Ratio and Proportion I Summary

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• If a and b are any two quantities of the same kind then the quotient  is called the ratio of a and b. It is expressed as a : b.

(read as a is to b).

• In a:b, a is called antecedent and b is called consequent.

• The value of ratio remains unchanged if both the consequent and antecedent are multiplied or divided by the same non zero quantity.

i.e. a:b=am:bm

• Comparison of ratios:

a:b>c:d if ad>bc i.e. ad-bc>0

a:b<0

a:b=c:d if ad=bc i.e. ad-bc=0

a:b>c:b a>c

a:b>a:c b

• Compounded ratio:

When two or more ratios are multiplied termwise, the ratio thus obtained is called their compounded ratio.

For a:b, c:d, the compounded ratio is ac:bd.

• Reciprocal ratio: The reciprocal ratio of a:b is  which is b:a.

• Duplicate ratio: It is the compounded ratio of two equal ratios.

The duplicate ratio of a:b is a²:b².

• Triplicate ratio: It is the compounded ratio of three equal ratio.

The triplicate ratio of a:b is a³:b³.

• Sub-duplicate ratio:

• Sub-triplicate ratio:

• Proportion:

Four quantities a, b, c, d are said to be in proportion if a:b=c:d.

a and d are called extremes.

b and c are called means.

d is called the fourth proportional to a, b and c.

• If a, b, c and d are in proportion i.e. a:b::c:d then ad=bc.

(The product of extremes is equal to the product of the means)

• Continued proportion: Two or more quantities are said to be in continued proportion if the ratio of the first to the second is equal to the ratio of the second to the third and so on.

a, b, c are in continued proportion if a:b=b:c.

• Mean proportional: If a, b, c are in continued proportion then b is called the mean proportional of a and c. c is called the third proportional to a and b.

Also b²=ac

• Results on proportion:

a) Invertendo If a:b=c:d then

b:a=d:c

b) Alternendo If a:b=c:d then

a:c=b:d

c) Componendo If a:b=c:d then

a+b:b=c+d:d

d) Dividendo If a:b=c:d then

a-b:b=c-d:d

e) Componendo and Dividendo

If a:b=c:d then

a+b:a-b=c+d:c-d