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Ratio and Proportion

Ratios

The knowledge of ratio and proportion is necessary for a family to run. The belongings of father must be divided among children proportionally. Or in other words the ratio should be the same.

The ratio is found of quantities
  • of the same kind
  • in the same units


A ratio of two quantities is the fraction that one quantity is of the other and is expressed in its lowest terms.
  • The ratio a:b is the fraction a/b. a:b is read as a is to b.
  • Terms of a ratio: In the ratio a:b, a is the first term and is called antecedent and b is second term and is called consequent.
  • A ratio remains unchanged if both of its terms are multiplied or divided by the same non-zero quantity.


Simplest form:

A ratio is said to be in the simplest form if H.C.F. of the antecedent and consequent is 1.

Comparison of ratios:

    To compare two ratios, compare its decimal fraction.

  1. Ratio of equality:

    • If antecedent of a ratio is equal to its consequent, then the ratio is called the ratio of equality.

  2. Ratio of greater Inequality:

    •  If antecedent of a ratio is greater than its consequent, then the ratio is called the ratio of greater inequality..

  3. Ratio of lesser inequality:

    •  If antecedent of a ratio is less than its consequent, then the ratio is called the ratio of lesser inequality.

Compounded ratios:

The compounded ratios of a:b and c:d is ac:bd.In compounded ratios the result is the term wise multiplication. The compounded ratios of a:b ,c:d and e:f is ace:bdf.

Reciprocal Ratio:

 If a≠0 and b≠0 then the reciprocal ratios of a:b ia 1/a:1/b i.e. b:a.

Duplicate Ratio:

 It is the compounded ratios of two equal ratios. Thus the duplicate ratios of a:b is aa:bb i.e. a2:b2.

Sub-Duplicate Ratio:

√a:√b is sub-duplicate ratio of a:b.

Triplicate Ratio:

It is the compounded ratios of three equal ratios. Thus the Triplicate ratios of a:b is aaa:bbb i.e. a3:b3.

Sub-Triplicate Ratio:

 a:b is sub-Triplicate ratio of a:b.

Continued Ratio:

 If A:B and B:C then A,B,C are said to be continued ratio A:B:C.

Proportion

If two ratios are equal then it is called a proportion.
a:b::c:d

If a/b = c/d then we say that a,b,c and d are in proportion and we write that
a:b::c:d

Terms of proportion:

a,b c and d are respectively called the first, second, third and fourth term of the proportion. a,b,c and d are also called the proportionals.  The first and fourth terms are called extremes while second and third terms are called means. d is also called the fourth proportional to a,b and c.

In proportion a and b should be of same kind and c and d should be of same kind. a and c may differ.

The, product of extremes = The product of means

Continued Proportion:

Three numbers a, b, c are said to be in continued proportion if a:b::b:c.
c is called the third proportional to a and b.

The non-zero quantities of same kind a,b,c,d,e,f,... are said to be in continued proportion if a/b=b/c =c/d=d/e=e/f =...

Mean proportional between a and c

If a,b and c are in continued proportional then b is called the mean proportional of a and c.
Product of means = product of extremes, we have
b×b = a×c ⇒ b2 = ac ⇒ b = √ac

Direct and Indirect Proportion

    If change in one quantity affect change in other then the changes are said to be in proportion.

  • Direct Proportion:

    • In increase in one quantity increases the other quantity or decrease in one quantity decreases the other quantity, the change is said to be in direct proportion.

  • Indirect Proportion:

    • In increase in one quantity decreases the other quantity or decrease in one quantity increases the other quantity, the change is said to be in indirect proportion and vary indirectly or inversely.




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