Ratio: ( a : b)
A ratio is the relation between two numbers which shows how much bigger one quantity is than another is obtained by dividing one quantity by other quantity of the same kind. The result obtained is an abstract number (quantity without any unit) integer or fraction.A ratio can exist only between two quantities of the same kind.A ratio can be written in the form of a : b, where a and b are any integers. It expresses a fraction. For example. 2 : 3 = \( 2 \over 3 \).
Antecedent : consequent
The first quantity of the ratio is called antecedent whereas the second quantity of the ratio is called consequent. I.e. -for a ratio of a : b , a is termed as antecedent or first term and b is termed as consequent or second term.Proportion: ( a : b :: c : d )
The equality between two ratios is called proportion. For example if \( { a \over b } = {c \over d }\) then it means \( a \over b \) is proportion with \( c \over d \) and can be written as a : b :: c : d where a and d are known as extremes and b and c are known as means.If four quantities are in proportion then the product of means is equals to the product of extremes. i.e in a : b :: c : d , \( a \times d = b \times c \)
Proportional Division
The process by which a quantity may be divided into parts which bear a give ratio to one another.is called proportional division and the parts are known as proportional parts.For example :
if we divide quantity y in ratio a : b : c then ,
First part = \( {a \over (a + b + c )} \times y \)
Second part = \( {b \over (a + b + c )} \times y \)
Third part = \( {c \over (a + b + c )} \times y \).
Example Questions :
Q.1 : In a Classroom there are 30 boys and 25 girls, what will be the ratio between girls and boys? (simple ratio)Solution :
here we have given,
No. of boys = 30
No. of girls = 25
to get ratio between girls and boys ,
antecedent = 25 and consequent = 30
girls : boys = \( {25 \over 30} = { 5 \over 6 } \)
∴ girls :boys = 5 : 6 Answer
Q.2 : Find out the ratio whose value is 2/3 and the antecedent is 18. (simple proportion)
Solution : Given, the value of the ratio is 2/3 which means 2/3 is the simplified form
here antecedent = 18 ,
let consequent = c
then \( {18 \over c } = {2 \over 3 } \)
=> \( 18 \times 3 = 2c \)
=> \( c = { 54 \over 2 } = 27 \)
=> ratio = 18/27 => 18 : 27 Answer
Q.3 : Find out the two quantities whose difference is 30 and ratio between them is 5/11.
Solution :
The difference of two quantities which are in ratio 5 : 11 is 11 - 5 = 6
if the difference between two quantities is 30 then we should multiply both the quantities by 30/6 = 5.
∴ 5 : 11 = \( 5 \times 5 : 11 \times 5 = 25 : 55 \) Answer.
Notes
1. Problems on Compound Proportion
Prakash Joshi 01 Jan 1970compound proportions contain three or more different kinds of quantities involving two or more simple proportions.
Rule to solve compound proportion problems:-
i) select all the quantities given in the problem such as man, work, hour and day, etc.
ii) put all the quantities in one line keeping the required quantity to the right-hand side.
iiI) without considering the quantities in (i) write I, II, III, and IV.
iv) below III put the last quantity in which the answer is wanted. below IV put x. put the sign of ratio : between III and IV and a sign of proportion:: between II and III.
v) now find out by careful inspection whether the quantities to be found out is greater or less than the third term.
if greater put the lesser of the two as the first term. if less put the greater of the two as the first term and the other as the second term.
divide the product of all the terms below II and III by the product of all the two as below I.
Example 1: if 5 men can do a piece of work in 20 days, in how many days will 10 men and 5 boys do the same work if one men does as much work as 2 boys.
Solution :
according to the above rule we write all in a line (men, boys, and days)
MEN BOYS DAYS
5 0 20
10 5 ?
here given a men work much of 2 boys so 1 men = 2boys.
5men +0boys = 20 days
10 men + 5 boys = ?
5 men = 20 days
1 men = 20 * 5 = 100 days
here days and men are inversey proportional.
10 men + 5/2 men ( as 1 men = 2boys) = 100 / (25/2) = 200/25 = 8 Days Answer.