Ratios, Percentages, Averages and Population Short Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Ratios, Percentages, Averages and Population short notes for CAT: 67 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

The Meaning of a Ratio

Ratio = Relative Size, Not Actual Size

If $A : B = 5 : 3$ , then

$A = 5 x, B = 3 x$

Ratio form	Possible actual values
$5 : 3$	$5, 3$ or $10, 6$ or $50, 30$

Memory hook: Ratio gives the shape of the numbers. The multiplier gives the size.

Part-to-Part and Part-to-Whole

Part-to-Part → Part-to-Whole

If $A : B = m : n$ , then total parts are $m + n$ .

$A = \frac{m}{m + n} \times T, B = \frac{n}{m + n} \times T$

Example

If $A : B = 3 : 5$ and $A + B = 64$ :

$8$ parts $= 64$

So $1$ part $= 8$ , hence $A = 24$ , $B = 40$ .

The Universal Ratio Setup

Universal Setup for Ratio Questions

Write ratio as variables: $m x, n x$ .

Identify the extra condition.

Make one equation.

Find $x$ , then answer the exact question.

CAT warning: The answer is often not

x

. It may be total, difference, new entrants, final shift, or a ratio.

Proportion: Equality of Two Ratios

Definition

Proportion

Two ratios are equal.

$\frac{A}{B} = \frac{C}{D}$

then

$A D = B C$

Ratios, Percentages, Averages and Population: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Ability NAT

The average of seven numbers is $9$ . What is the total of the seven numbers?

Correct Answer:

Step-by-Step Solution

Key idea: the total method says convert an average into a hidden total by multiplying average by count.

Step 1: Use $Total = Average \times Count$ .

Step 2: Average $= 9$ , Count $= 7$ .

Step 3: Total $= 9 \times 7 = 63$ .

Answer: $63$ .

Question 2 · Quantitative Ability MCQ

In competitive exams, when a problem provides the average of a set and the number of items in the set, what is the most powerful "hidden" quantity you should immediately calculate?

A.
The total sum
B.
The median
C.
The mode
D.
The standard deviation

Correct Answer:

Step-by-Step Solution

Key idea: this is a total-method trigger question, recognisable because both the average and the count are explicitly given.

Step 1: The formula connecting average, total, and count is $Average = \frac{Total}{Count}$ .

Step 2: Rearranging this gives $Total = Average \times Count$ .

Step 3: In average problems, the total sum is the most useful hidden quantity because it allows you to combine groups, track changes, or apply constraints.

Answer: The total sum.

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Ratios, Percentages, Averages and Population Short Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Ratios, Percentages, Averages and Population short notes for CAT: 67 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice q

A question from this chapter

Question 1

The average of seven numbers is 9 . What is the total of the seven numbers?

Question 2

In competitive exams, when a problem provides the average of a set and the number of items in the set, what is the most powerful "hidden" quantity you should immediately calculate?

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