# Properties of Proportions

1. If a:b :: c:d, then b:a :: d:c by invertendo.

So, if  then by invertendo

Example:

2. If a:b :: c:d, then a:c :: b:d by alternendo.

If  then by alternendo

Example:

3. If a:b :: c:d, then (a + b):b :: (c + d):d by componendo.

If , then by componendo .

Example:

.

4. If a:b :: c:d, then (a â€“ b) : b :: (c â€“ d) : d by dividendo.

If , then by dividendo

Example:

5. If a:b::c:d, then (a + b):(a â€“ b)::(c + d):(c â€“d) by componendo and dividend.

If , then by componendo and dividendo, .

Example:

6. If a:b::c:d::e:f, then a:b::c:d::e:f::(a + c + e):(b + d + f) by addendo.

Example:

7. If a:b::c:d::e:f, then a:b::c:d::e:f::(a â€“ c â€“ e):(b â€“ d â€“ f) by subtrahendo.

If , then by subtrahendo

Example:

Example
What must be added to each of 6, 17, 27 and 59, so that the sums are in proportion?
Solution
Let x be the number to be added:
(6 + x ) : (17 + x ) :: (27 + x ) : (59 + x )
Using dividendo, we get

If we had not used dividendo, we would have had to cross-multiply and solve as shown below: (6 + x ) (59 + x ) = (27 + x ) (17 + x ) â‡’ x = 5

Note: In the above example, the second method looks easier. But we might end up with a quadratic/cubic equation after cross-multiplication. Hence, use of componendo and dividendo usually saves a lot of time by eliminating the need to solve the equation in a quadratic form.