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Lesson 15 min· Section 1 of 4

The straight-up lesson, built for the exam.

Ratio & Proportion

Formula sheet

Everything for this chapter, in one place.

The basics


A ratio $a:b$	means the quantities are $ak$ and $bk$ for some $k$
In lowest terms	$k$ is the size of one "part"
$a:b = c:d$	$\iff ad = bc$
Componendo–dividendo	$\dfrac{a}{b} = \dfrac{c}{d} \implies \dfrac{a+b}{a-b} = \dfrac{c+d}{c-d}$

Chaining two ratios


Given $a:b$ and $b:c$	scale both so the shared term matches
Formally	$a:b:c = a_1c_1 : b_1c_1 : b_2c_2$ after matching on $b$
Three or more	chain one link at a time, always through the shared term

Proportion


Direct	$y = kx$ — doubling one doubles the other
Inverse	$y = \dfrac{k}{x}$ — doubling one halves the other
To a power	$y \propto x^2$ means $y = kx^2$
Combined	$y \propto \dfrac{xz}{w}$ means $y = \dfrac{kxz}{w}$

What changes a ratio, and what does not


Multiplying both parts by $m$	the ratio is unchanged
Adding the same amount to both	the ratio moves toward $1:1$
A transfer between the two	the total is unchanged — use it as the anchor
Each part changing by its own %	treat as two separate multipliers

Sub-groups and overall ratios


Group ratio and sub-ratios given	assign separate letters to each sub-group, then force the group ratio
Interest, output or cost	if two ratios multiply (e.g. amount $\times$ rate), the result ratio is the product term by term
Removing one group's share	subtract its amounts from the totals, then re-form the ratio

Integer conditions


$a:b$ in lowest terms	the real values are $ak$ and $bk$ with $k$ a positive integer
A total	must be a multiple of the sum of the parts
A bound on the total	gives a bound on $k$ , and the extreme answer comes from the extreme $k$

Why this chapter matters

0.7

questions in a typical Quant section

19th

of 25 chapters

≈ 2 of 66 marks a section ride on it.

Asked in17 of 24 papers

20172025

Year by yearper section

2017: 1.32017: 1.3 per section
2018: 0.72018: 0.7 per section
2019: 12019: 1 per section
2020: 0.32020: 0.3 per section
2021: 0.32021: 0.3 per section
2022: 0.72022: 0.7 per section
2023: 0.72023: 0.7 per section
2024: 12024: 1 per section
2025: 12025: 1 per section

Question type21 asked

MCQ 71%Type-in 29%

Type-in answers carry no negative marking.

From every Ratio & Proportion question in CAT papers since 2017.