All About Ratio And Proportion
Ratio
is a mathematical term used to compare two similar quantities expressed
in the same units. The ratio of two terms ‘x’ and ‘y’ is denoted by x :
y. In ratio x : y , we can say that x as the first term or antecedent
and y, the second term or consequent.
In general, the ratio of a number x to a number y is defined as the quotient of the numbers x and y i.e. x/y.
Example: The ratio of 25 km to 100 km is 25:100 or 25/100, which is 1:4 or 1/4, where 1 is called the antecedent and 4 the consequent.
Note that fractions and ratios are same; the only difference is that ratio is a unit less quantity while fraction is not.
Compound Ratio
Ratios
are compounded by multiplying together the fractions, which denote
them; or by multiplying together the antecedents for a new antecedent,
and the consequents for a new consequent. The compound of a : b and c : d
is i.e. ac : bd.
Properties of Ratio:
☑ a
: b : c = A : B : C is equivalent to a / A = b /B = c /C, this is an
important property and has to be used in ratio of three things.
☑
i.e. the inverse ratios of two equal ratios are equal. This property is called Invertendo.
☑
i.e. the ratio of antecedents and consequents of two equal ratios are equal. This property is called Alternendo.
☑
This property is called Componendo.
☑
☑
This property is called Componendo - Dividendo.
☑
Suppose any given quantity ‘a’ is to be divided in the ratio of m : n.
Then,
Proportion
When two ratios are equal, the four quantities composing them are said to be in proportion.
If a/b=c/d, then a, b, c, d are in proportions.
This is expressed by saying that ‘a’ is to ‘b’ is to ‘c’ is to ‘d’ and the proportion is written as
(product of means = product of extremes)
If there is given three quantities like a, b, c of same kind then we can say it proportion of continued.
a : b = b : c the middle number b is called mean proportion. a and c are called extreme numbers.
So, b2 = ac. (middle number)2 = ( First number x Last number ).
Application: These
properties have to be used with quick mental calculations; one has to
see a ratio and quickly get to results with mental calculations.
Example:
should quickly tell us that
should quickly tell us that
Q. A certain amount was to be distributed
among A, B and C in the ratio 2 : 3 : 4, but was erroneously distributed
in the ratio 7 : 2 : 5. As a result of this, B received Rs. 40 less.
What is the actual amount?
(b) Rs. 270
(c) Rs. 230
(d) Rs. 280
(e) None of these
Q. Mixture of milk and water has been kept
in two separate containers. Ratio of milk to water in one of the
containers is 5 : 1 and that in the other container 7 : 2. In what ratio
the mixtures of these two containers should be added together so that
the quantity of milk in the new mixture may become 80%?
(a) 2 : 3
(b) 3 : 2
(c) 4 : 5
(d) 1 : 3
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