Important Formulas for CAT 2021 Preparation

CAT Quantitative Aptitude section is designed to test the candidate’s mathematical aptitude and numeracy skills. Hence, Recounting theorems and formulas by heart, and knowing how and when to apply them is an essential part of CAT preparation. These formulas will help you solve question papers from previous years and mock test series that are vital to enhancing performance and quality during the exam. 

Latest Update: CAT 2021 notification is expected to be out by July 2021 on the official website iimcat.ac.in. CAT 2021 Exam is expected to be conducted in the last week of November 2021. IIM Ahmedabad is the exam conducting body for CAT 2021. Check CAT 2021 Exam Schedule

For preparing well for CAT 2021, here's a list of important formulas you need to master. Several formulas are also given with ways in which they can be used in problems. Check CAT Quantitative Aptitude Preparation Tips

CAT Important Formulas for Mensuration

CAT Important Formulas for Algebra

Formulas for Logarithms, Surds, and Indices

One of the simplest topics in the Quantitative Aptitude section of CAT is Logarithms, Surds, and Indices. Although there are a huge number of formulae, the basic concepts are quite easy to understand and implement. There are no shortcuts to consider and there is a limited scope of the questions that can be asked. This section's accuracy in answering questions is very high and well-prepared students continue to score very well here.

Indices

Suppose X,Y > 0 and m,n are rational numbers, then, 

• Xm × Xn = Xm+n

• X0 = 1

• Xm / Xn = Xm-n

• (Xm)n = Xmn

• Xm ×Ym =(X×Y)m

• Xm / Ym =(X / Y)m

• X-m = 1/Xm

Suppose X and Y are positive real numbers and a,b are rational numbers, then, 

• Surds

Surds are irrational numbers that involve a root like . Like surds are two surds that have the same number under the radical sign. These can be added or subtracted.

In order to find must be written as , where .

• Logarithms

When , x is defined as the logarithm of N to the base a. This is shown as:

Note: The logarithm of zero or a negative number is not defined. 

When 0 < a < 1,

When a > 1,

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Formulas for Permutations and Combinations

The more questions you answer, the stronger you get with this topic. Take a look at the formula list and comprehend the formulae. However, the best way to tackle this topic is to solve questions. From this topic, answer as many questions as you can. You will start to see that all of them are basically variations of the same few themes that are mentioned in the formula list.

• Arrangement: n items may be arranged in n! ways. 

• Permutation: This is a way of choosing and arranging r items out of a set n objects a

Combination:

• This is a way of choosing and r items out of n, where the arrangement does not matter. It is represented as

• Choosing r items from n is similar to choosing (n-r) items out of n.

• The total number of choices that can be made from n distinct items is represented as

Partitioning :

• The number of ways to partition n identical items in r different slots is shown as .

• The number of ways to partition n identical items in r different slots such that every slot receive a minimum of 1 is shown as .

• The number of ways to partition n dissimilar items in r distinct slots is given as .

• The number of ways to partition n dissimilar items in r different slots such that the arrangement matter is given as .

Arranging with repetitions

When x items from n items get repeated, the number of ways of arranging n items is n! ways. Suppose a, b, and c items are x! n! that are related within n items, then they can be arranged in a! b! c! ways.

Rank of a word

• To obtain the rank of a word in the alphabetical list of the word’s permutations, begin with alphabetical arrangement of n letters. Suppose there are x letter greater than the first letter of the word, there will be a minimum of x* (n-1)! words above the word. 

• Eliminate the first affixed letter from the set. If there are y letters above the second letter, then will be y* (n-2)! words higher before the word and so on. 

• Therefore, the rank of the word is x* (n-1)! + y* (n-2)! … +!

Integral Solutions

• The number of positive integral solutions to , where .

• The number of non-negative integral solutions to , where .

Circular Arrangements 

• The number of ways to arrange n items around a circle is 1 for n = 1,2 and (n-1)! for

• Suppose it is a bracelet or necklace that could be flipped over, the possibilities would be (n-1)!/2. 

Derangements 

When n different items are arranged, the number of which they could be arranged such that they do not take up their intended place is .

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Formulas for Simple and Compound Interest 

Some of the simplest problems in the quantitative aptitude section of CAT exam are those of Simple Interest (S.I) and Compound Interest (C.I) The number of concepts in these topics is small, and most of the problems could be solved by direct application of the formulae. The principal and the interest (that occurs every period) remain constant in Simple Interest. In Compound Interest, after each compounding period, the interest received over the duration is rolled back to the current principal.

Thus, the principal and the interest change after each compounding period over a period of time. With a positive interest rate and time period (>1 year) for the same principal, the compound interest on the loan is always higher than the simple interest.

• Simple Interest 

Amount (A) = Principal (P) + Interest (I)

The Simple Interest (I) that has occurred over a period of time (T) for a rate of interest per annum R is, . 

• Compound Interest 

If money is borrowed at Compound Interest (I) for N number of years, the Amount to be paid is, . The Interest is A-P, i.e, .

When the interest is compounded half yearly, the Amount is, .

When the interest is compounded quarterly, the Amount is . 

• Simple Interest and Compound Interest 

• If the rate of interest is R1% for the first year, R2% for the second year and R3% for the third year, the Amount is . 

• If there is a difference between C.I and S.I at the same interest rate for a certain amount, then .

• If the interest is compounded yearly but the time is in fraction form, then, assume that . Then, the Amount is, . 

• Suppose R is the rate per annum, the current worth of Rs. K that is due in N years will be given as, 

Current worth = . 

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Formulas for Profit, Loss, and Discount

The amount paid to buy an item, or the cost of making an item is known as Cost Price (C.P). The price at which a commodity is sold is called the Selling Price (S.P). The price the article is marked with is called Marked Price (M.P).

• When S.P > C.P, Profit (P) = S.P - C.P. 

• When C.P > S.P, Loss (L) = C.P - S.P. 

• %Profit = Profit/C.P x 100 

• %Loss = Loss/C.P x 100 

• Discount = M.P - S.P

(Note: when do discount is provided, M.P = S.P)

• %Discount = Discount/M.P x 100 

• The total rise in price owing to two subsequent increased of X% and Y% is expressed as .

• Suppose two items are sold at the same price Rs. X, one with a profit of P% and the other at a loss of P%. There will then be an overall loss of . The absolute loss value will be . 

Formulas for Geometry 

Geometry is one amongst the most difficult sections without preparation and one of the simplest with preparation. This section will take a lot of time to master, with so many formulas to learn and remember. Learn and recall formulas and try visualizing and solving as many formula-related questions as you can.

• The distance between two points that have the coordinates is expressed as . 
• The slope, m = (when , the lines are perpendicular to each other).
• The midpoint between the two points and is .
• If two lines are parallel, their slopes will be equal, that is, .
• If two lines are perpendicular, the product of their slopes would be -1, that is, . 
• When two intersecting lines have the slopes and , the angle between the two lines would be .
• The length of the perpendicular from a point on the AX + BY + C = 0 line is .
• The distance between two parallel lines namely, and is given as .
• General equation of a line: Ax + By = C
• Slope intercept: y = mx + c, where c is the y-intercept
• Point slope:
• Intercept:
• Two point:
• General equation of a circle: ; circle centre is (-g, -f); radius of circle = . When the origin is the centre of the circle, its equation will be .

• Pythagoras theorem

In a right-angled triangle ABC, .

• Apollonius Theorem 

When AD is the media to side BC is a triangle ABC, . 

• Midpoint Theorem 

The line that connects the midpoint of any two sides of a triangle is seen to be parallel to the third side. It is also half the length of the third side. 

When X is the midpoint of CA and y the midpoint of CB, XY will be parallel to AB and XY = ½ * AB. 

• Basic Proportionality Theorem 

When a line is parallel to one side of a triangle and it intersects the other two sides at two points, it splits the two sides into the ratio of the respective sides.

Suppose in a triangle ABC, D and E are the points that lie on AB and BC. 

Then, AD/DB = EC/BE. 

• Angle Bisector Theorem 

When PS is the angle bisector for angle P, RS/QS = PR/RS.

• Cyclic Quadrilateral 

Area

• Equilateral Triangle 

When x is the side of an equilateral traingle, 

• Isosceles Triangle 

When a, b, and c are the lengths of sides BC, AC, and AB respectively, 

Area =

• Special Triangles 

• Direct Common Tangents 

• Transverse Common Tangent 

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Formulas for Mensuration

Mensuration is one of the most scoring topics in CAT QA section. Candidates should be aware of all the formulas for area and volume of various geometrical figures in order to solve the questions asked in the exam.

Areas of Geometrical Figures

• Triangle = ½ * base * height 

• Rectangle = length * width 

• Trapezoid = ½ * sum of bases * height 

• Parallelogram = base * height 

• Circle =

• Rhombus = ½ * product of diagonals

• Square =

• Kite = ½ * product of diagonals

Volumes of Solids 

• Cube =

• Cuboid = length * base * height 

• Prism = area of base * height 

• Cylinder =

• Pyramid = ⅓ * area of base * height 

• Cone =

• Cone Frustum =

• Sphere =

• Hemisphere =

• Total Surface Area of Solids 

• Prism = 2 * base area + base perimeter * height 

• Cube =

• Cuboid = 2(lh + bh + lb) 

• Cylinder =

• Pyramid = ½ * perimeter of base * slant height + area of base 

• Cone (when l is the slant height) =

• Cone Frustum =

• Sphere =

• Hemisphere =  

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Formulas for Ratio and Proportions

A ratio can be represented either as fraction a/b or using the a: b notation. In each of these representations, 'a' is the antecedent and ′b′ is called the consequent. The amounts of the products should be of the same character for a ratio to be described. A ratio does not have to be positive. If we deal with quantities of products, however, their ratios will be positive. 

• A ratio remains the same if the same non-zero number multiplies or splits both the previous and the consequent. 

• You can compare two ratios in their fraction notation, just as we compare real numbers.

• When a, b, and x are positive, 

• When two ratios a/b and c/d are equal, 

Invertendo:

Alternendo:

Componendo:

Dividendo:

Componendo-Dividendo:

For all real p, q, r, s such that pa+qb≠0 and rc+sd≠0,

Duplicate Ratios

The duplicate ratio of a:b is

The sub-duplicate ratio of a:b is

The triplicate ratio of a:b is

The sub-triplicate ratio of a:b is

Variations

Formulas for Number System

The most important topic in the quantitative section is Number Systems. It is a rather vast topic and from this segment, a large number of questions appear in CAT each year. Knowing basic tricks such as rules of divisibility, HCF and LCM, prime number and remaining theorems will help significantly boost the ranking.

• 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2
• (1² + 2² + 3² + ….. + n²) = n ( n + 1 ) (2n + 1) / 6
• (1³ + 2³ + 3³ + ….. + n³) = (n(n + 1)/ 2)²
• The sum of first n odd numbers = n²
• The sum of first n even numbers = n (n + 1)
• (a – b)² = (a² + b² – 2ab)
• (a + b)² = (a² + b² + 2ab)
• (a + b)(a – b) = (a² – b²)
• (a + b)² = (a² + b² + 2ab)
• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
• (a³ – b³) = (a – b)(a² + ab + b²)
• (a³ + b³) = (a + b)(a² – ab + b²)
• (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)
• When a + b + c = 0, then a³ + b³ + c³ = 3abc
• (a + b)n = an + (nC1)an-1b + (nC2)an-2b² + … + (nCn-1)abn-1 + bn
• LCM × HCF = Product of the Numbers
• LCM of Co-prime Numbers = Product Of The Numbers

Formulas for Quadratic Equations

The concept is very easy and the students should be interested in this subject. Some simple formulae and definitions are convenient. Techniques such as the elimination of options and the assumption of value can help to quickly resolve questions from this subject.

• Roots =

• Sum of Roots = -b/a

• Product of Roots = c/a

Formulas for Trigonometry

Trigonometric Identities

• Sine=Opposite/Hypotenuse

• Secant=Hypotenuse/Adjacent

• Cosine=Adjacent/Hypotenuse

• Tangent=Opposite/Adjacent

• Cosecant=Hypotenuse/Opposite

• Co−Tangent=Adjacent/Opposite

Reciprocal identities

• CosecΘ=1/sinΘ

• secΘ=1/cosΘ

• cotΘ=1/tanΘ

• sinΘ=1/CosecΘ

• cosΘ=1/secΘ

• tanΘ=1/cotΘ

Mixtures and Alligations 

• Alligation

This is the rule that helps one to find the ratio in which two or more ingredients must be combined at the specified price to produce a mixture of the desired amount. 

• Mean Price

The expense of a mixture unit quantity is called the mean price. 

• Alligation Rule

If there are two ingredients combined then, (Quantity of cheaper / Quantity of dearer) = (C.P. of dearer – Mean Price / Mean price – C.P. of cheaper)

Formulas for Set Theory and Functions 

The Law of Demorgan is the basic and most important formulation for sets, defined as, 

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

The relation, R⊂A×AR⊂A×A, is known as: 

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

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

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

When any relation R is symmetric, reflexive, and transitive in a specific set A, that relation is called an equivalence relation.

Formulas for Probability 

• Sample Space

If we do an experiment, then the sample space is called the set S of all potential outcomes. 

• Event

The event is called any subset of a sample space.

• Probability 

Let S be the sample and E be an event. P is the probability of occurrence of an event. 

Thus, P(E) =n(E) / n(S). 

Formulas for Percentages

In order to find out 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)

Formulas for Speed, Time, and Distance

• Distance = Speed x Time

• Time = Distance/Speed

• Speed= Distance/Time

• Average Speed = Total Distance Travelled/Total Time Taken

• Suppose A can do work in n days, then A’s 1 day’s work = 1/n

• When A’s 1 day’s work =1/n, A can complete the work in n days.

Formulas for Progressions

• Arithmetic Mean =

Sum =

Nth term =

• Geometric Mean =

Sum =

Nth term =

• Harmonic Mean =