Introduction Proportions - CAT PDF Download

When two ratios are equal, the four quantities composing them are said to be proportionals. Thus if a/b = c/d, then a, b, c, d are proportionals. This is expressed by saying that a is to b as c is to d, and the proportion is written as

a:b::c:d or a:b = c:d

The terms a and d are called the extremes while the terms b and c are called the means.

Thus, we can say, Product of Extremes = Product of Means 

Example: What is the value of x in the following expression? 

5/8 =  x/12 

Solution:

5/8 =  x/12 

X = 60/8 = 7.5  

It can be calculated with the help of percentages also. In this question, the percentage increase in the denominator is 50%, so the numerator will also increase by 50%. 

Operations in Proportions

If four quantities a, b, c and d form a proportion, many other proportions may be deduced by the properties of fractions. The results of these operations are very useful. These operations are                    

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

Alternando: If a/b = c/d, then a/c = b/d                         

Componendo: Ifa/b=c/d,then (a+b)/b=(c+d)/d                         

Dividendo: Ifa/b=c/d,then (a-b)/b=(c-d)/d                    

Componendo and Dividendo: Ifa/b=c/d then (a + b)/(a – b) = (c + d)/(c – d)

Variation

Two quantities A and B are said to be varying with each other if there exists some relationship between A and B such that the change in A and B is uniform and guided by some rule. 

Some typical examples of variation

Area (A) of a circle = pi. R², where R is the radius of the circle. Area of a circle depends upon the value of the radius of a circle, or, in other words, we can say that the area of a circle varies as the square of the radius of a circle. 

At a constant temperature, the pressure is inversely proportional to the volume. 

If the speed of any vehicle is constant, then the distance traversed is proportional to the time taken to cover the distance. 

Essentially there are two kinds of proportions that two variables can be related by:

Direct Proportion:

When it is said that A varies directly as B, you should understand the following implications:                    

(a) Logical implication: When A increases B increases

(b) Calculation implication: If A increases by 10%, B will also increase by 10

(c) Graphical implications: The following graph is representative of this situation.

(d) Equation implication: The ratio A/B is constant. 

Inverse Proportion:

When A varies inversely as B, the following implication arise. 

(a) Logical implication: When A increases B decreases 

(b) Calculation implication: If A decreases by 9.09%, B will increase by 10%.

(c) Graphical implications: The following graph is representative of this situation. 

(d) Equation implication: The product A X B is constant. 

Example: The height of a tree varies as the square root of its age (between 5 and 17 years). When the age of a tree is 9 years, its height is 4 feet. What will be the height of the tree at the age of 16? 

Solution: Let us assume the height of the tree is H and its age is A years. 

So, H ∝ √A, or, H = K x √A Now, 4 = K x √9 K = 4/3 

So, height at the age of 16 years = H = K x √A = 4/3 x 4 = 

16/3 = 5 feet 4 inches. 

Practice Question:

Total expenses at a hostel is partly fixed and partly variable. When the number of students is 20, total expense is Rs 15,000 and when the number of students is 30, total expense is Rs 20,000. What will be the expense when the number of students is 40? 

(a) 20,000

(b) 25,000

(c) 30,000

(d) 32,0000

Solution:

Expenses = F + K. V; where F is the fixed cost and V is the number of students. 

Rs 15,000 = F + K.20 ------------(1) 

Rs 20,000 = F + K.30 ------------(2) 

Solving (1) and (2), 

Rs 5,000 = 10. K 

= K = Rs 500 

So, F = Rs 5000 

So, F + 40 K = Rs 5,000 + 40 x 500 = Rs 25,000 

Example: Rs. 5783 is divided among Sherry, Berry, and Cherry in such a way that if Rs.28, Rs.37and Rs.18be deducted from their respective shares, they have money in the ratio 4:6:9. Find Sherry’s share.                 
(a) Rs.1256
(b) Rs.1228
(c) Rs.1456
(d) Rs.1084
Solution: The problem clearly states that when we reduce 28, 37 and 18 rupees, respectively from Sherry’s, Berry’s and Cherry’s shares, the resultant ratio is: 4:6:9.                 
Thus, if we assume the reduced values as 4x, 6x and 9x, we will have -> 
Sherry’s share -> 4x + 28, Berry’s share -> 6x + 37 and Cherry’s share -> 9x + 18 and thus we have
(4x + 28) + (6x + 37) + (9x + 18) = 5783 -> 19x = 5783 – 83 = 5700
Hence, x = 300.
Hence, Sherry’s share is Rs. 1228. 

Note: For problems based on this chapter we are always confronted with ratios and proportions between different number of variables. For the above problem we had three variables which were in the ratio of 4 : 6 : 9. When we have such a situation we normally assume the values in the same proportion, using one unknown ‘x’ only (in this example we could take the three values as 4x, 6x and 9x, respectively).

Then, the total value is represented by the addition of the three giving rise to a linear equation, which on solution, will result in the answer to the value of the unknown ‘x’.

However, the student should realise that most of the time this unknown ‘x’ is not needed to solve the problem. This is illustrated through the following alternate approach to solving the above problem:

Assume the three values as 4, 6 and 9 Then we have

(4 + 28) + (6 + 37) + (9 + 18) = 5783 -> 19 = 5783–83 = 5700 -> 1 = 300

Hence, 4 + 28 = 1228.

While adopting this approach the student should be careful in being able to distinguish the numbers in bold as pointing out the unknown variable.

Practice Question:

If 10 persons can clean 10 floors by 10 mops in 10 days, in how many days can 8 persons clean 8 floors by 8 mops?

(a) 12 1⁄2 days
(b) 8 days 

(c) 10 days
(d) 8 1/3 days 

Sol: Do not get confused by the distractions given in the problem. 10 men and 10 days means 100 man-days are required to clean 10 floors. That is, 1 floor requires 10 man-days to get cleaned. Hence, 8 floors will require 80 man-days to clean.

Therefore, 10 days are required to clean 8 floors. 

Practice Question: 

Three quantities A, B, C are such that AB = KC, where K is a constant. When A is kept constant, B var- ies directly as C; when B is kept constant, A varies directly C and when C is kept constant, A varies inversely as B. Initially, A was at 5 and A:B:C was 1:3:5. Find the value of A when B equals 9 at constant C.
(a) 8
(b) 8.33
(c) 9
(d) 9.5
Solution:  Initial values are 5, 15 and 25. Thus we have 5 X 15 = K X 25.
Hence, K = 3.
Thus, the equation is AB = 3C.
For the problem, keep C constant at 25. Then, A X 9 = 3 X 25.
i.e., A = 75/9 = 8.33

Example: The ratio of water and milk in a 30-litre mixture is 7 : 3. Find the quantity of water to be added to the mixture in order to make this ratio 6 :1. 
(a) 30
(b) 32
(c) 33
(d) 35
Solution:
Solve while reading —> As you read the first sentence,
you should have 21 litres of water and 9 litres of milk in your mind.
In order to get the final result, we keep the milk constant at 9 litres.
Then, we have 9 litres, which corresponds to 1
Hence, '7' corresponds to 6. Solving by using the unitary method we have 54 litres of water to 9 litres of milk.
Hence, we need to add 33 litres of water to the original mixture.
Alternatively, we can solve this by using options. you should try to do the same.

This document is the final document of EduRev’s notes of Ratios & proportions chapter. In the next document we have tried to solve the different CAT previous year questions & curated some of the best tests for you to practice - as a CAT aspirant you should be well aware of types of questiosn and should have a good practice of questions. Hence we suggets you to practice all the tquestions in the upcoming documents & Tests.