QA has been a daunting section for many CAT aspirants over the years, especially non-engineers. With so many topics and sub-topics, it becomes difficult to remember all the formulas. Having all such formulas in one place definitely helps you to go through them before your exam.

So, here's the CAT Quant formula Cheat Sheet, which covers all the formulae of all the topics in the QA syllabus.

All the best!

**NUMBER SYSTEM**

1. 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2

2. (1² + 2² + 3² + ….. + n²) = n ( n + 1 ) (2n + 1) / 6

3. (1³ + 2³ + 3³ + ….. + n³) = (n(n + 1)/ 2)²

4. (a + b)n = an + (nC1)an-1b + (nC2)an-2b² + … + (nCn-1)abn-1 + bn

5. Sum of first n odd numbers = n²

6. Sum of first n even numbers = n (n + 1)

7. a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – bc – ca)

8. The product of n consecutive integers is always divisible by n!

9. The sum of any number of even numbers is always even

10. The sum of even number of odd numbers is always even

11. The sum of odd number of odd numbers is always odd

12. If N is a composite number such that N = ap . bq . cr .... where a, b, c are prime factors of N and p, q, r .... are positive integers, then

a. the number of factors of N is given by the expression (p + 1) (q + 1) (r + 1) ...

b. it can be expressed as the product of two factors in 1/2 {(p + 1) (q + 1) (r + 1).....} ways

c. if N is a perfect square, it can be expressed

(i) as a product of two DIFFERENT factors in 1/2 {(p + 1) (q + 1) (r + 1) ... - 1 } ways

(ii) as a product of two factors in 1/2 {(p + 1) (q + 1) (r + 1) ... +1} ways

d. sum of all factors of N =

e. the number of co-primes of N =

f. sum of the numbers in (e) =

g. It can be expressed as a product of two factors in 2^{n}^{–1}, where ‘*n*’ is the number of different prime factors of the given number N

**ARITHMETIC**

*Ratio, Proportion and Variation*

1. If a : b : : c : d, then ad = bc

2. If a : b : : c : d, then a + b : b : : c + d : d

3. If a : b : : c : d, then a - b : b : : c - d : d

4. If a : b : : c : d, then a + b : a - b : : c + d : c - d

5. If then k=

*Simple Interest and Compound Interest*

I = Interest, P is Principle, A = Amount, n = number of years, r is rate of interest

**1. Interest under**

a) Simple interest, I = Pnr/100

b) Compound interest, I =

**2. Amount under**

a) Simple interest, A =

b) Compound interest, A = P

**3. Effective rate of interest when compounding is done k times a year**

*Mixture and Alligation*

**1.** If p1, p2 and p are the respective concentrations of the first mixture, second mixture and the final mixture respectively, and q1 and q2 are the quantities of the first and the second mixtures respectively, then Weighted Average (p)

p =

**2.** If C is the concentration after *n* dilutions, V is the original volume and x is the volume of liquid replaced each time then

C =

*Profit, Loss and Discount*

**1.** Profit/Gain = (S.P.) – (C.P.)

**2.** Profit % = Profit/(C P)×100

**3.** S P = (100+gain % )/100 ×C P

**4.** C P = 100/(100+gain %)×S P

**5.** Loss = (C.P.) – (S.P.)

**6.** Loss % = Loss/(C.P.)×100

**7.** S P = (100-loss %)/100×C P

**8.** C P = 100/(100-loss %)×S P

*Time, Speed and Distance*

**Distance** = Speed x Time

**Time** = Distance/Speed

**Relative Speed =**

*Percentage*

To find what percentage of x is y: y/x × 100

Increase N by S % = N( 1+ S/100 )

Decrease N by S % = N (1 – S/100)

*Time and Work*

If A can do a piece of work in n days, then A’s 1 day’s work = 1/n

If A’s 1 day’s work =1/n, then A can finish the work in n days.

*Average*

*Race*

**Linear Race:**

**Circular Race:**

**ALGEBRA**

*Quadratic Equations*

1. If a, b and c are all rational and x + is an irrational root of ax2 + bx + c = 0, then x - is the other root

2. If α and β are the roots of ax² + bx + c = 0, then α + β = and αβ =

3. When a > 0, ax² + bx + c has a minimum value equal to at x =

4. When a < 0, ax² + bx + c has a maximum value equal to at x =

*Logarithm*

**GEOMETRY**

**1.** In a triangle ABC, if AD is the angular bisector, then

**2.** In a triangle ABC, if E and F are the points of AB and AC respectively and EF is parallel to BC, then

**3.** In a triangle ABC, if AD is the median, then AB2 + AC2 = 2(AD2 + BD2)

**4.** In parallelogram, rectangle, rhombus and square, the diagonals bisect each other

**5.** Sum of all the angles in a polygon is (2n – 4)90

**6.** Exterior angle of a polygon is

**7.** Interior angle of a polygon is

**8.** Number of diagonals of a polygon is

**9.** The angle subtended by an arc at the centre is double the angle subtended by the arc in the remaining part of the circle

**10.** Angles in the same segment are equal

**11.** The angle subtended by the diameter of the circle is 90°

**MENSURATION**

**1. Plane Figures**

**2. Solids**

### C0-ordinate Geometry, Functions and Graphs, Trigonometry

**1.** If a point P(x, y) divides the line segment joining A(x1, y1) and B(x2, y2) in the ratio m : n, then x = and y = , positive sign for internal division and negative sign for external division

**2.** The area of a triangle with the vertices at (0, 0), (x1, y1) and (x2, y2) is Δ =

**3.** The coordinates of the centroid C(x, y) of a triangle ABC formed by joining the points

A(x1, y1); B(x2, y2) and C(x3, y3) are given by

**4.** The slope of line with points (x1, y1) and (x2, y2) lying on it is m =

**5.** If m1 and m2 are the slopes of two lines L1 and L2 respectively, then the angle ‘θ’ between them is given by tanθ =

**6.** The equation of the x-axis is y = 0 and that of y-axis is x = 0

**7.** The equation of a line parallel to x-axis is of the form y = b and that of a line parallel to y-axis is of the form x = a (a and b are some constants)

**8.** Point slope form of a line: y – y1 = m (x – x1)

**9.** Two point form of a line:

**10.** Slope intercept form of a line: y = mx + b

**11.** Intercept form of a line :

**12.** Two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are

(i) parallel if or m1= m2

(ii) perpendicular if a1 a2 + b1 b2 = 0 or m1 m2 = -1

**13.** The distance between two parallel lines of the form ax + by +c1 = 0 and ax + by + c2 = 0 is given by

**14.** If ax + by + c = 0 is the equation of a line, then the perpendicular distance of a point (x1, y1) from the line is given by

**15.** sine rule : = 2R, where R is the circumradius of triangle ABC

**16.** cosine rule : cosA = , similarly cosB and cosC can be defined

**PROGRESSIONS**

*Arithmetic Progression (A.P)*

a is the first term, d is the last term and n is the number of terms

1. Tn = a + (n – 1)d

2. Sn = =

3. Tn = Sn – S(n-1)

4. Sn = A.M * n

*Geometric Progression (G.P)*

a is the first term, r is the common ratio and n is the number of terms

5.

6.

*Harmonic Progression (H.P)*

7. H.M of a and b =

8. A.M > G.M > H.M

9. (G.M)² = (A.M) (H.M)

10. Sum of first n natural numbers Σn =

11. Sum of squares of first n natural numbers ∑n²=

12. Sum of cubes of first n natural numbers ∑n³ = = (∑n)²

**PERMUTATIONS & COMBINATIONS, PROBABILITY**

**1.** n (A∪B) = n (A) + n (B) – n (A∩B)

**2.** If A and B are two tasks that must be performed such that A can be performed in 'p' ways and for each possible way of performing A, say there are 'q' ways of performing B, then the two tasks A and B can be performed in p × q ways

**3.** The number of ways of dividing (p + q) items into two groups containing p and q items respectively is

**4.** The number of ways of dividing 2p items into two equal groups of p each is , when the two groups have distinct identity and , when the two groups do not have distinct identity

**5. **

**6.** The total number of ways in which a selection can be made by taking some or all out of (p + q + r + .....) items where p are alike of one kind, q alike of a second kind, r alike of a third kind and so on is {(p + 1) (q + 1) (r + 1) ....} - 1

**7.** P(Event) = and 0 ≤ P(Event) ≤ 1

**8.** P(A ∩ B) = P(A) × P(B), if A and B are independent events

**9.** P(A ∪ B) = 1, if A and B are exhaustive events

**SET THEORY**

The Demorgan’s Law is the basic and most important formula for sets, which is defined as

(A ∩ B) ‘ = A’ U B’ and (A U B)’ = A’ ∩ B’

The relation R⊂A×AR⊂A×A is said to be called as:

• Reflexive Relation: If a R a ∀∀ a ∈∈ A.

• Symmetric Relation: If aRb, then bRa ∀∀ a, b ∈∈ A.

• Transitive Relation: If aRb, bRc, then aRc ∀∀ a, b, c ∈∈ A.

If any relation R is reflexive, symmetric and transitive in a given set A, then that relation is known as equivalence relation.

*Source:*

- Handa Ka Funda
- HitBullsEye
- TIME
- Cracku