Chapter 2 of 4

Quantitative Aptitude

In the GATE Civil syllabus under General Aptitude · 7 parts

Data Interpretation · Percentage · Ratio, Proportion & Partnership · Permutation, Combination & Probability · Mensuration — 2D Figures · Mensuration — 3D Solids · Elementary Statistics

CIVILGYAN TEST SERIES · civilgyantests.com — free study notes for civil engineering exams

📑 Contents (45 sections)

Part 1 of 7

Data Interpretation

Last reviewed 22 Sept 2026 · 5 min read

What a DI set really tests

Not arithmetic — reading. Nearly every mark lost in this section comes from one of four things: the unit in the heading (lakh, crore, thousand), the row or column read one line off, "percentage increase" mistaken for "percentage point", or an average taken over the wrong number of years.

Read in this order: title → unit → row and column headings → the question. Only then look at the numbers.

The four presentations

Type	What it is good at	Watch out for
Table	exact values, several variables	which way the years run
Bar chart	comparing quantities	the scale, and bars that do not start at zero
Line graph	trend over time	a broken axis exaggerating a rise
Pie chart	shares of one whole	two pies with different totals

A caselet presents the same data as a paragraph of prose; convert it into a small table before answering anything.

The five questions

FormulaThe recurring calculations

Total of a row or column — plain addition
Average $= \frac{total}{number of entries}$ (count the entries, do not assume five)
Ratio — reduce before dividing
Percentage share $= \frac{part}{whole} \times 100$
Percentage change $= \frac{new - old}{old} \times 100$ — always over the old value

For pie charts: $1% = 3.6°$ , so a sector of $54°$ is $15%$ . Two pie charts can only be compared in absolute terms if both totals are given.

Working quickly

Keep the unit out of the arithmetic: if everything is "₹ lakh", work in plain numbers and put the unit back at the end.
Use fraction equivalents rather than long division: $\frac{1}{8} = 12.5%$ , $\frac{1}{7} \approx 14.3%$ , $\frac{1}{6} \approx 16.7%$ , $\frac{1}{3} \approx 33.3%$ .
When options are far apart, approximate: $\frac{85}{180}$ is a little under half, so 47% is right and 43% is not.
Do not recompute a total you already worked out for the previous question — DI sets are built so each answer feeds the next.

Part 2 of 7

Percentage

Last reviewed 16 Sept 2026 · 5 min read

Meaning

Per cent means "per hundred": $x % = \frac{x}{100}$ .

Conversions

Fraction → %: multiply by 100 ( $3/8 = 37.5%$ ).
% → fraction: divide by 100 and simplify ( $12.5% = 1/8$ ).
Decimal ↔ %: move the decimal point two places ( $0.075 = 7.5%$ ).

Key fraction equivalents

%	Fraction	%	Fraction
50	1/2	12.5	1/8
33.33	1/3	11.11	1/9
25	1/4	10	1/10
20	1/5	9.09	1/11
16.67	1/6	8.33	1/12
14.28	1/7	6.25	1/16
66.67	2/3	37.5	3/8
75	3/4	62.5	5/8
40	2/5	87.5	7/8

Basic formulas

FormulaPercentage relations

x% of N $= \frac{x}{100} \times N$
What % is A of B? $= \frac{A}{B} \times 100$
Number whose x% is A $= \frac{A \times 100}{x}$
Percentage change $= \frac{New - Old}{Old} \times 100$
After increase of x%: New = Old × $(1 + \frac{x}{100})$ ; after decrease: Old × $(1 - \frac{x}{100})$

Percentage point — absolute difference between percentages (from 20% to 25% is a 5 percentage point increase, but a 25% increase).

Comparison formulas

Formula"More than / less than"

If A is x% more than B, then B is less than A by $\frac{x}{100 + x} \times 100%$
If A is x% less than B, then B is more than A by $\frac{x}{100 - x} \times 100%$

Successive percentage changes

FormulaNet effect

Two successive changes of a% and b% (use negative for decrease):

Net change = a + b + \frac{ab}{100} %

Increase x% then decrease x% → net loss of $x^{2} /100$ %
Apply repeatedly for three changes (combine two, then the third)

Product constancy (price and consumption)

FormulaExpenditure = price × consumption

If price increases by r%, consumption must decrease by

\frac{r}{100 + r} \times 100%

to keep expenditure constant; if price decreases by r%, consumption can increase by $\frac{r}{100 - r} \times 100%$ .

Area of rectangle with length and breadth changed by a% and b%: net change $= a + b + ab /100$ %; square/circle with side/radius changed by x%: area changes by $2 x + x^{2} /100$ %.

Population and depreciation

FormulaGrowth and depreciation

Population after n years (growth r% p.a.): $P_{n} = P (1 + \frac{r}{100})^{n}$
Population n years ago: $P / (1 + \frac{r}{100})^{n}$
Value after depreciation of r% p.a.: $V_{n} = V (1 - \frac{r}{100})^{n}$

Worked examples

Worked ExampleExample 1 — basic

(a) Find 37.5% of 640. (b) What percent of 250 is 45? (c) 18 is 12% of what number?

Solution. (a) $\frac{3}{8} \times 640 =$ 240; (b) $\frac{45}{250} \times 100 =$ 18%; (c) $\frac{18 \times 100}{12} =$ 150

Worked ExampleExample 2 — more than / less than

A's salary is 25% more than B's. By what percent is B's salary less than A's?

Solution. $\frac{25}{125} \times 100 =$ 20%

Worked ExampleExample 3 — successive change

The price of cement rises by 20% and then falls by 10%. Find the net change.

Solution. $20 - 10 + \frac{20 \times ( - 10 )}{100} =$ +8%

Worked ExampleExample 4 — equal increase and decrease

A number is increased by 10% and then decreased by 10%. Net change?

Solution. $- \frac{1 0 ^{2}}{100} =$ −1% (a decrease of 1%)

Worked ExampleExample 5 — consumption adjustment

The price of steel increases by 25%. By what percent must a builder reduce consumption to keep expenditure unchanged?

Solution. $\frac{25}{125} \times 100 =$ 20%

Worked ExampleExample 6 — area change

The length of a rectangular plot is increased by 20% and breadth decreased by 20%. Find the change in area. What if the side of a square increases by 10%?

Solution. Rectangle: $20 - 20 - 4 =$ −4%; square: $20 + 1 =$ +21%

Worked ExampleExample 7 — population

A town's population of 50 000 grows at 4% per year. Find the population after 2 years.

Solution. $50000 \times 1.0 4^{2} = 50000 \times 1.0816 =$ 54 080

Worked ExampleExample 8 — elections

In an election between two candidates, the winner got 58% of valid votes and won by 4800 votes. Find the total valid votes.

Solution. Margin = 58% − 42% = 16% → $0.16 V = 4800$ → V = 30 000

Worked ExampleExample 9 — examinations

A student needs 40% to pass. She scored 180 marks and failed by 20 marks. Find maximum marks.

Solution. Pass marks = 200 = 40% → maximum 500

Worked ExampleExample 10 — depreciation

A machine worth ₹ 5 lakh depreciates at 10% per year. Find its value after 3 years.

Solution. $5 \times 0. 9^{3} = 5 \times 0.729 =$ ₹ 3.645 lakh

Worked ExampleExample 11 — percentage points

Unemployment rose from 6% to 7.5%. Express the rise in percentage points and percent.

Solution. 1.5 percentage points; $\frac{1.5}{6} \times 100 =$ 25%

Frequently tested points

x% = x/100; know fraction equivalents (1/8 = 12.5%, 1/6 = 16.67%, 1/7 ≈ 14.28%).
% change = (new − old)/old × 100.
A is x% more than B → B is x/(100 + x) × 100% less than A; x% less → x/(100 − x) × 100% more.
Successive changes: a + b + ab/100; equal rise and fall → −x²/100 %.
Constant expenditure: price up r% → consumption down r/(100 + r) × 100%.
Square/circle area change for x% change in side/radius: 2x + x²/100.
Population $P (1 + r /100)^{n}$ ; depreciation $V (1 - r /100)^{n}$ .
Percentage point vs percent.

Common MistakeCommon mistakes

Calculating percentage change on the new value instead of the original.
Adding successive percentages directly (10% up and 10% down is not zero change).
Confusing percentage points with percent change.

Revision SummaryChapter summary

Percentages express parts per hundred and interconvert with fractions and decimals, with key equivalents enabling fast calculation.
Basic relations find a percentage of a quantity, a quantity from its percentage and percentage change.
Comparison formulas convert "more than" into "less than" statements, and successive changes combine as a + b + ab/100.
Product constancy links price and consumption, and area changes follow the same successive-change rule.
Growth and depreciation compound over time, with applications to elections, examinations and budgets.

Part 3 of 7

Ratio, Proportion & Partnership

Last reviewed 22 Sept 2026 · 6 min read

Meaning of a ratio

A ratio compares two quantities of the same kind by division: $a : b = \frac{a}{b}$ , where $a$ is the antecedent and $b$ the consequent. A ratio has no unit, so 4 kg : 800 g must first be written in one unit — 4000 g : 800 g = 5 : 1.

Multiplying or dividing both terms by the same non-zero number does not change a ratio: $6 : 9 = 2 : 3$ .
$a : b : c$ means the three quantities are in the proportion $a$ , $b$ , $c$ of one common part $k$ — write them as $ak$ , $bk$ , $c k$ and one equation finds $k$ .

Comparing two ratios

Cross-multiply: $a : b > c : d$ when $a d > b c$ . To compare $5 : 7$ and $7 : 10$ : $5 \times 10 = 50$ and $7 \times 7 = 49$ , so $5 : 7$ is greater.

Derived ratios

Name	For the ratio $a : b$
Duplicate	$a^{2} : b^{2}$
Sub-duplicate	$a : b$
Triplicate	$a^{3} : b^{3}$
Sub-triplicate	$3 a : 3 b$
Inverse	$b : a$
Compounded of $a : b$ and $c : d$	$a c : b d$

Proportion

FormulaProportion and its consequences

Four quantities are in proportion when $a : b = c : d$ , written $a : b :: c : d$ .

a \times d = b \times c (product of extremes = product of means)

Fourth proportional to $a$ , $b$ , $c$ is $\frac{b c}{a}$
Third proportional to $a$ and $b$ is $\frac{b ^{2}}{a}$
Mean proportional between $a$ and $b$ is $ab$
$a$ , $b$ , $c$ are in continued proportion when $b^{2} = a c$

FormulaComponendo and dividendo

If $\frac{a}{b} = \frac{c}{d}$ , then

\frac{a + b}{a - b} = \frac{c + d}{c - d}

This turns an equation such as $\frac{3 x + 2}{3 x - 2} = \frac{5}{3}$ into one step instead of cross-multiplication.

Part 4 of 7

Permutation, Combination & Probability

Last reviewed 22 Sept 2026 · 5 min read

Counting first

FormulaFundamental principle

If one job can be done in $m$ ways and, independently, a second in $n$ ways, then both together can be done in $m \times n$ ways (AND → multiply). If a job can be done by either of two exclusive methods, in $m$ or in $n$ ways, the total is $m + n$ (OR → add).

$n! = n (n - 1) \dots 2 \cdot 1$ , with $0! = 1$ .

Permutations — order matters

FormulaArrangements

^{n} P_{r} = \frac{n !}{( n - r )!}

All $n$ objects: $n!$
With repetition allowed, $r$ places from $n$ types: $n^{r}$
$n$ letters of which one letter repeats $p$ times and another $q$ times: $\frac{n !}{p ! q !}$
Circular arrangements of $n$ objects: $(n - 1)!$ , and $\frac{( n - 1 )!}{2}$ when clockwise and anticlockwise are the same (a garland or necklace)
Objects that must stay together: tie them into one block, arrange the blocks, then arrange within the block

Combinations — order does not matter

FormulaSelections

^{n} C_{r} = \frac{n !}{r ! ( n - r )!},^{n} C_{r} =^{n} C_{n - r},^{n} C_{r} = \frac{^{n} P _{r}}{r !}

$^{n} C_{0} =^{n} C_{n} = 1$ ; $^{n} C_{1} = n$
$^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$
Total subsets of an $n$ -set: $2^{n}$
"At least one" is usually easiest as total $-$ none

A quick test: a committee is a combination (Ram and Shyam is the same committee as Shyam and Ram), while a president-and-secretary pair is a permutation.

Part 5 of 7

Mensuration — 2D Figures

Last reviewed 22 Sept 2026 · 5 min read

Quadrilaterals

FormulaAreas and perimeters

Figure	Area	Perimeter
Rectangle	$l \times b$	$2 (l + b)$ , diagonal $l^{2} + b^{2}$
Square	$a^{2} = \frac{d ^{2}}{2}$	$4 a$ , diagonal $a 2$
Parallelogram	base $\times$ height	$2 (a + b)$
Rhombus	$\frac{1}{2} d_{1} d_{2}$	$4 a$ , with $a = \frac{1}{2} d_{1}^{2} + d_{2}^{2}$
Trapezium	$\frac{1}{2} (a + b) h$	sum of the four sides

Triangles

FormulaFour ways to an area

Base and height: $\frac{1}{2} bh$
Three sides (Heron): $s = \frac{a + b + c}{2}$ , area $= s (s - a) (s - b) (s - c)$
Equilateral of side $a$ : $\frac{3}{4} a^{2}$ , height $\frac{3}{2} a$
Right-angled: $\frac{1}{2} \times$ product of the two legs

Circle, sector, segment and ring

FormulaCircle formulas

For radius $r$ and an angle $θ$ in degrees:

Area = π r^{2}, Circumference = 2 π r

Arc = \frac{θ}{360} \times 2 π r, Sector area = \frac{θ}{360} \times π r^{2} = \frac{arc \times r}{2}

Minor segment = sector - triangle, Ring (annulus) = π (R^{2} - r^{2})

Use $π = \frac{22}{7}$ when the radius is a multiple of 7, and $3.14$ otherwise.

Paths and borders

A path of width $w$ outside a rectangle $l \times b$ : outer size $(l + 2 w) \times (b + 2 w)$ ; path area $=$ outer $-$ inner.
A path inside: inner size $(l - 2 w) \times (b - 2 w)$ .
Two crossing paths through the middle of a field: area $= l w + b w - w^{2}$ — the overlap is counted once.

Part 6 of 7

Mensuration — 3D Solids

Last reviewed 22 Sept 2026 · 5 min read

The standard solids

FormulaVolumes and surfaces

Solid	Volume	Curved / lateral surface	Total surface
Cuboid	$l bh$	$2 h (l + b)$	$2 (l b + bh + h l)$
Cube	$a^{3}$	$4 a^{2}$	$6 a^{2}$
Cylinder	$π r^{2} h$	$2 π r h$	$2 π r (r + h)$
Cone	$\frac{1}{3} π r^{2} h$	$π r l$	$π r (r + l)$
Sphere	$\frac{4}{3} π r^{3}$	—	$4 π r^{2}$
Hemisphere	$\frac{2}{3} π r^{3}$	$2 π r^{2}$	$3 π r^{2}$

Diagonal of a cuboid $= l^{2} + b^{2} + h^{2}$ ; of a cube $= a 3$ .

FormulaCone, frustum, prism and pyramid

Cone slant height: $l = r^{2} + h^{2}$
Frustum of a cone (radii $R$ and $r$ , height $h$ ): $V = \frac{π h}{3} (R^{2} + R r + r^{2})$ , slant $l = h^{2} + (R - r)^{2}$ , curved surface $= π l (R + r)$
Hollow cylinder (outer $R$ , inner $r$ ): $V = π h (R^{2} - r^{2})$
Prism: $V =$ base area $\times$ height; lateral surface $=$ base perimeter $\times$ height
Pyramid: $V = \frac{1}{3} \times$ base area $\times$ height; lateral surface $= \frac{1}{2} \times$ base perimeter $\times$ slant height

Capacity and units

1 m^{3} = 1000 litres, 1 litre = 1000 cm^{3}

Part 7 of 7

Elementary Statistics

Last reviewed 22 Sept 2026 · 6 min read

Frequency distributions

Raw data is grouped into classes with frequencies. For a class such as 20–30: the class mark (midpoint) is 25, the class size is 10, and the cumulative frequency is the running total of the frequencies up to that class. In an exclusive distribution (10–20, 20–30) the upper limit belongs to the next class; in an inclusive one (10–19, 20–29) it does not, and the limits must be corrected by half the gap before any formula is used.

The three averages

FormulaMean

Ungrouped: $\overset{x}{ˉ} = \frac{\sum x}{n}$
Grouped: $\overset{x}{ˉ} = \frac{\sum f x}{\sum f}$ with $x$ the class mark
Assumed mean: $\overset{x}{ˉ} = A + \frac{\sum f d}{\sum f}$ , where $d = x - A$
Step deviation: $\overset{x}{ˉ} = A + h \frac{\sum f u}{\sum f}$ , where $u = \frac{x - A}{h}$
Combined mean of two groups: $\frac{n _{1} x ˉ _{1} + n _{2} x ˉ _{2}}{n _{1} + n _{2}}$

FormulaMedian and mode

Ungrouped (data arranged in order):

$n$ odd → the $(\frac{n + 1}{2})$ th value; $n$ even → the mean of the two middle values.
Mode is the most frequent value; a set may have none or several.

Grouped:

Median = l + \frac{\frac{N}{2} - c f}{f} \times h, Mode = l + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h

where $l$ is the lower limit of the median or modal class, $N = \sum f$ , $c f$ the cumulative frequency before it, $f$ its frequency, $h$ the class size, and $f_{0}, f_{1}, f_{2}$ the frequencies of the classes before, at and after the modal class.

Empirical relation: $Mode = 3 Median - 2 Mean$ .

Which one to use: the mean uses every value but is dragged by extremes; the median ignores extremes and suits skewed data such as incomes; the mode is the only average for a non-numeric category.

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