 |
| Some important results |
 |
| 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)  |
| |
 |
| |
| 1st method: |
| |
| 9x - 4x = 13y + 11y |
| |
| 5x = 24 y |
| |
 |
| |
Squaring both sides,  |
| |
 |
| |
(Componendo and Dividendo) |
| |
Again  |
| |
 |
| |
 |
| |
 |
| |
 |
| |
Ans: (i) (ii)  |
| |
| |
| |
| 2nd Method: |
| |
from 1st method |
| |
| Let x = 24 k |
| |
| and y = 5k |
| |
(i)  |
| |
 |
| |
 |
| |
 |
| |
(ii)  |
| |
 |
| |
 |
| |
 |
| |
Ans: (i) , (ii) |
| |
| 3rd 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 |
| |
 |
| |