENDO.

(6) If , then each of these ratios equals where l,m,n are real numbers.

Let

Similarly c = dk and e = fk

(This method is called k-method. It will be freely used in this topic.)

If 9x - 11y = 4x + 13y, find

(i) , (ii)

1^st method:

9x - 4x = 13y + 11y

5x = 24 y

Squaring both sides,

(Componendo and Dividendo)

Again

Ans: (i) (ii)

2^nd Method:

from 1^st method

Let x = 24 k

and y = 5k

(i)

(ii)

Ans: (i) , (ii)

3^rd Method:

We can also solve the problem by substituting and get the same result.

If a : b = c : d, then prove that

Let

or a = bk.

Similarly, c = dk

LHS …(i)

RHS

…(ii)

From (i) and (ii),

LHS = RHS Hence proved.

Which number should be added to each of the numbers 7, 22 and 62 so that the sums would be in continued proportion?

Let the number added to each of them be 'x'.

7 + x, 22 + x and 62 + x are in continued proportion.

(22 + x)² = (7 + x) (62 + x)

484 + 44x + x² = x² + 69x + 434

44x - 69x = 434 - 484

-25x = -50

x = 2

2 should be added.

If

show that

Let

(equal ratios results)

…(i)

Similarly, we can obtain …(ii)

From (i) and (ii), we get the required result