We use numbers in everyday life quite frequently. One of the uses of numbers is for comparison. When two things of same kind are attributed numerical values, we are able to compare them. This comparison is expressed in phrases like ‘is greater than',' is multiple of’ etc.
Let us take an example , a familiar situation in which Vignette scored 17 runs while Vijay amassed 51 runs in an inning of cricket. Then we say,
1) Vijay scored 34 runs more than Vignette or Vignette scored 34 runs less than Vijay or
(2) Vijay scored three times as many runs as Vignette or we say that Vignette scored only one third of the runs scored by Vijay. When we compare in the way as
(3), we are finding the ratio between the two numbers. In short, the ratio between two quantities ‘a’ and ‘b’( where b>0 is the value of fraction a/b in its lowest terms) Let us revise a few things about ratio.

The phrase, ‘the ratio of 17 to 51’ is written as ’17:51’ and read as ’17 is to 51’ While comparing two quantities in terms of ratio, we must bear in mind the following:
1) The two quantities must be of same kind.
2) The units of measurement of the two quantities must be the same.
3) As the ratio denotes how many times is one quantity of the other, it is a pure number( without any unit of measurement)
For example: 4m : 80 cm=400cm: 80cm=5:1 1hr 30 min : 2 hrs 15 min=90 min:135min=2:3 The numbers involved in a ratio are called its ‘terms’ 1

That expression of a ratio, both of whose terms do not have common factor other than one, is called the ratio in its lowest terms. Thus by canceling the common factor of the two terms 105 and 135, we obtain the lowest form f the ratio 105:135 as 7:9. Percentage is a special kind of a ratio . it is a ratio having its second term 100. Please note down certain important aspects of ratios.
1) when we consider the ratio of two numbers as A:B ,then the first number need not be ‘a’ and the second number need not be ‘b’.They can be ka and kb, where k is any non zero multiple of a and b.
2) Two ratios a : b and c: d (a/b and c/d) are said to be equal if a x d = b x c.
3) If two ratios are A:B and B:C, they are briefly written as a:b:c.
For example: when we say that the ratio of measures of angle A angle B and angle C of triangle ABC are three is to four is to five( 3 :4 :5) We really mean to say that m< A : m<B=3:4 and m<B : m<C= 4:5. That is, we write the ratio 3:4 and 4:5 in brief as 3:4:5. Also ,when we say that ratio of four numbers is a : b : c : d, then we mean that the ratio of three pairs of numbers taken in order are a, b, c, c, d.

PROPORTION:- Proportion is a very familiar and an important mathematical concept. Let us try to understand this. Suppose a fruit seller tells you that the price of oranges is Rs.20 a dozen, that sirs 20 for 12 oranges If you want to buy six oranges, how do you determine their cost? As the number of oranges is half of one dozen, their cost also has to be half. Therefore, the cost of 6 oranges is half of rs.20 i.e. Rs 10. In other words, you think that the cost of oranges in proportion to their number.
Mathematically, proportion is defined as follows: When a/b=c/d, the numbers a,b,c are in proportion. When a,b,c are in proportion ,they are respectively called the first, the second, the third and the fourth proportional. a and d are called the extremes, while b and c are called the means. 2 Ratio and Proportion For example,14/21=18/27 Therefore 14,21,18,27 are in proportion. 14 and 27 are the extremes while 21 and 18 are the means. You know that when a/b=c/dither the products a x d and b x c are equal. So when four numbers are in proportion, the product of extremes is equal to the product of means. The concept of proportion need not be restricted to only two equal ratios. It may be extended thus. If a/b=c/d=e/f…..,then a, b, c, d, e… are said to be in proportion.
For example: 5/9= 20 / 36=15/27. So 5,9,20,36,15,27 are in proportion. Continued proportion Consider the ratios 25:20 and 20:16. These ratios are equal . So, the numbers 25,20,20,16 are in proportion.
The means of this proportion are equal. So, we say, the numbers25,20,16 are in continued proportion. Generally ,when a / b = / c then a,b,c are in continued proportion, also, when a/b=c, we get b2 = a x c When a, b, c are in continued proportion ,b is called the ‘geometric mean’ or ‘ mean proportional’ between a and c. From the above discussion, we get four equivalent statements.
(1) a/b=c
(2) b2 =a x c
(3) a,b,c are in continued proportion and
(4) b is the geometric mean of a and c