GATE General Aptitude Formula Sheet: Every Quant Formula

By MD ANISH AHAMADUpdated 9 Oct 20269 min read

The GATE General Aptitude section needs only school arithmetic, but it needs it fast and without slips. This formula sheet collects every quantitative formula the 15-mark section draws on, the same for every branch. Each group gives the formulas, one worked example and the trap that costs marks.

In this guide

Key takeaways
What the aptitude section tests, and where formulas fit
Percentages and successive change
Profit, loss and discount
Ratio, averages and mixtures
Simple and compound interest
Time, speed and distance
Time and work, and pipes
Number system: divisibility, HCF, LCM, remainders and unit digits
Counting and basic probability
Logarithms and exponents
Progressions and series
Mensuration of standard shapes
Clocks and calendars
Quick revision

Key takeaways

General Aptitude carries 15 marks in every GATE 2027 paper: 10 questions, five of 1 mark and five of 2 marks.
Successive percentage changes multiply; they never simply add.
Profit is a percentage of the cost price, while discount is a percentage of the marked price.
In time and work, and in pipes, you add rates, never times.
Average speed over equal distances is the harmonic mean, not the plain average.
"At least one" is always one minus "none", in counting and in probability alike.

What the aptitude section tests, and where formulas fit

The GATE 2027 brochure gives the GA section 10 questions and 15 marks in all test papers: five of 1 mark and five of 2 marks. Marking is common to every paper, so the GATE exam pattern and marking scheme guide applies to every branch.

The official GA syllabus has four sections: verbal, quantitative, analytical and spatial aptitude. Formulas live almost entirely in the quantitative section: ratios, percentages, powers, exponents and logarithms, permutations and combinations, series, mensuration and geometry, statistics and probability. For the other parts and the time plan, read the GATE General Aptitude preparation guide.

Remember: The brochure bans papers, data books and calculators in the hall. You get only the on-screen calculator and a scribble pad, so these formulas must live in your head.

Percentages and successive change

A percentage is a fraction with 100 as the denominator. A change of $r\%$ multiplies a value by the factor $\displaystyle 1 + \frac{r}{100}$ , and a fall uses a negative $r$ .

Situation	Formula
Two successive changes	$\displaystyle a + b + \frac{ab}{100}\ \%$
Rise needed to undo a fall of $r\%$	$\displaystyle \frac{r}{100 - r} \times 100\%$
A is $r\%$ more than B, so B is less than A by	$\displaystyle \frac{r}{100 + r} \times 100\%$
Rise of $x\%$ then fall of $x\%$	net loss of $\displaystyle \frac{x^{2}}{100}\%$

Worked example. A price rises by 25% and then falls by 20%. The net change is $\displaystyle 25 - 20 + \frac{25 \times (-20)}{100} = 0$ , because $1.25 \times 0.8 = 1$ . So a 25% rise undoes a 20% fall, the second row with $r = 20$ .

Trap. Adding the percentages. A 20% rise followed by a 20% fall is a 4% loss, not zero.

Profit, loss and discount

The cost price (CP) is what the seller paid. The marked price (MP) is the label, and the selling price (SP) is what the buyer pays.

Quantity	Formula
Profit or loss per cent	$\displaystyle \frac{\text{SP} - \text{CP}}{\text{CP}} \times 100$
Selling price from profit $p\%$	$\displaystyle \text{SP} = \text{CP}\left(1 + \frac{p}{100}\right)$
Selling price from discount $d\%$	$\displaystyle \text{SP} = \text{MP}\left(1 - \frac{d}{100}\right)$
Mark-up needed	$\displaystyle \frac{\text{MP}}{\text{CP}} = \frac{100 + p}{100 - d}$
Dishonest weight (gives $w$ less per unit)	$\displaystyle \text{gain}\% = \frac{w}{\text{true} - w} \times 100$

Worked example. A trader buys an item for ₹600, marks it 50% above cost and allows a 20% discount. Then:

\begin{aligned} \text{MP} &= 600 \times 1.5 = 900 \\ \text{SP} &= 900 \times 0.8 = 720 \\ \text{profit} &= \frac{720 - 600}{600} \times 100 = 20\% \end{aligned}

Trap. Taking the discount on the cost price. Discount is always a share of the marked price, profit a share of the cost price.

Ratio, averages and mixtures

A ratio compares two quantities of the same kind. An average is total over count; alligation is a shortcut for a mixture's average.

Rule	Formula
Combine $a:b$ and $b:c$	scale both to the same $b$ , then read $a:b:c$
Average	$\displaystyle \bar{x} = \frac{\text{sum}}{n}$
Weighted average	$\displaystyle \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}$
Alligation, mean $m$ between $c_1 < m < c_2$	$\displaystyle \frac{n_1}{n_2} = \frac{c_2 - m}{m - c_1}$
Remove $x$ of $V$ and refill with water, $n$ times	pure part left $\displaystyle = V\left(1 - \frac{x}{V}\right)^{n}$

Worked example. Rice at ₹40 a kg is mixed with rice at ₹60 a kg to make rice worth ₹52 a kg. Cheap to dear is $(60 - 52) : (52 - 40) = 2 : 3$ .

Trap. Averaging two averages. If 20 students average 60 and 30 students average 70, the class average is $\displaystyle \frac{1200 + 2100}{50} = 66$ , not 65.

Simple and compound interest

Simple interest is earned on the principal $P$ alone. Compound interest is also earned on interest already added.

Quantity	Formula
Simple interest	$\displaystyle \text{SI} = \frac{P \cdot r \cdot t}{100}$
Compound amount	$\displaystyle A = P\left(1 + \frac{r}{100}\right)^{n}$ , and $\text{CI} = A - P$
Half-yearly compounding	rate $\displaystyle \frac{r}{2}$ , periods $2n$
CI minus SI, 2 years	$\displaystyle P\left(\frac{r}{100}\right)^{2}$
CI minus SI, 3 years	$\displaystyle P\left(\frac{r}{100}\right)^{2}\left(3 + \frac{r}{100}\right)$

Worked example. On ₹10,000 at 10% for 2 years, $\text{CI} = 10000 \times 1.1^{2} - 10000 = 2100$ and $\text{SI} = 2000$ . The difference, 100, matches $10000 \times 0.1^{2}$ .

Trap. Using the 2-year difference for 3 years. Here the 3-year difference is 310, not 300.

Time, speed and distance

Speed is distance per unit time. Relative speed is the speed of one body as seen from the other.

Situation	Formula
Basic	$d = v \times t$ ; $\displaystyle 1\ \text{km/h} = \frac{5}{18}\ \text{m/s}$
Average speed, equal distances	$\displaystyle \frac{2xy}{x + y}$
Relative speed	same direction $u - v$ ; opposite $u + v$
Train crossing a pole, or a platform $P$	$\displaystyle t = \frac{L}{v}$ , or $\displaystyle t = \frac{L + P}{v}$
Two trains crossing	$\displaystyle t = \frac{L_1 + L_2}{u \pm v}$
Boat speed $b$ , stream $s$	downstream $b + s$ , upstream $b - s$ ; $\displaystyle b = \frac{D + U}{2}$ , $\displaystyle s = \frac{D - U}{2}$

Worked example. Trains of 150 m and 100 m run towards each other at 54 km/h and 36 km/h. The relative speed is $\displaystyle 90 \times \frac{5}{18} = 25$ m/s, so they cross in $\displaystyle \frac{250}{25} = 10$ s. Moving the same way, they take $\displaystyle \frac{250}{5} = 50$ s.

Trap. Averaging the speeds. A car that goes at 40 km/h and returns at 60 km/h averages $\displaystyle \frac{2 \times 40 \times 60}{100} = 48$ km/h, not 50.

Time and work, and pipes

A worker who finishes a job in $a$ days has a rate of $\displaystyle \frac{1}{a}$ of the job per day. Rates add when people or pipes work together.

Situation	Formula
A and B together	$\displaystyle \frac{1}{a} + \frac{1}{b}$ per day, time $\displaystyle \frac{ab}{a + b}$
Men, days, hours	$\displaystyle \frac{M_1 D_1 H_1}{W_1} = \frac{M_2 D_2 H_2}{W_2}$
Pipes	inlets $\displaystyle +\frac{1}{a}$ , outlets $\displaystyle -\frac{1}{c}$

Worked example. Pipe A fills a tank in 6 hours and pipe B in 12. Pipe C empties it in 8 hours. With all three open:

\begin{aligned} \text{net rate} &= \frac{1}{6} + \frac{1}{12} - \frac{1}{8} \\ &= \frac{4 + 2 - 3}{24} = \frac{1}{8} \text{ tank per hour} \end{aligned}

So the tank fills in 8 hours.

Trap. Adding the times. Two workers taking 6 and 12 days finish together in 4 days, not 18 or 9.

Number system: divisibility, HCF, LCM, remainders and unit digits

The HCF is the largest number dividing both numbers; the LCM is the smallest number both divide. The unit digit is the last digit.

Rule	Formula or test
Divisible by 3, 9	digit sum divisible by 3, 9
Divisible by 4, 8	last two digits by 4; last three by 8
Divisible by 11	alternate-digit sums differ by a multiple of 11
Two numbers	$\text{HCF} \times \text{LCM} = a \times b$
Fractions	$\displaystyle \text{HCF} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}$ , $\displaystyle \text{LCM} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}}$
Remainders	$(ab) \bmod n = \big[(a \bmod n)(b \bmod n)\big] \bmod n$
Fermat, $p$ prime not dividing $a$	$a^{p-1} \equiv 1 \pmod{p}$
Number of divisors of $p^{a} q^{b}$	$(a + 1)(b + 1)$
Unit-digit cycles	2: 2, 4, 8, 6; 3: 3, 9, 7, 1; 7: 7, 9, 3, 1; 8: 8, 4, 2, 6; 4: 4, 6; 9: 9, 1

Worked example. For the unit digit of $7^{2027}$ , divide 2027 by 4 to get remainder 3. The third term of 7, 9, 3, 1 is 3, so $7^{2027}$ ends in 3.

Trap. A remainder of zero. $7^{2028}$ takes the fourth term, so it ends in 1.

Counting and basic probability

A permutation counts arrangements, where order matters. A combination counts selections, where it does not. Probability is favourable outcomes over equally likely total outcomes.

Rule	Formula
Arrangements and selections	$\displaystyle {}^{n}P_{r} = \frac{n!}{(n - r)!}$ , $\displaystyle {}^{n}C_{r} = \frac{n!}{r!\,(n - r)!}$
Repeated items	$\displaystyle \frac{n!}{p!\,q!}$
Circular seating; necklace	$(n - 1)!$ ; $\displaystyle \frac{(n - 1)!}{2}$
Union	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
At least one	$1 - P(\text{none})$
Conditional; independent	$\displaystyle P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ ; $P(A \cap B) = P(A)\,P(B)$

Worked example. Pick a committee of 3 from 5 men and 4 women. The probability of at least one woman is:

1 - \frac{{}^{5}C_{3}}{{}^{9}C_{3}} = 1 - \frac{10}{84} = \frac{37}{42}

Trap. Counting "at least one" directly. One woman, then any two of the other eight, gives $4 \times 28 = 112$ , but the true count is $84 - 10 = 74$ . The direct count repeats committees with two or three women.

Logarithms and exponents

An exponent counts how often a base multiplies itself. A logarithm asks which power of the base gives a number.

Rule	Formula
Exponents	$a^{m} a^{n} = a^{m+n}$ , $(a^{m})^{n} = a^{mn}$ , $\displaystyle a^{-n} = \frac{1}{a^{n}}$ , $a^{1/n} = \sqrt[n]{a}$
Product, quotient, power	$\log ab = \log a + \log b$ , $\displaystyle \log \frac{a}{b} = \log a - \log b$ , $\log a^{n} = n \log a$
Change of base	$\displaystyle \log_b a = \frac{\log a}{\log b}$ ; $\log_b a \cdot \log_a b = 1$ ; $a^{\log_a x} = x$
Digits in $N$	$\lfloor \log_{10} N \rfloor + 1$

Worked example. Since $\log_{10} 2 \approx 0.30103$ , $\log_{10} 2^{50} \approx 15.05$ . So $2^{50}$ has 16 digits.

Trap. Inventing rules. $\log(a + b)$ does not split, and $\log ab$ is a sum, not $\log a \times \log b$ .

Progressions and series

An arithmetic progression (AP) adds a fixed difference $d$ each step. A geometric progression (GP) multiplies by a fixed ratio $r$ .

Series	Formula
AP term and sum	$a_n = a + (n - 1)d$ ; $\displaystyle S_n = \frac{n}{2}\big[2a + (n - 1)d\big] = \frac{n}{2}(a + l)$
Number of terms	$\displaystyle n = \frac{l - a}{d} + 1$
GP term and sum	$a r^{n-1}$ ; $\displaystyle S_n = \frac{a(r^{n} - 1)}{r - 1}$ ; $\displaystyle S_\infty = \frac{a}{1 - r}$ for $\vert r \vert < 1$
Natural numbers, squares, cubes	$\displaystyle \frac{n(n+1)}{2}$ , $\displaystyle \frac{n(n+1)(2n+1)}{6}$ , $\displaystyle \left[\frac{n(n+1)}{2}\right]^{2}$

Worked example. The two-digit multiples of 7 run from 14 to 98. There are $\displaystyle \frac{98 - 14}{7} + 1 = 13$ terms, and their sum is $\displaystyle \frac{13}{2}(14 + 98) = 728$ .

Trap. Forgetting the $+1$ in the term count. From 14 to 98 is 12 gaps but 13 terms.

Mensuration of standard shapes

Mensuration measures length, area and volume.

Shape	Area or surface	Volume
Triangle	$\displaystyle \frac{1}{2}bh$ ; Heron $\sqrt{s(s-a)(s-b)(s-c)}$ ; equilateral $\displaystyle \frac{\sqrt{3}}{4}a^{2}$	—
Trapezium; circle; sector	$\displaystyle \frac{1}{2}(a + b)h$ ; $\pi r^{2}$ ; $\displaystyle \frac{\theta}{360}\pi r^{2}$	—
Cube; cuboid	$6a^{2}$ ; $2(lb + bh + hl)$	$a^{3}$ ; $lbh$
Cylinder	curved $2\pi r h$	$\pi r^{2} h$
Cone, $l = \sqrt{r^{2} + h^{2}}$	curved $\pi r l$	$\displaystyle \frac{1}{3}\pi r^{2} h$
Sphere; hemisphere	$4\pi r^{2}$ ; total $3\pi r^{2}$	$\displaystyle \frac{4}{3}\pi r^{3}$ ; $\displaystyle \frac{2}{3}\pi r^{3}$

Worked example. A metal sphere of radius 6 cm is melted into cones of radius 3 cm and height 4 cm. Volume is conserved, so:

\text{cones} = \frac{\frac{4}{3}\pi \times 6^{3}}{\frac{1}{3}\pi \times 3^{2} \times 4} = \frac{288}{12} = 24

Trap: Scaling every length by $k$ scales area by $k^{2}$ and volume by $k^{3}$ . Doubling a sphere's radius gives four times the surface and eight times the volume.

Clocks and calendars

The hour hand moves 0.5° a minute and the minute hand 6°, so the minute hand gains 5.5° a minute. An odd day is a day left over after whole weeks.

Fact	Formula or value
Angle at $H$ hours $M$ minutes	$\theta = \vert 30H - 5.5M \vert$ ; use $360 - \theta$ if $\theta > 180$
Hands coincide	every $\displaystyle \frac{720}{11}$ minutes; 22 times a day
Odd days	ordinary year 1, leap year 2
Centuries	100 years 5; 200 years 3; 300 years 1; 400 years 0
Leap year	divisible by 4; a century year only if divisible by 400

Worked example. At 3:40, $\theta = \vert 30 \times 3 - 5.5 \times 40 \vert = \vert 90 - 220 \vert = 130^{\circ}$ .

Trap. Treating 1900 as a leap year. It is a century year not divisible by 400.

In one line: Write factors, rates and complements before any arithmetic, and most GA traps vanish.

Formulas earn marks only with practice. Each of the six GATE 2027 books has a General Aptitude chapter with practice questions in the current GATE style. The GATE CSE formula sheet and GATE ECE formula sheet cover subject papers, and the virtual calculator guide helps you avoid on-screen slips.

Quick revision

GA is 10 questions and 15 marks in every paper: five of 1 mark, five of 2 marks.
Successive changes: $\displaystyle a + b + \frac{ab}{100}$ ; a rise and fall of $x\%$ loses $\displaystyle \frac{x^{2}}{100}\%$ .
Profit is on cost price, discount on marked price.
Average speed over equal distances is $\displaystyle \frac{2xy}{x + y}$ ; convert km/h with $\displaystyle \frac{5}{18}$ .
Work and pipes: add rates, subtract outlets, then invert.
CI minus SI for 2 years is $\displaystyle P\left(\frac{r}{100}\right)^{2}$ .
Unit digits cycle in fours; a zero remainder takes the fourth term.
"At least one" is $1 - P(\text{none})$ ; clock angle is $\vert 30H - 5.5M \vert$ .

Frequently asked questions

How many marks does General Aptitude carry in GATE 2027?

General Aptitude carries 15 marks in every GATE 2027 paper. It has 10 questions, five worth 1 mark and five worth 2 marks, and the other 85 marks belong to your subject. Only MCQs carry negative marking, one third of the mark for a 1-mark question and two thirds for a 2-mark question. Confirm the current pattern at gate2027.iitm.ac.in.

Is the GATE General Aptitude syllabus the same for every branch?

Yes. There is one General Aptitude syllabus for GATE 2027 with four sections, namely verbal, quantitative, analytical and spatial aptitude, and the GA section carries 15 marks in all test papers. A CSE, ECE, EE, ME, Civil or DA candidate therefore revises the same aptitude formulas. The official GA syllabus PDF is on the GATE 2027 website.

Can I take a formula sheet into the GATE exam hall?

No. The GATE 2027 brochure prohibits papers, loose sheets, printed or handwritten material, data books, tables and calculators inside the hall. You get an on-screen virtual scientific calculator and a scribble pad from the invigilator. So the aptitude formulas must be in your memory, and practising them with the virtual calculator before the exam saves time on the day.

What is the formula for successive percentage change?

Two successive changes of a per cent and b per cent give a net change of a plus b plus ab over 100 per cent, with a fall written as a negative number. Equivalently, multiply the factors, such as 1.25 times 0.8. A rise of x per cent followed by a fall of x per cent is always a net loss of x squared over 100 per cent, never zero.

How do you find the angle between the hands of a clock?

At H hours and M minutes the angle between the hands is the absolute value of 30H minus 5.5M degrees. The hour hand moves 0.5 degree a minute and the minute hand 6 degrees a minute, so the minute hand gains 5.5 degrees each minute. If the result is above 180 degrees, subtract it from 360 to get the smaller angle.

How do you find the unit digit of a large power?

Unit digits repeat in cycles of at most four. For 2 the cycle is 2, 4, 8, 6; for 3 it is 3, 9, 7, 1; for 7 it is 7, 9, 3, 1; for 8 it is 8, 4, 2, 6. Divide the exponent by 4 and take that position in the cycle. A remainder of zero means the fourth term, not the first.

Sources

Dates, fees and the syllabus are set by the GATE 2027 organising institute and can change. Always confirm at gate2027.iitm.ac.in.