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Ratio Proportion & Variation

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☼ Ratio  :   A ratio is a comparison of  two numbers or quantities by division. The  ratio of  ‘a’ to ‘b’  is written as,  a : b ( read as a is to b )  =    =  a :  b  ( Where, b \ne  0 )

In the ratio  a : b,  a   and   b  are called the terms of  the ratio;  ‘a’ is the antecedent, ‘b’ is the consequent.

■ Important  notes  :

 ▪ The ratio of  two quantities can be obtained only if  their measures are taken in the same units.

e.g.   2  kg : 100  gm  =  2000 gm  :  100  gm  =  20 : 1

 ▪ The salaries of  A  and  B  are in the ratio  a : b  do not mean  that A’s salary = a and  B’s  salary  =  b.

But the meaning of  a : b  is  A’s  salary  =  ak  and  B’s  salary  =  bk.

 ▪ If  a > b, then   a : b  is called a ratio of  greater inequality. And some positive number is added to each term of  a : b, then the ratio is diminished.

 ▪ If  a < b, then   a : b  is called a ratio of  less inequality. And some positive number is added  to each term of  a : b, then the ratio is increased.

■ Some special type of  Ratios :

 (i) Zero ratio  :  If  the antecedent (a)  in a ratio is zero, the ratio is called a zero ratio. If  a = 0, then   a : b is called a zero ratio.

e.g.    0  :  5   is a zero ratio.

 (ii) Non – zero ratio  :  If  a   0,  then   a : b is called a zero ratio.

 (iii) Inverse  ratio  :  If  a : b  is  a  non-zero  ratio, then  b : a is called the inverse ratio of  a : b.

 (iv) Equal  ratio  :  The  ratios  a : b  and  c : d  are called equal ratios if  ad = bc.

e.g.  for  3 : 6 and 4 : 8,  3 \times 8  =  6 \times 4. Hence,  3 : 6  and  4 : 8 are equal.

 (v) Compound  Ratio  :  If   two  ratio  a  :  b  and  c  :  d  are  in  different  units, then also  the  compound  ratio  is  ac  :  bd

  e.g. Find  compound ratio of   2 : 3  and  9  :  10.

  Soln. compound ratio  =  \frac{2\times9}{3 \times 10}  =   \frac{3}{5}   =   3  :  5

 (vi) Duplicate  Ratio  :  a  :  b   is  called  duplicate  ratio  of   a  :  b.

  e.g. Find  the  duplicate ratio of  2  :  3.

  Soln. Duplicate ratio  =   2^{2}:3^{2}   =    4  :  9.

 (vii) Triplicate  Ratio  :  a^{3}:b^{3}   is  called  triplicate  ratio  of   a  :  b.

  e.g. Find  the  triplicate ratio of  5  :  2.

  Soln. Triplicate ratio  =   5^{3}:2^{3}    =    125  :  8.

 (viii) Sub-duplicate  Ratio  :  \sqrt{16}:\sqrt{9}  is  called  sub-duplicate  ratio  of   a  :  b.

  e.g. Find  the  sub-duplicate  ratio of  16  :  9.

  Soln. Sub-duplicate  ratio  =   \sqrt[3]{a}:\sqrt[3]{b}  =    4  :  3.

 (ix) Sub-triplicate  Ratio  :   :    is  called  sub-triplicate  ratio  of   a  :  b.

  e.g. Find  the  sub-triplicate  ratio of    27  :  125.

  Soln. Sub-triplicate ratio  =   \sqrt[3]{27}:\sqrt[3]{125}   =    3  :  5.

■ Properties of  Ratios  :

 (i) If  \frac{a}{b}   =  \frac{c}{d}  = \frac{e}{f} , then  it can also be expressed as a : c : e  =  b : d : f.

 (ii) If   \frac{a}{b}  =  \frac{c}{d}  =  \frac{e}{f}  =  \frac{g}{h}  =  . . . , then  each  ratio  =  \frac{\textup{ma + nc + ke + lg + ...}}{\textup{mb + nd + kf + lh + ...}}  

\large \mapsto   \frac{a}{b}  =  \frac{c}{d}  , then each ratio = \frac{a+c}{b+d}= \frac{a-c}{b-d}    i.e.    \frac{a}{b}  =  \frac{c}{d} = \frac{a+c}{b+d}= \frac{a-c}{b-d}

 

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