RATIO AND PROPORTION CONCEPTS,FORMULAS,DEFINITIONS,EXAMPLES,TRICKS

Ratio and Proportion is very important chapter for all competitive exam and everything is included in it like all ratio,proportion,compounded ratio,types of proportion and most important questions.    

     

RATIO AND PROPORTION

RATIO : -

The ratio of two quantities x and y is the fraction a/b and we write it as x : y.

In the ratio x: y, we call x as the first term or antecedent and y, the second term or consequent 
i.e. ratio 4: 9 represents with antecedent =4, consequent = 9

Types of Ration

duplicate ratio = a^2 : b^2

triplicate ratio = a^3 : b^3

 

sub-duplicate ratio= √a : √b

sub-triplicate ratio= 3√a : 3√b

TYPES OF PROPORTION

* Fourth Proportion = ( a,b,c ) = bc/a

* Third Proportion = ( a,b,c ) = (b^2)/a

* Mean  Proportion =  ( a,b,c ) = root(ab)

  Qes.  If p: q = 2:3 and q:r = 5:7 then what is the ratio p:q:r

  Ans.       p:q=2:3

                q:r=5:7

Multiply by 3/5 so as to make the ratio term of q Common,

p:q = 5 x 3/5:7 x 3/5

q:r= 3:21/5

p:q:r =2:3:21/5

= 2x5:3x5:21/5 x 5

  Hence, p:q:r =10:15:21

IMPORTANT TRICK

For the questions on similar lines as the above, the following method comes handy for faster analysis of the question.

point 1:-                 p : q : r

                               2 : 3 :                    
                                     5 : 7

point 2:-                p : q : r
                               2 : 3 -- 3                     [ 3 has to repeat the right side from q point ]

                               5 -- 5 : 7                     [ 5 has to repeat the left side from q point ]

                 =  (2×5): (5×3): (3×7)

                       =  10 : 15 : 21                         
 

COMPOUNDED RATIO

The compounded ratio of the ratios (P:Q), (R:S), (T:U) will be (PRT : QSU)

Qes.  What is the equivalent compound ratio of 17:23: 115: 153 :: 18:25

Ans.  We know, compound ratio of the ratios  (P:Q), (R:S), (T:U) will be (PRT : QSU)

Thus, the compound ratio of (170: 230), (115: 153), (18:25)

 =    (170 x 115 x 18) / (230 x 153 x 25) = 2 : 5

PROPORTION : -

The equality of two ratios is called proportion.

If a :b = c : d, we write, a: b::c: d and we say that a, b, c, d are in proportion.
 Here, a and d are called extremes, while b and c are called mean terms.

Product of means = Product of extremes 
Thus,   a : b : : c : d 
    ⇒  (b x c) = (a x d)

TYPES OF PROPORTION

Third Proportion
If a : b = b : c, then c is called the third proportion to a and b.

Fourth Proportion
If a : b = c : d, then d is called the fourth proportion to a, b, c.

Mean Proportion
Mean proportion of a and b will be √ab

Qes . What is third proportion to 25 and 20?

 Ans . When c is the third proportion to a and b, relation between a and b is:

             b²= ac

           Let third proportion to 25 and 20 be x, then,

             25: 20:: 20: x 

         ⇒ 25x=20^2     

         ⇒ ( 20 multiply 20 ) divide 25 = 16

            Third proportion to 25 and 20 is 16

Qes. Find the mean proportional between 9 & 3?

Ans. As we know that, mean proportional = √(ab)

        ⇒ √(9 x 3)

        ⇒ 5.196 

        So, the mean proportional of 9 and 3 = 5.196

Qes. Mean proportional of 4 and 9 is a and third proportional of 18 and a is b. Find the fourth proportional of b, 12, 14.

Ans.

Mean proportional of 4 and 9 = a

     ⇒ a^2 = 4 x 9

     ⇒ a=6

Third proportional of 18 and 6 = b

  ⇒ 6^2 = 18 x b 

   ⇒b=2

Forth proportional of 2, 12 and 14

  ⇒2/12 = 14/x

  ⇒x =84

Qes.   If 3:9:5:x

Ans.   If 3:9:5:x

          3/9 = 5/x

          x= 5 x 9/3

         x= 15

Some Important Questions :-

Qes. A bag has coins of Rs. 1, 50 Paise and 25 Paise in ratio of 10:9: 4. What is the
        worth of the bag if the total number of coins in the bags is 72?

Ans.⇒ Number of Rs. 1 Coins = 10/18 x 72 = 40

       ⇒ Number of 50 Paise coins = 9/18 x 72=36

       ⇒ Number of 25 Paise coins = 4/18 x 72=16

     ⇒ Total worth of the bag = (40 x 1) + (36 x 0.5) + (16 x 0.25) = 40 + 18+ 4 =Rs.62          

Qes. If 18: 13.5 :: 32: x and (x + y): y :: 18:10, then what is the value of y?

Ans. 18: 13.5 :: 32:x

   ⇒x = (32 x 13.5)/18

   ⇒x = 24

 Now,

    (x + y): y : : 18:10

    ⇒ (24+y) :y: : 9:5

    ⇒ 5(24+y)=9y

    ⇒120+5y=9y

    ⇒ 4y=120

    ⇒ y = 30

Qes. There are a certain number of Rs.10, Rs.20 and Rs.50 notes available in a bag. The ratio of the number of notes of Rs.10, Rs.20 and Rs.50 is 9:4:6. The total amount available in a bag is Rs.2350. The amount of Rs.10 and Rs.50 in a bag is -

Ans. Let the number of notes of Rs.10, Rs.20 and Rs.50 be 9a, 4a and 6a respectively.

Given,

 ⇒10 × 9a + 20 × 4a + 50 x 6a = 2460

 ⇒ 410a = 2460

 ⇒a=5

Number of notes of Rs.10 = 9x5 = 45

Number of Notes of Rs.20 = 4 x5 = 20

Number of notes of Rs.50 = 6 x5 = 30

Required amount = 10 x 45+ 50 x 30 = Rs.1950

Qes. Mr. mani divides Rs. 169 such that 4 times the 1st share, thrice the 2nd share and twice the third share amount to the same. Then the value of the 2nd share is:

Ans. Given:

Total amount = Rs. 169

now,

Let the share of P, Q and R is 4P: 3Q: 2R.

P : Q : R = 1/4:1/3: 1/2 = 3:4:6 

The value of the 2nd share = (4/13) x 169= Rs. 52