# Properties of Proportions

- If
*a:b :: c:d, then b:a :: d:c by invertendo.**So, if then by invertendo***Example:**

- If
*a:b :: c:d, then a:c :: b:d by alternendo.**If then by alternendo***Example:**

- If
*a:b :: c:d, then (a**+**b):b :: (c**+ d):d by componendo.**If , then by componendo .***Example:****.** - If
*a:b :: c:d, then (a â€“ b) : b :: (c â€“ d) : d by dividendo.**If , then by dividendo***Example:**

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

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

- 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:**

ExampleWhat must be added to each of 6, 17, 27 and 59, so that the sums are in proportion?SolutionLet*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.