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Ratio

Ratio can be understood in two ways:
  1. Ratio as a bridging element
  2. Ratio as a multiplier

Ratio as a Bridging Element

Ratio as a bridging element helps us in establishing the relationship between more than two quantities. This can be further understood with the following example:
 
Suppose conversion rate of our currency Rupee is given with respect to US dollar and also with respect to Pound sterling. If we have to find the conversion ratio of US dollar with respect to pound sterling, we can do it by making rupee as the bridge between US dollar and pound sterling.
 
Example-1
The ratio of the age of A and B is 2:5 and ratio of the age of B and C is 3:4. What is the ratio of the age of A, B and C?
Solution
Since B is the common platform which associates A and C, so we will try to make B equal in both the cases.
 
Age of A:Age of B = [2:5] × 3
 
Age of B:Age of C = [3:4] × 5
 
Or, Age of A:Age of B = 6:15...(1)
 
Age of B:Age of C = 15:20...(2)
 
Since ratio of B is same in both the cases, hence, age of A:Age of B:Age of C = 6:15:20.
 
 
Example-2
Given that
 
 
Salary of A:Salary of B = 1:2
 
Salary of B:Salary of C = 3:4
 
Salary of C:Salary of D = 5:6
 
Salary of D:Salary of E = 7:8
 
Salary of E:Salary of F = 9:10
 
What is the ratio of the salaries of A, B, C, D, E, and F?
Solution
Salary of A:Salary of B: Salary of C: Salary of D: Salary of E: Salary of F = (1 × 3 × 5 × 7 × 9):(2 × 3 × 5 × 7 × 9):(2 × 4 × 5 × 7 × 9):(2 × 4 × 6 × 7 × 9):(2 × 4 × 6 × 8 × 9):(2 × 4 × 6 × 8 × 10).
 
(Understand the above mechanism with the help of the method given in Example 2. In these cases, this method can be used as a shortcut to find the ratios in the following way: For A, take all the leftmost digits, and now keep shifting towards the right digits by quitting one by one all the leftmost digits. So, B = Right digit of 1st ratio and so on for C, D, E and F.)
 
 
Example-3
If A:B = 3:4,
 
B:C = 5:7
 
C:D = 10:11

What is the ratio of A:D?
Solution
A = 3 × 5 × 10 and D = 4 × 7 × 11
 
So, the ratio = 150:308
 
Alternatively, (A/B) × (B/C) × (C/D) = (3/4) × (5/7) × (10/11) = (3 × 5 × 10)/(4 × 7 × 11) = 150:308
 
 
Example-4
A, B, C and D purchase a gift worth Rs 60. A pays 1/2 of what others are paying, B pays 1/3rd of what others are paying and C pays 1/4th of what others are paying. What is the amount paid by D?
Solution
Since A is paying 1/2 of what others are paying, so A is paying 1/3rd of the total amount.
 
(To understand this, let us assume that B, C and D are paying Rs 2x. So A is paying Rs x. The total amount being paid by A, B, C and D = 3x = Rs 60, hence, the amount paid by A = x/3x = 1/3rd of the total.)
 
So, the amount paid by A = 60/3 = Rs 20
 
Similarly, B is paying 1/4th of the total and C is paying 1/5th of the total.
 
Hence, the amount paid by B and C are Rs 15 and Rs 12 respectively.
 
So, the amount paid by D = Rs 13
 

Ratio as a Multiplier

The moment we say that the ratio of two numbers A and B is 5:1, what we mean to say that A is 5 times of B.

It can also be seen that A:B:C in A/2:B/3:C/4 = K is not same as A:B:C = 1/2:1/3:1/4 since multiplier of A, B and C are not the same in both the cases.

Ratio of A:B:C in A/2:B/3:C/4 = K can be calculated in the following way
 
Since A/2 = B/3 = C/4 = K, so A = 2K, B = 3K and C = 4 K
 
Hence, the ratio of A:B:C = 2:3:4
 
While calculating the ratio of A, B and C in A:B:C = 1/2:1/3:1/4, we will multiply each of A, B and C by the LCM of the denominator of all the ratios, i.e., 12.
 
So, A:B:C = 6:4:3
 
Example-5
10 persons can cut 8 trees in 12 days. How many days will 8 persons take to cut 6 trees?
Solution
Let us see this question in a changed perspective
 
Suppose if the question is—10 persons can cut 8 trees in 12 days. How many days will 10 persons take to cut 4 trees?
 
Answer to this question is:Since the amount of work is getting halved, so the number of days will also get halved.
 
There are three factors, namely, the number of men, the number of days and the number of trees, which are responsible for the final answer.
 
Since the number of men are less in the final situation, so more number of days will be required. Hence, multiplier = 10/8 (had there been 12 persons, multiplier would have been 10/12).
 
The number of trees are less in the final situation, so less number days will be required. So, multiplier = 6/8
 
Hence, the total number of days = 12 × 10/8 × 6/8 = 90/8 = 11.25 days
 

Example-6
A train approaches a tunnel AB. Inside the tunnel, a cat is sitting at a point that is 3/8th of the distance of AB measured from the entrance A. When the train whistles, the cat runs. If the cat moves to the entrance of the tunnel A, the train catches the cat exactly at the entrance. If the cat moves to the exit B, the train catches the cat exactly at the exit. What is the ratio of the speed of the train and the speed of the cat?
Solution
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Initially, this was the position of the train and the cat. Now, let us assume that the cat is moving towards exit B. The moment the cat covers 3/8th of AB distance in the direction of exit B, the train will be at the entrance A.
 
 
Now, if the cat moves in the direction of exit B, the train is catching up with the cat at the exit B. So, in the time cat covers 2/8th distance, the train is covering the whole distance from A to B.
 
So, the ratio of the distance covered by train and the distance covered by the cat = 4:1
 
So, the ratio of speed = 4:1
 

Example-7
Pranesh can do a work in 15 days. In how many days will the work be completed by his brother Saket if efficiency of Saket is 60% more than that of Pranesh?
Solution
Since the ratio of efficiency of Pranesh and Saket = 100:160 = 5:8 the number of days taken by Pranesh and Saket will be in the ratio of 8:5
 
Since Pranesh takes 15 days to do this work, Saket will take 15 × 5/8 = 9.37 days
 




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