The concept of ratio, proportion and variation is an important one for the aptitude examinations. Questions based on this chapter have been regularly asked in the CAT exam (direct or application based). In fact, questions based on this concept regularly appear in all aptitude tests (XLRI, CMAT, NMIMS, SNAP, NIFT, GRE IRMA, MAT, bank PO, etc.).
Proportion
When two ratios are equal, the four quantities composing them are said to be proportionals. Thus if a/ b = c/ d, then a, b, c, d are proportionals. This is expressed by saying that a is to b as c is to d, and the proportion is written as
a : b : : c : d
a : b = c : d
•The terms a and d are called the extremes while the terms b and c are called the means.
•If four quantities are in proportion, the product of the extremes is equal to the product of the means.
Let a, b, c, d be the proportionals.
Then by definition a/ b = c/ d
ad = bc
Hence if any three terms of proportion are given, the fourth may be found. Thus if a, c, d are given, then b = ad/ c.
•If three quantities a, b and c are in continued proportion, then a : b = b : c
ac = b2
n this case, b is said to be a mean proportional between a and c; and c is said to be a third proportional to a and b.
•If three quantities are proportionals the first is to the third is the duplicate ratio of the first to the second.
That is: for a : b : : b : c
a : c = a2 : b2
•If four quantities a, b, c and d form a proportion, many other proportions may be deduced by the properties of fractions. The results of these operations are very useful. These operations are
1. Invertendo: if a/ b = c/ d then b/ a = d/ c
2. Alternando: If a/ b = c/ d, then a/ c = b/ d
3. Componendo: If a/ b = c/ d, then =
(a+b)/b = (c+d) d
4. Dividendo: If a/ b = c/ d,
then = (a-b)/b = (c-d) d
5. Componendo and Dividendo: If a/ b = c/ d, then (a + b)/( a – b) = (c + d)/( c – d)
Two Kinds of Proportions
(1) Direct Proportion
When it is said that A varies directly as B, you should understand the following implications:
(a) Logical implication: When A increases B increases
(b) Calculation implication: If A increases by 10%, B will also increase by 10%
(c) Equation implication: The ratio A/ B is constant.
(2) Inverse Proportion:
When A varies inversely as B, the following implication arise.
(a) Logical implication: When A increases B decreases
(b) Calculation implication: If A decreases by 9.09%, B will increase by 10%.
(c) Equation implication: The product A ¥ B is constant.
A quantity ‘A’ is said to vary directly as another ‘B’ when the two quantities depend upon each other in such a manner that if B is changed, A is changed in the same ratio.
Note: The word directly is often omitted, and A is said to vary as B.
The symbol µ is used to denote variation. Thus, A µ B is read “A varies as B”.
If A µ B then, A = KB where K is any constant.
Thus to find K = A/ B, we need one value of A and a corresponding value of B.
where K = 3/ 12 = 1/ 4 fi A = Bx (1/ 4).
A quantity A is said to vary inversely as another B when A varies directly as the reciprocal of B. Thus if A varies inversely as B, A = m/ B, where m is constant.
A quantity is said to vary jointly as a number of others when it varies directly as their product. Thus A varies jointly as B and C, when A = mBC.
If A varies as B when C is constant, and A varies as C when B is constant, then A will vary as BC when both B and C vary.
The variation of A depends partly on that of B and partly on that of C. Assume that each letter variation takes place separately, each in its turn producing its own effect on A.