Percentages, profit and loss
| Percentage change | (new - old) / old x 100 |
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| Successive changes a% and b% | a + b + ab/100 percent |
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| Profit percentage | (SP - CP) / CP x 100 |
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| Selling price after profit p% | CP x (100 + p) / 100 |
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| Two items sold at the same price, one at x% gain and one at x% loss | Overall loss of x^2/100 percent |
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| Marked price and discount d% | SP = MP x (100 - d) / 100 |
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| Equivalent single discount for d1% and d2% | d1 + d2 - d1 d2 / 100 |
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| Common fractions | 12.5% = 1/8, 16.67% = 1/6, 33.33% = 1/3, 37.5% = 3/8, 66.67% = 2/3 |
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Practise percentages →
Interest and growth
| Simple interest | SI = P R T / 100 |
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| Compound interest amount | A = P (1 + R/100)^n |
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| CI minus SI for 2 years | P (R/100)^2 |
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| Population growth or depreciation | P (1 +/- r/100)^n |
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Ratio, mixtures and averages
| If a : b = c : d | ad = bc, and (a + b) : (a - b) = (c + d) : (c - d) |
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| Alligation rule | Cheaper quantity : dearer quantity = (dearer - mean) : (mean - cheaper) |
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| Average | Sum of values / number of values |
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| Average of n consecutive integers | (first + last) / 2 |
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| Weighted average | Sum of (weight x value) / sum of weights |
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Time, speed and distance
| Distance | Speed x time |
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| Convert km/h to m/s | Multiply by 5/18 |
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| Average speed for equal distances at a and b | 2ab / (a + b) |
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| Relative speed | Add when moving towards each other, subtract when moving in the same direction |
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| Train crossing a pole | Distance = length of train |
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| Train crossing a platform | Distance = length of train + length of platform |
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| Boat downstream and upstream | Boat + stream, boat - stream |
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| Boat speed in still water | (downstream + upstream) / 2 |
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Practise time, speed and distance →
Time and work, pipes
| Work rate | 1 / time taken |
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| A and B together | 1/a + 1/b per day, so time = ab / (a + b) |
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| Man days | Men x days x hours is constant for the same work |
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| Pipe that empties | Counts as a negative rate |
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Practise time and work →
Algebra
| (a + b)^2 | a^2 + 2ab + b^2 |
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| a^2 + b^2 | (a + b)^2 - 2ab |
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| a^2 - b^2 | (a - b)(a + b) |
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| a^3 + b^3 | (a + b)(a^2 - ab + b^2) |
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| Quadratic ax^2 + bx + c = 0 | x = (-b +/- sqrt(b^2 - 4ac)) / 2a |
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| Sum and product of roots | -b/a and c/a |
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| Nature of roots | Discriminant > 0 two real roots, = 0 equal roots, < 0 no real roots |
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| Arithmetic progression | nth term = a + (n - 1)d, sum = n/2 [2a + (n - 1)d] |
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| Geometric progression | nth term = a r^(n-1), sum = a (r^n - 1) / (r - 1) |
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| Logarithms | log(ab) = log a + log b, log(a/b) = log a - log b, log(a^n) = n log a |
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| Modulus | |x - a| = k gives x = a + k or x = a - k |
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Practise algebra →
Number system
| Number of factors of p^a q^b | (a + 1)(b + 1) |
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| Sum of factors of p^a q^b | (p^(a+1) - 1)/(p - 1) x (q^(b+1) - 1)/(q - 1) |
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| HCF x LCM | Product of the two numbers |
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| Trailing zeros in n! | floor(n/5) + floor(n/25) + floor(n/125) + ... |
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| Sum of first n natural numbers | n(n + 1)/2 |
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| Sum of squares of first n numbers | n(n + 1)(2n + 1)/6 |
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| Sum of first n odd numbers | n^2 |
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| Divisibility by 3 and 9 | Sum of digits divisible by 3 or 9 |
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| Divisibility by 11 | Difference of sums of alternate digits is 0 or a multiple of 11 |
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| Unit digit cycles | Powers of 2: 2, 4, 8, 6. Powers of 3: 3, 9, 7, 1. Powers of 7: 7, 9, 3, 1. Powers of 4 and 9 have cycle 2. |
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Practise number system →
Geometry and mensuration
| Triangle area | 1/2 x base x height, or sqrt(s(s - a)(s - b)(s - c)) |
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| Equilateral triangle area | (sqrt 3 / 4) a^2 |
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| Circle | Area = pi r^2, circumference = 2 pi r |
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| Sector | Area = (angle / 360) x pi r^2, arc = (angle / 360) x 2 pi r |
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| Right triangle circumradius | Half of the hypotenuse |
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| Inradius of a triangle | Area / s |
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| Cylinder | Curved surface = 2 pi r h, volume = pi r^2 h |
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| Cone | Volume = 1/3 pi r^2 h, slant height = sqrt(r^2 + h^2) |
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| Sphere | Surface = 4 pi r^2, volume = 4/3 pi r^3 |
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| Polygon interior angle sum | (n - 2) x 180 degrees |
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| Diagonals of an n sided polygon | n(n - 3)/2 |
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| Distance between two points | sqrt((x2 - x1)^2 + (y2 - y1)^2) |
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Permutations, combinations and probability
| Permutations | nPr = n! / (n - r)! |
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| Combinations | nCr = n! / (r! (n - r)!) |
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| Arrangements with repeated items | n! / (p! q! ...) |
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| Circular arrangements of n items | (n - 1)! |
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| Probability | Favourable outcomes / total outcomes |
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| At least one | 1 - probability of none |
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| Independent events | P(A and B) = P(A) x P(B) |
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Practise probability →
VARC checklist
- Read for the main claim and the turning point (but, however, yet) before the questions.
- Answer from the passage, not from general knowledge.
- Reject options with extreme words unless the passage uses them.
- In para jumbles, find the opening sentence and the pairs that cannot be separated.
- Spend no more than 7 minutes on a dense passage.
Practise reading comprehension →
DILR checklist
- Scan every set for 2 to 3 minutes and start with the clearest one.
- Draw the skeleton (grid, positions, table) before reading conditions twice.
- Estimate before you calculate when options are far apart.
- Branch only where the conditions leave a real choice.
- Abandon a set that is not opening up after about 8 minutes.
Practise DI tables →
Exam day checklist
- Marking: +3 for a correct answer, -1 for a wrong multiple choice answer, no negative marking for type in the answer questions.
- Three sections of 40 minutes each, with no returning to a finished section.
- Attempt the easy questions first inside each section.
- Keep rough work organised by question number.
- Check the official CAT website for the current pattern before the exam.
Edited by BITS Pilani and IIM alumni. Found a mistake? Tell us through the FAQ page and we will correct it.
