Continuous ratio

Consider the following ratio: .

Note that your means are equal, so it is called continuous ratio. Like this:

Continuous proportion is any proportion of equal means.

Generally speaking, a continuous proportion can be represented by:

Proportional Third

Dice two natural numbers The and B, non-null, is called third proportional of these numbers the number x such that:


  • Determine the third proportional of numbers 20 and 10.
    We indicate by x the third proportional and we set the proportion:
    (applying the fundamental property)
    20 x = 10. 10
    20x = 100

    x = 5
    So the third proportional is 5.

Geometric mean or proportional mean

Given a continuous ratio , the number B is called geometric mean or proportional average in between The and ç. Example:

  • Determine the positive geometric mean between 5 and 20.

    5 20 = b. B
    100 = b2
    B2 = 100
    b =

    b = 10
    Therefore, the positive geometric mean is 10.
Next: Proportion Properties