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Learn Quantitative Aptitude concepts, understand formulas, work through solved examples and practise questions — for TSPSC, APPSC, Telangana Police, AP Police, SSC, UPSC, banking, railway and other State and Central government exams.
Quantitative Aptitude tests a candidate's numerical ability — arithmetic, algebra, geometry and data interpretation — through problems that mirror everyday calculations: percentages, profit and loss, interest, time and work, and speed and distance. Nearly every competitive exam in India (TSPSC, APPSC, SSC, UPSC, banking and railway) includes a Quantitative Aptitude or Numerical Ability section.
The subject rewards a specific way of studying: learn the underlying concept first, memorise the core formula, solve it once the standard (longer) way to be sure you understand why it works, and only then learn the shortcut for it. A shortcut you don't understand will fail you on the one question it doesn't apply to.
Because most aptitude sections are timed at roughly a minute per question, both speed and accuracy matter — a fast wrong answer costs more than a slow right one wherever negative marking applies. Build speed by memorising squares, cubes, tables and fraction-percentage equivalents; build accuracy by solving the full method before relying on a shortcut. Use the Formula Sheet below for quick revision, work through the Solved Examples, then practise with the MCQ bank and mock test on this page.
| Divisible by | Rule |
|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) |
| 3 | Sum of digits is divisible by 3 |
| 4 | Last two digits form a number divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 8 | Last three digits form a number divisible by 8 |
| 9 | Sum of digits is divisible by 9 |
| 11 | Difference between the sum of digits at odd places and even places is 0 or divisible by 11 |
HCF × LCM = Product of two numbers (true only for exactly two numbers).
x% of N = N × x / 100
What percent is a of b = (a / b) × 100
Percentage change = (New − Old) / Old × 100
Successive changes of a% and b% combine to a + b + ab/100 (use a negative sign for a decrease)
Learning fraction equivalents (1/8 = 12.5%, 1/6 = 16⅔%, 3/8 = 37.5% etc.) turns many percentage calculations into quick mental arithmetic instead of long division.
Ratio a : b = a / b (compares two quantities of the same kind)
Proportion a : b :: c : d holds when a × d = b × c
Direct proportion: as one quantity increases, the other increases in the same ratio
Inverse proportion: as one quantity increases, the other decreases in the same ratio (their product stays constant)
Partnership problems are ratio problems in disguise: when partners invest for different periods, profit is shared in the ratio of (investment × time) for each partner.
Average = Sum of observations / Number of observations
Average speed for equal distances at speeds x and y = 2xy / (x + y)
New average after a change: add (total change) / (number of observations) to the old average
Profit = Selling Price (SP) − Cost Price (CP), when SP > CP
Loss = CP − SP, when CP > SP
Profit% / Loss% = (Profit or Loss / CP) × 100
Discount = Marked Price (MP) − SP; Discount% = (Discount / MP) × 100
Successive discounts of a% and b% combine using the same successive-change formula as percentages
Simple Interest (SI) = (Principal × Rate × Time) / 100
Amount = Principal + Interest
Compound Interest (CI) = P(1 + R/100)T − P
For half-yearly compounding, use Rate/2 and Time × 2 in the CI formula (similarly Rate/4 and Time × 4 for quarterly)
For 2 years, CI − SI = P × (R/100)2
If A completes a work in a days, A's one-day work = 1/a
Combined one-day work of A and B = 1/a + 1/b; days to finish together = ab / (a + b)
Pipes and cisterns: an inlet pipe adds a positive rate of work, an outlet (emptying) pipe subtracts a negative rate
Efficiency and days worked are inversely proportional — a person who is twice as efficient takes half the time to do the same work.
Speed = Distance / Time
Convert km/h to m/s: multiply by 5/18. Convert m/s to km/h: multiply by 18/5
Relative speed (same direction) = difference of speeds; (opposite directions) = sum of speeds
Boats and streams: Downstream speed = boat speed + stream speed; Upstream speed = boat speed − stream speed
Sum of angles of a triangle = 180°; of a quadrilateral = 360°
An angle in a semicircle is always 90°
Pythagoras theorem (right triangle): (hypotenuse)² = (base)² + (height)²
| 2D Shape | Area | Perimeter |
|---|---|---|
| Square | a² | 4a |
| Rectangle | l × b | 2(l + b) |
| Triangle | ½ × base × height | sum of sides |
| Circle | πr² | 2πr (circumference) |
| Parallelogram | base × height | 2(a + b) |
| Trapezium | ½ × (sum of parallel sides) × height | sum of sides |
| 3D Shape | Volume | Total Surface Area |
|---|---|---|
| Cube | a³ | 6a² |
| Cuboid | l × b × h | 2(lb + bh + hl) |
| Cylinder | πr²h | 2πr(r + h) |
| Cone | ⅓πr²h | πr(r + l), where l = √(r² + h²) |
| Sphere | ⅔πr³ | 4πr² |
| Hemisphere | ⅔⁄₂πr³ | 3πr² |
Data Interpretation questions reuse the same Percentage, Ratio and Average formulas above — applied to values read from a table or chart instead of stated directly in the question. Read the axis labels and units carefully before calculating; the most common DI mistake is misreading a scale (e.g. figures given "in thousands" or "in lakhs").
Additional topics that appear less frequently, with their core formula for quick reference. Detailed pages for these are being added.
Solve in BODMAS order: Brackets, Of, Division, Multiplication, Addition, Subtraction.
Cheaper qty / Dearer qty = (Dearer price − Mean price) / (Mean price − Cheaper price)
Profit is shared in the ratio of (investment × time period) of each partner.
(a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; a² − b² = (a + b)(a − b)
Probability of an event = (Favourable outcomes) / (Total possible outcomes)
Permutation nPr = n! / (n − r)!; Combination nCr = n! / (r! × (n − r)!)
Arithmetic Progression, nth term = a + (n − 1)d; Geometric Progression, nth term = a × r(n−1)
A single-page recap of the most-used formula from each topic above — use this for last-minute revision before an exam.
x% of N = N × x / 100
Profit% = (SP − CP) / CP × 100
a : b :: c : d ⇒ a × d = b × c
Average = Sum of terms / Number of terms
SI = PRT/100; CI = P(1 + R/100)T − P
Days together = ab / (a + b)
Speed = Distance / Time; km/h × 5/18 = m/s
Cheaper : Dearer = (Dearer − Mean) : (Mean − Cheaper)
a² − b² = (a + b)(a − b)
Sum of angles of a triangle = 180°
Area of a circle = πr²; Volume of a cylinder = πr²h
P(event) = Favourable outcomes / Total outcomes
A student secures 480 marks out of 750. What percentage of marks did the student get?
Given: Marks obtained = 480, Total marks = 750
Formula: Percentage = (Marks obtained / Total marks) × 100
Solution: (480 / 750) × 100 = 64%
Answer: 64%
Shortcut: 750 / 100 = 7.5, so 480 / 7.5 = 64 directly.
A shopkeeper buys an article for ₹800 and sells it for ₹920. Find the profit percentage.
Given: CP = ₹800, SP = ₹920
Formula: Profit% = (SP − CP) / CP × 100
Solution: Profit = 920 − 800 = 120. Profit% = (120 / 800) × 100 = 15%
Answer: 15% profit
A can complete a piece of work in 12 days and B can complete the same work in 15 days. In how many days will they complete it together?
Given: A's time = 12 days, B's time = 15 days
Formula: Days together = (a × b) / (a + b)
Solution: (12 × 15) / (12 + 15) = 180 / 27 = 20/3 = 6⅔ days
Answer: 6⅔ days
Check: one-day work together = 1/12 + 1/15 = 5/60 + 4/60 = 9/60 = 3/20, so time = 20/3 days — matches.
A train 150 m long crosses a pole in 15 seconds. Find the speed of the train in km/h.
Given: Distance = 150 m, Time = 15 s
Formula: Speed = Distance / Time, then convert m/s to km/h by multiplying by 18/5
Solution: Speed = 150 / 15 = 10 m/s. In km/h: 10 × 18/5 = 36 km/h
Answer: 36 km/h
Practise questions covering Number System, Percentage, Ratio, Average, Profit & Loss, Interest, Time & Work, Speed & Distance, Algebra, Geometry, Mensuration and Data Interpretation.
Practice Aptitude Questions →A 20-question multiple choice test on Quantitative Aptitude.
Start Aptitude Mock Test →Practise Quantitative Aptitude in the pattern commonly followed by these examinations. CompeteBits does not claim a question came from a specific official examination unless the source is verified. For verified, question-by-question previous year papers with worked solutions, see Previous Papers and the Study Materials arithmetic pages linked above.
Quick-access cards for last-minute revision.
Quantitative Aptitude is the numerical ability section of competitive exams, testing arithmetic, algebra, geometry and data interpretation through problems like percentages, profit and loss, interest, time and work, and speed and distance.
The most important topics are Number System, Percentage, Ratio and Proportion, Average, Profit and Loss, Simple and Compound Interest, Time and Work, Time-Speed-Distance, Geometry, Mensuration and Data Interpretation.
Memorise squares up to 30, cubes up to 15, multiplication tables up to 20, and fraction-percentage equivalents. Regular timed practice builds calculation speed over time.
Solve problems using the full method before relying on shortcuts, double-check units and what the question actually asks, and practise enough questions per topic to avoid careless mistakes.
Percentage, Profit and Loss, Simple and Compound Interest, Time and Work, and Speed-Distance formulas are used most often, both directly and inside Data Interpretation questions. See the Formula Sheet on this page.
SSC exams test Number System, Percentage, Ratio, Average, Profit and Loss, Interest, Time and Work, Speed and Distance, Algebra, Geometry, Mensuration and Data Interpretation at a moderate difficulty level. Practise topic-wise, then attempt full mock tests.
Banking exams emphasise speed alongside accuracy, with a strong focus on Data Interpretation, Simplification, Approximation, Percentage and Interest. Practise timed sets to build calculation speed.
TSPSC and APPSC exams cover core arithmetic topics such as Percentage, Ratio, Average, Profit and Loss, Time and Work, and Time-Speed-Distance, usually at a foundational to moderate difficulty level.
You can practice Quantitative Aptitude multiple-choice questions on CompeteBits, covering Number System, Percentage, Ratio, Average, Profit and Loss, Interest, Time and Work, Speed and Distance, Geometry and Mensuration.
You can attempt a timed Quantitative Aptitude mock test on CompeteBits with instant scoring and correct/wrong answers.