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| 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. |
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(1) If  then bc = ad |
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| This property is known as INVERTENDO. |
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(2) If  , then ad = bc |
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| This property is known as ALTERNENDO. |
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(3) If we add 1 to both sides of  then  |
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| This property is known as COMPONENDO. |
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(4) If we subtract 1 from both sides of  then  |
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| This property is known as DIVIDENDO. |
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| (5) If result (3) is divided by the result (4), then |
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or   |
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| This property is known as COMPONENDO & DIVIDENDO. |
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(6) If  , then each of these ratios equals  where l,m,n are real numbers. |
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Let  |
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  |
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| Similarly c = dk and e = fk |
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| (This method is called k-method. It will be freely used in this topic.) |
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| If 9x - 11y = 4x + 13y, find |
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(i)  , (ii)  |
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| 1st method: |
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| 9x - 4x = 13y + 11y |
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| 5x = 24 y |
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Squaring both sides,  |
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  |
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 (Componendo and Dividendo) |
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Again  |
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Ans: (i)  (ii)  |
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| 2nd Method: |
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 from 1st method |
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| Let x = 24 k |
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| and y = 5k |
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 (i)  |
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(ii)  |
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Ans: (i)  , (ii)  |
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| 3rd Method: |
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We can also solve the problem by substituting  and get the same result. |
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If a : b = c : d, then prove that  |
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Let  |
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  or a = bk. |
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| Similarly, c = dk |
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LHS  …(i) |
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RHS  |
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 …(ii) |
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| From (i) and (ii), |
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| LHS = RHS Hence proved. |
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| Which number should be added to each of the numbers 7, 22 and 62 so that the sums would be in continued proportion? |
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| Let the number added to each of them be 'x'. |
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 7 + x, 22 + x and 62 + x are in continued proportion. |
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 (22 + x)2 = (7 + x) (62 + x) |
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| 484 + 44x + x2 = x2 + 69x + 434 |
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| 44x - 69x = 434 - 484 |
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| -25x = -50 |
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 x = 2 |
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 2 should be added. |
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If  |
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 show that  |
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Let  |
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  (equal ratios results) |
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  …(i) |
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Similarly, we can obtain  …(ii) |
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 From (i) and (ii), we get the required result |
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