Quantitative Aptitude Notes Part 7( Ratio and proportion)

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Ratio and proportion

What is Ratio?
A ratio is a relationship between two numbers by division of the same kind. The ration of a to b is written as a : b = a / b In ratio a : b , we can say that a as the first term or antecedent and b, thesecond term or consequent.

 

What is Proportion?
The idea of proportions is that two ratio are equal.
If a: b = c: d, we write a: b: : c: d,
Example: 3 / 15 = 1 / 5
a and d called extremes, where as b and c called mean terms

 

Example:
If P : Q : R = 2 : 3 : 4 , Then P / Q : Q / R : R / P = ?
Answer:
P : Q : R = 2 : 3 : 4 . Let P = 2k, Q = 3k, R = 4k. Then,
= =,  = = and = =.

SO, : : = : : = 8 : 9 : 24 .

Example:
If a : b = 3 : 7 and b : c = 5 : 9, find a : b : c .
Answer:  A : B = 3 : 7, B : C = 5 : 9
= ( 5 x 7 / 5 ) : ( 9 x 7 / 5 ) = 7 : 63 / 5 .
= A : B : C = 3 : 7 : 63 / 5 = 15 : 35 : 63 .

 

 

Example:
If 75 is divided into three parts proportional to 3, 5, 8, 9 then the smallest part is :
Answer:
ratio is = 3 : 5 : 8 : 9, sum of ratio terms is = 25.
So the smallest part is = ( 75 x 3 / 25 ) = 9 .

 

Example:
The Salary of Three friend A, B, C are divided into ratio 5 : 6 : 8.If the increment has given of 10%, 20%, 25%, Find the new ratio of three friend salary ?
Answer:
Step 1: We assume ration as 5x, 6x, 8x
now the increment of new salary is A = 110 / 100, B = 120 / 100, C = 125 / 100.
Step 2: A,s new salary is 110 X 5x / 100 = 55x / 10.
B,s new salary is 120 X 6x / 100 = 36x / 5.
C,s new salary is 125 X 8x / 100 = 10.
Step 3:New ratio is 55x /10: 36x / 5 : 10.

 

Example:
Find the mean proportional between given two number that is 64 and 49.
Answer:
The mean proportion of two numbers is
Root of 64 and 49 is √8 x √ 7 = 8 x 7 = 56.
So, the mean proportional is 56.

 

 

Example:
Which of the following ratio is the greatest ?
7 : 15 , 15 : 23 , 17 : 25 , 21 : 29
Answer:
7 / 15 = 0.466
15 / 23 = 0.652
17 / 25 = 0.68
21 / 29 = 0.724
So, 0.724 is greatest and therefore, 21 : 29

 

Example:
What number has to be added to each term of 3 : 5 to make the ratio 5 : 6 :
Answer :
Let the number to be added x , Then
3 + x / 5 + x = 5 / 6
6 ( 3 + x ) = 5 ( 5 + x )
x = ( 25 – 18 ) = 7
So , the number to be added is 7 .

 

Example:
A money bag contains 50 p, 25 p, and 10 p coins in the ratio 5 : 9 : 4, and the total amounting to Rs.206.
Find the individual number of coins of each type.
Answer: 

Step 1: Let the number of 50 p,25 p, and 10 p coins be 5x, 9x, 4x respectively.

 
Then, 5x / 2 + 9x / 4 + 4x / 10 = 206
= 50x + 45x + 8x = 4120= 103x = 4120
= x = 40.
Step 2: Number of 50 p coins is ( 5 x 40 = 200 ),
Number of 25 p coins is( 9 x 40 = 360 ),
Number of 10 p coins ( 4 x 40 = 160 ),

 

Example:
In a bottle mixture of 80 liters and the ratio of milk and water is 3 : 2. If this mixture ratio is to be 2 : 3.  What the quantity of water to be further added?
Answer:
Step 1: Quantity of Milk ( 80 x 3 / 5 ) = 48 liters, So Quantity of water in it  ( 80 – 48 ) = 32 liters.
Step 2: New Ratio required 2: 3, Let x water to be added, Then Milk: Water is = 48: (32+x)
=48 / (32 + x).
Step 3: Now 48 / (32 + x) = 2: 3
48 / (32 + x) = 2 / 3
2x = 144 – 64
x = 80/2
=40 liters.

 

Example:
The ratio of copper and zinc is 9 : 3 in an alloy . If quantity of zinc is 24 kg. The quantity of copper is would be?

 

 

Answer:
ratio of copper and zinc is 9 : 3
Let the required quantity of copper be x kg.
So, 9: 3: 24: x
9x = 24 x 3
x = 72 / 9
x = 8 kg

 

Example: In a liquid mixture of 60 liters, the ratio of milk and water is 2:1. If this ratio is to be 1: 2, then the quantity of water to be further added is:
Answer:
So Quantity of milk = (60 x 2 / 3) = 40 liters.
Water in it = (60 – 40) = 20 liters.
New ratio required = 1: 2.
Let quantity of water to be added further be x liters
milk: water = 40 / (20 + x)
Now,
40 / ( 20 + x ) = 1 / 2
20 + x = 80
x = 60 litres
So, 60 litres of water to be added further.

 

Example:
20% alcohol present in a 15 litres mixture and rest of water  . If 3 litres of water be mixed with it , the percentage of alcohol in the new mixture would be  ?

 

 

Answer:
20% present in a 15 litres mixture
So, 20 x 15 / 100 = 3 litres .
Water in it = (15 – 3) 12 liters.
New quantity of mixture = (15 + 3) = 18 liters.
Percentage of alcohol in new mix is = (3 x 100 / 18) % = 50 / 3.

 

 

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