Maths formulas are very important in all competitive exams. Formulas are very important in SSC CGL,CHSL exams,CAT,IBPS. Below we provide some important maths formula it will helpful to aspirants to crack aptitude section in various competitive exams.
Number Systems
- 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
- Sum of first n odd numbers = n²
- Sum of first n even numbers = n (n + 1)
Algebra
- (a + b)2 = a2 + 2ab + b2
- (a – b)2 = a2 – 2ab + b2
- a2 – b2 = (a – b)(a + b)
- a2 + b2 = (a + b)2 – 2ab
- (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
- (a +b+ c+…)2=a2+b2+c2+⋯+2(ab +ac+ bc +⋯)
- (x+ y+ z)2=x2+y2+z2+2xy+2yz+2xz
- (x +y−z)2=x2+y2+z2+2xy−2yz−2xz
- (x− y+ z)2=x2+y2+z2−2xy−2yz+2xz
- (x−y−z)2=x2+y2+z2−2xy+2yz−2xz
- (a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)
- (a – b)3 = a3 – 3a2b + 3ab2 – b3 = a3 – b3 – 3ab(a – b)
- 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-1 b + (nC2)an-2 b² + … + (nCn-1)abn-1 + bn
- x3+y3+z3−3xyz=(x+ y+ z)(x2+y2+z2−xy−yz−xz)
- x3+y3=(x+ y)(x2−xy+y2)
- x3−y3=(x−y)(x2+xy+y2)
- (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
- (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4
- a4 – b4 = (a – b)(a + b)(a2 + b2)
- a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)
- x2+y2+z2−xy−yz−zx=1/2[(x−y)2+(y−z)2+(z−x)2]
- x2+y2=1/2[(x+ y)2+(x−y)2]
- (x +a)(x +b)(x +c)=x3+(a +b+ c)x2+(ab +bc+ ca)x+ abc
Profit, Loss and Discount
- Profit (or) Gain = SP – CP
- Profit % = (Profit/CP) × 100
- SP = (100 + gain % )/100 × CP
- CP = 100/(100 + gain %) × SP
- Loss = CP – SP
- Loss % = Loss/(CP) × 100
- SP = (100 – loss %)/100 × CP
- CP = 100/(100 – loss %) × SP
LCM and HCF
LCM × HCF = Product of the numbers
LCM of co-prime numbers = Product of the numbers
Speed, Time and Distance
Distance = Speed × Time
Time = Distance/Speed
Speed= Distance/Time
Average Speed= Total Distance / Total Time
1 km/hr = 5/18 m/sec
1 m/sec = 18/5 km/hr
If the ratio of the speeds of A and B is a: b, then the ratio of the times taken by them to cover the same distance is 1/a: 1/b =b: a
If a man covers a specific distance at x km/hr and an equal distance at y km/hr. Then, the average speed for the total journey will be (2xy /x+y) km/hr.
Time and Distance Boat and Stream
When it comes to Time and distance &Boat and Streams questions there is a vast range of sub concepts involved in it. It is important for any candidates to understand the prominent formulas of Boat and Stream questions to score well in any competitive exams.
Formulas: For Upstream and Downstream
Upstream = (u−v) km/hr.
Downstream = (u+v)Km/hr.
Speed of Boat in Still Water = ½ (Downstream Speed + Upstream Speed)
Average Speed of Boat = {(Upstream Speed × Downstream Speed) / Boat’s Speed in Steal Water}
Speed of Stream = ½ (Downstream Speed – Upstream Speed)
Average Speed of Boat = {(Upstream Speed × Downstream Speed) / Boat’s Speed in Steal Water}.
Percentages
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.
Averages
Average = (Sum of observations/Number of observations)
Simple Interest and Compound Interest
Simple Interest is reliable on the principal amount considered for a deposit or a loan. On the other hand, compound interest is regarded as a principal amount and the interest functions over every period.
Formulas For Simple Interest
SI = P x R x T/100
Principal = Simple Interest ×100/ R × T
Rate of Interest = Simple Interest ×100 / P × T
Time = Simple Interest ×100 / P × R
If the rate of Simple interest differs from year to year, then
Simple Interest = Principal × (R1+R2+ R3…..)/100
( SI=Simple Interest P=Principal Amount (This the amount invested)T=Number of years R=Rate of interest (per year) in percentage).
Formulas For Compound Interest
The difference between the amount and the money borrowed is called the compound interest for a given period of time
Let principal =P; time =n years; and rate = r% per annum and let A be the total amount at the end of n years, then A = P*[1+ (r/100)]n;
CI = {P*[1+ (r/100)]n -1}
When compound interest reckoned half-yearly, then r% become r/2% and time n becomes 2n; A= P*[1+ (r/2*100)]2n
For the quarterly, A= P*[1+ (r/4*100)]4n
The difference between compound interest and simple interest over two years is given by,
Pr2/1002 or P(r/100)2
The difference between compound interest and simple interest over three years is given by,
P(r/100)2*{(r/100)+3}… Read more at: https://www.bankersadda.com/maths-formulas-for-bank-exams-2023/
Permutation and Combination
Permutation Formula: A permutation is the choice of r things from a set of n things without replacement. Order matters in permutation.
Combination Formula: A combination is the choice of r things from a set of n things without replacement. The order does not matter in combination.
Mixtures and Alligations
Alligation: It is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of the desired price.
Mean Price: The cost of a unit quantity of the mixture is called the mean price.
Rule of Alligation:
If two ingredients are mixed, then
(Quantity of cheaper / Quantity of dearer) = (C.P. of dearer – Mean Price / Mean price – C.P. of cheaper)
Coordinate Geometry
The Distance Between two Points A(x1, y1) and B(x2, y2):
AB² = (x2 – x1)² + (y2 – y1)²
The Midpoint of a Line Joining Two Points
The midpoint of the line joining the points (x1, y1) and (x2, y2) is:
[(x1 + x2)/1, (y1 + y2)/2]
The Equation of a Line Using One Point and the Gradient
The equation of a line which has gradient m and which passes through the point (x1, y1) is:
y – y1 = m(x – x1)
Ratio and Proportion:
When we get involved in comparing two quantities, either encounter the difference (a-b) or we go for division. When we go for dividing two numbers to acquire the magnitude of one in comparison to another we name it as a ratio. The ration of two digits will be represented as ‘a:b’. The formulas to remember for Ratio and Proportion are listed below.
Ratio Formula: a:b->a/b
Proportion Formula: a:b :: c:d -> a/b=c/d
Properties of Ratio
Ratio remains the same when you multiply or divide both the quantities with the same non- zero number.
a:b=pa:pb=qa:qb
a:b=a/p:b/p=a/q:a/q
Where p=q≠0
We can compare two ratios as real number in their fraction form
a:b=p:q ⇔ aq=bp
a:b>p:q ⇔ aq>bp
If two ratios are equal
a:b=p:q ⇔ b/a=q/p (invertendo)
a:b=p:q ⇔ a/p=b/q (Altertendo)
a:b=p:q ⇔ (a+p)/p=(b+q)/q (componendo)
a:b=p:q ⇔ (a-p)/p=(b-q)/q (dividendo)
Properties Of Proportion:
Product of extremes = product of means i.e., ad = bc
a, b, c, d,…. are in continued proportion means, a:b = b:c = c:d
a:b = b:c then b is called mean proportional and b2 = ac
The third proportional of two numbers, a and b, is c, such that, a:b = b:c
d is fourth proportional to numbers a, b, c if a:b = c:d.
The two terms ‘b’ and ‘c’ are known as ‘means or mean terms’, on the other hand the terms ‘a’ and ‘d’ are known as ‘extremes or extreme terms.’
Mensuration Formulas for 2-D Figures
The area of 2-D figures is always calculated in square units and the perimeter is always calculated in units. The below table will give you the complete list of areas and perimeters of different 2-D figures such as square, triangle (scalene, isosceles, equilateral, right), trapezium, parallelogram, rhombus, circle, etc.
| Mensuration Formulas for 2-Dimensional Figures | ||
| Shape | Area | Perimeter |
| Circle | πr² | 2 π r |
| Square | (side)² | 4 × side |
| Rectangle | length × breadth | 2 (length + breadth) |
| Scalene Triangle | √[s(s−a)(s−b)(s−c),
Where, s = (a+b+c)/2 |
a+b+c (sum of sides) |
| Isosceles Triangle | ½ × base × height | 2a + b (sum of sides) |
| Equilateral Triangle | (√3/4) × (side)² | 3 × side |
| Right Angled Triangle | ½ × base × hypotenuse | A + B + hypotenuse, where the hypotenuse is √A²+B² |
| Parallelogram | base × height | 2(l+b) |
| Rhombus | ½ × diagonal1 × diagonal2 | 4 × side |
| Trapezium | ½ h(sum of parallel sides) | a+b+c+d (sum of all sides) |
Mensuration Formulas for 3-D Figures
Under 3-D Figures, we can calculate the total surface area which is equal to curved surface area+ area of top and bottom. Curved surface area is also known as lateral surface area, and is measured in square units. Total surface area is also measured in square units whereas volume is measured in cubic units.
| Mensuration Formulas for 3-Dimensional Figures | |||
|---|---|---|---|
| Shape | Area | Curved Surface Area (CSA)/ Lateral Surface Area (LSA) |
Total Surface Area (TSA) |
| Cone | (1/3) π r² h | π r l | πr (r + l) |
| Cube | (side)³ | 4 (side)² | 6 (side)² |
| Cuboid | length × breadth × height | 2 height (length + breadth) | 2 (lb +bh +hl) |
| Cylinder | π r² h | 2π r h | 2πrh + 2πr² |
| Hemisphere | (2/3) π r³ | 2 π r² | 3 π r² |
| Sphere | 4/3πr³ | 4πr² | 4πr² |
Important Terms Related to Mensuration Formula
Before understanding the mensuration formulas, we need to understand certain terms. These are
● Perimeter: This is measured in units such as m, cm, etc and it is the measure of or sum of the continuous length of the boundary of a figure.
● Area: This is measured in square units such as m², cm², etc and it is the surface enclosed in a figure.
● Volume: This is measured in cubic units such as m³, cm³, etc, and is nothing but the space occupied by an object.
● Curved/Lateral Surface area: This is measured in square units such as m², cm², etc and it is the area of the curved surface in a figure.
● Total Surface area: This is measured in square units such as m², cm², etc and it is the area of the total surface in a figure including the top and bottom portions.
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