Quantitative Aptitude Ratio and Proportion Study Material
1. Ratio: The ratio of two quantities of the same kind and in the same units is a comparison by division of the measure of two quantities.
In other words, ratio of two quantities of the same kind is the relation between their measures and determines how many times the one quantity is greater than or less than the other quantity.
Ratio : (a) Ratio A ratio is a comparison of two numbers (quantities) by division. The ratio of a to b is written as,
a : b = = a + b
In the ratio a : b. a and b are called the terms of the ratio; V is the antecedent ‘b‘ is the consequent. A ratio is a number, so to find out the ratio of two quantities, they must be expressed in the same units
(b) Proportion A proportion is an expression which states that two ratios are equal.
c.g. 3/12 = 1/4 is a proportion. It can also be expressed as 3 : 12 = 1 : 4 or 3 : 12 :: I: 4.
Each quantity in proportion is called term or proportional. The first and the last terms arc called the extremes, whereas the second and the third terms are called the middle terms. When four quantities are in proportion, the last quantity is said to be fourth propartional to the other three and also we find, product of middle terms = product of extremes y
2nd term x 3rd term = 1st term x 4th term
e.g. In 4:8= 12 : 24,
Properties of Ratio
(a) In a ratio, two quantities arc compared. So, the quantities must be of the same kind, i.e. they must be expressed in the same units.
(b) The ratio of two quantities determines how many times one quantity is contained by the other.
(c) The order of the terms in a ratio a : b is very important. Since 4 : 5 is different from 5 : 4.
Dividing a Given Number in The Given Ratio
Let ‘A’ be the given number. The given ratio is a1: a2
Here A is to be divided in the ratio a1 : a2.
It implies that A is divided in two parts such that value of first part: value of second part = a1 : a2.
first part = (a1/a1+a2)x A = first term of ratio x (Sum of parts/ Sum of terms of ratio)
Second part = (a1/a1+a2)x A = Second term of ratio x (Sum of parts/ Sum of terms of ratio)
Since, A has been divided into two parts, so, first part + second part = A.
Example: Two numbers are in the ratio 8:9. If the sum of the numbers is 119, find the numbers.
Solution: Since the sum of two numbers is 119, so, the problem implies that 119 is divided in two parts in the ratio 8:9.
first number = (8/8+9) x119 = 56
second number = (9/8+9)x 119 = 63
Note: These relations are also true for dividing a given number into more than two ratios (i.e. more than two parts) When any number A is divided in more than one ratio such as a : b : c : d :_
value of any part = (its related ratio term/a+b+c+……..) x A
third part = (c/a+b+c+…) x A
Example: Dividing Rs 3,200 among P, Q. R in the ratio 5:2:9, find the amount received by Q.
Solution: Amount received by Q = its related ratio term/sum of ratio terms x Total amount
= (2/5 + 2 + 9)x 3200
= Rs 400
Kinds of Ratios
(i) Ratio of greater inequality : A ratio a: b is called a ratio of greater inequality if
antecedent ‘a’ > consequent ‘b’
Ex. :7:4 , 9:5 ,3:2
(ii) Ratio of less inequality : A ratio a : b is said to be a ratio of less inequality if a < b.
Ex. 4:7 ,5 :9 2: 3.
(iii) Ratio of equality: A ratio a: b is said to be a ratio of
equality if a = b.
Ex. 2:2, 4:4, 5:5
Compounded Ratio: If two or more ratios are multiplied term wise, i.e, the antecedents to that of the consequents, the ratio thus obtained is called their compounded ratio.
(i) The compounded ratio of a : b and c : d is ac : bd.
(ii) The compounded ratio of a : b, c : d, and e : f is ace: bdf.
Duplicate Ratio : It is the compounded ratio of two equal ratios.
Thus, the duplicate ratio of a : b is aa : bb
Ex.4/9 is called the duplicate ratio of 2/3 .
Triplicate Ratio : It is the compounded ratio of three equal ratios.
Thus, the triplicate ratio of a : b is aaa : bbb,
Ex. 8/27 is called triplicate ratio of 2/3 .
Sub-duplicate Ratio: A ratio x : y is the sub-duplicate ratio of a ratio a: b if the duplicate ratio of x : y is a : b.
Ex. 4/5 is the sub-duplicate ratio of 16/25