Proportions
Proportions are a method of relating one quantity to another. The equality of two ratios is called proportion. The symbol for proportion is : : .
Thus, if a : b = c : d then we say that a, b, c, d are in proportion and a : b :: c : d. Here a, b, c, d are called first, second, third, fourth terms of the proportion respectively. Here ‘a’ and ‘d’ are called extremes, ‘b’ and ‘c’ are called means.
Geometrical mean of a, b, c: Geometrical mean of a, b, c is or b^{2} = ac. In this case ‘b’ is called mean proportional or geometrical mean of a, b, c.
Direct proportion: If two quantities are so related to each other that an increase (or decrease) in the first causes an increase (or decrease) in the second, then the two quantities are said to vary directly.
Indirect proportion: If two quantities are so related to each other that an increase (or decrease) in the first causes a decrease (or increase) in the second, then the two quantities are said to vary indirectly or inversely.
Efficiency is indirectly proportional to number of days D taken to complete the work. Then mathematically, or , where k is a constant.
Examples on ratio definitions:

The ratio between Rs.50 and Rs.10 is 50 : 10 = 50/10 = 5/1 = 5 : 1

The ratio between 5 kg and 15 kg is 5 : 15 = 5/15 = 1/3 = 1 : 3

The ratio between Rs.1 and 5 kg is not possible, since they are of different units.
Examples on important property:
5/15 = (5 * 8)/(15 * 8) shows that 5 : 15 is equal to 40 : 120
3/7 = [3/5]/[7/5] shows that 3 : 7 is equal to 3/5 : 7/5.
Example on ratio in simplest form:
Find out the ratio of cherries to the grapes and what is the simplest form of that ratio ?
Example on compound ratio
Find the compound ratio of following figures ?
Example on compound ratio:
If P : Q = 8 : 6 and Q : R = 6 : 5 then P : Q : R = 8 : 6 : 5.
Example on direct proportion:
If the cost of one book is Rs.6 then the cost of 10 books is Rs.60 and 15 books cost is Rs.90....
So, if number of books are increased automatically cost also increases i.e, the number of books is in direct proportion with the cost of books.
(Ratio of number of books) :: (Ratio of cost of books)
Example on indirect proportion: Time taken to finish a work is inversely proportional to number of persons at work (More persons, less is the time taken to finish a work).
(Ratio of men) :: (Inverse ratio of time taken).