Bank exams Maths


 1.Number system 
1. When sum and difference of two numbers (X and Y) are given, then
X = (sum + difference)/2
Y = (sum + difference)/2 

2. Difference between  two digits of  two digit number is  = (Difference in original and
interchanged number)/9 

3. Sum of first n odd numbers is n2 

4. Sum of first n even numbers n(n+1) 

5. Sum of squares of first n natural numbers is n(n+1)(2n+1)/6

6. Sum of cubes of first n natural numbers is [n(n+1)/2]2  

Algebra  :
1. (a+b)2 = a2 + 2ab + b2 
2. (a-b)2  = a2 - 2ab + b2 
3. (a+b)2 = (a-b)2 + 4ab 
4. (a-b)2  = (a+b)2 - 4ab 
5. (a+b)3 = a3+b3+3ab(a+b) = a3+b3+3a2b+3ab2 
6. (a-b)3 = a3- b3-3ab(a-b) = a3- b3 - 3a2b + 3ab2 
7. a3+b3  =(a+b)3 - 3ab(a+b) 
8. a3- b3   =(a-b)3 + 3ab(a-b)
9. a2 -b2   = (a-b)(a+b) 
10. a3 +b3  = (a+b)(a2-ab+b2) 
11. a3 - b3  = (a-b)(a2+ab+b2) 
12. am x an   = am+n 
13. am / an   = am – n 
14. (a/b)(m / n) =(b/a)-(m / n) 
15. am / b – n = am x bn 
Ratio and Proportion 
1. If four quantities are in proportion, then Product of Means = Product of Extremes. 
In the proportion a:b::c:d, we have bc = ad 
2. If a:b::c:x, x is called the fourth proportional of a, b, c.                             
a/b = c/x or, x = bc/a.  
3. If two numbers are in a:b ratio and the sum of these numbers is x, then numbers will be
ax/(a+b) and bx/(a+b) respectively   Formulae 
4. If three numbers are in the ratio a:b:c and the sum of these numbers is x, then these numbers
will be ax/(a+b+c), bx/(a+b+c) and cx/(a+b+c) respectively 
5. The ratio of two numbers is a : b. If n is added to each of these numbers, the ratio becomes c
: d. The two numbers will be given as an(c-d)/(ad-bc) and bn(c-d)/(ad-bc) respectively 
6. The ratio of two numbers is a : b. If n is subtracted from each of these numbers, the ratio
becomes c : d. The two numbers are given as an(d-c)/(ad-bc) and bn(d-c)/(ad- bc)
respectively 
7. If the ratio of two numbers is a: b, then the numbers that should be added to each of the
numbers in order to make this ratio c:d is given by (ad-bc)/(c-d) 
8. If the ratio of two numbers is a:b, then the number that should be subtracted from each of
the numbers in order to make this ratio c:d is given by  (bc-ad)/(c-d) 
9. The CP of the item that is cheaper is CPcheaper and the CP of the item that is costlier (dearer) is
CPDearer. The CP of unit quantity of the final mixture is called the Mean Price and is given by
πΆπ‘ƒπ‘šπ‘’π‘Žπ‘› π‘π‘Ÿπ‘–π‘π‘’    =  πΆπ‘ƒπ‘β„Žπ‘’π‘Žπ‘π‘’π‘Ÿ − πΆπ‘ƒπ‘šπ‘’π‘Žπ‘› π‘π‘Ÿπ‘–π‘π‘’ πΆπ‘ƒπ‘šπ‘’π‘Žπ‘› π‘π‘Ÿπ‘–π‘π‘’ − πΆπ‘ƒπ‘β„Žπ‘’π‘Žπ‘π‘’π‘Ÿ  
Percentage 
1. a % of b = a x b/100  
2. If A is x% more than B, then B is less than A by
%100 100         x x 
Formulae
3. If A is x% less than B, then B is more than A by  
%100 100        x x 
4. If A is x% of C and B is y% of C, then A = x/y  B 
5. If two numbers are respectively x% and y% more than a third number, then first number is
%100
100 100
y x of the second number and the second number  is % 100 100 100          x y of the first
number        
  6. If two numbers are respectively x% and y% less than a third number, then the first number
is % 100 100 100            y x of the second number and the second number is % 100 100 100          x y of the first
number 
7. If the price of a commodity decreases by P %, then the increase in consumption so that the
expenditure remains same is % 100 100        P P 
8. If the price of a commodity increases by P%, then the reduction in consumption so that the
expenditure remains same is % 100 100        P P 
9. If a number is changed (increased/decreased) successively by x% and y%, then net% change
is given by [x+y+(xy/100)]%, which represents increase or decrease in value according as the
sign is positive or negative 
10. If two parameters A and B are multiplied to get a product and if A is changed  by x% and another parameter B is changed by y%, then the net% change in the product (A  B) is given
[x+y+(xy/100)]%
Formulae 
11. In an examination, the minimum pass percentage is x%. If a student secures y marks and
fails by z marks, then the maximum marks in the examination is 100(y+z)/x 
12. If the present population of a town (or value of an item) be P and the population (or value of
item) changes at r% per annum, then population (or value of item) after n years = nr P        100 1 and the Population (or value of item) n years ago = nr P        100 1 
13. If a number A is increased successively by x% followed by y% and then by z%, then the final
value of A will be    
   
100
1
100
1
100
1
zyx A 
Averages  
1. Average = Sum of quantities/ Number of quantities 
2. Sum of quantities = Average  Number of quantities 
3. The average of first n natural numbers is (n +1)/2 
4. The average of the squares of first n natural numbers is  (n +1)(2n+1)/6 
5. The average of cubes of first n  natural numbers is n(n +1)2/4 
6. The average of first n odd numbers is given by (last odd number +1)/2 
7. The average of first n even numbers is given by (last even number + 2)/2 
8. The average of first n consecutive odd numbers is n
Formulae   
9. The average of squares of first n consecutive even numbers is 2(n+1)(2n+1)/3 
10. The average of squares of consecutive even numbers till n is (n+1)(n+2)/3 
11. The average of squares of squares of consecutive odd numbers till n is n(n+2)/ 3. 
12. If the average of n consecutive numbers is m, then the difference between the smallest and
the largest number is 2(m-1) 
13. If the number of quantities in two groups be n1 and n2 and their average is x and y
respectively, the combined average is  (n1x +n2y)/( n1 + n2) 
14. The average of n quantities is equal to x. When a quantity is removed, the average becomes
y. The value of the removed quantity is n(x-y) + y 
15. The average of n quantities is equal to x. When a quantity is added, the average becomes y.
The value of the new quantity is n(y-x) + y  
Profit and Loss 
1. Gain = SP – CP 
2. Loss = CP - SP 
3. Gain on Rs. 100 is Gain per cent 
4. Gain% = (Gain  100)/CP 
5. Loss on Rs. 100 is Loss per cent 
6. Loss% = (Loss  100)/CP
Formulae 
7. When the Cost Price and Gain per cent are given:
SP = [(100+Gain %)/100] x CP 
8. When the Cost Price and Loss per cent are given:
SP = [(100-Loss %)/100] x CP 
9. When the Selling Price and Gain per cent are given:
CP = [100/(100+Gain %)] x SP 
10. When the Selling Price and Loss per cent are given:
   CP = [100/(100-Loss %)] x SP 
11. When p articles are sold at the cost of q similar articles, the                                 
Profit/Loss % = [(q-p)/p]x100 
12. If two articles are sold at the same price with a profit of x % on one and a loss of x % on the
other, the net loss % = (x2/100)% 
13. If two articles bought at the same price are sold with a profit of x % on one and a loss of x %
on the other, then overall there will be No Profit No Loss   
Simple and Compound Interest 
1. Simple Interest, SI = PTR/100 
2. Principal, P = 100  SI/RT 
3. Rate, R = 100  SI/PT
Formulae 
4. Time, T = 100  SI/RP 
5. Amount, A = P + SI = P + (PTR)/100 
6. If a certain sum of money becomes n times itself at R% p.a. simple interest in T years, then  T = [(n-1)/R]  100 years 
7. If a certain sum of money becomes n times itself in T years at a simple interest, then the time T’ in which it will become m times itself is given by T = (m-1/n-1)  T years 
8. If a certain sum of money P lent out at SI amounts to A1 in T1 years and to A2 in T2 years,
then  
P = (A1T2-A2T1)/(T2-T1),    R = (A1 – A2)/(A1T2 - A2T1)  100% 
9. If a certain sum of money P lent out for a certain time T amounts to A1 at R1% per annum
and to A2 at R2% per annum, then 
P = (A2R1-A1R2)/(R1-R2)  T = (A1-A2)/(A2R1-A1R2)  100 years 
10. Compound Interest,  CI = 𝑃 1 + 𝑅 100
  𝑛 − 𝑃    = 𝑃  1 + 𝑅 100
𝑛
− 1   
11. Amount, A = 𝑃 1 + 𝑅 100
𝑛
 ,  if interest is payable annually 
12. Amount, A = 𝑃 1 + 𝑅′ 100
𝑛′
, R’= R/2, n’ = 2n; if interest is payable half-yearly 
13. Amount, A =𝑃 1 + 𝑅′′ 100
𝑛′′
, R’’= R/4, n’’ = 4n; if interest is payable quarterly 
14. When time is fraction of a year, say 43 4
 years, then Amount, 
      A = 𝑃 1 +
𝑅 100
4
× 1 +
3 4
 π‘…
100
  
15. When Rates are different for different years, say, R1, R2, R3 for 1st , 2nd  & 3rd  years
respectively, then, Amount = 𝑃 1 +
𝑅1 100
  1 +
𝑅2 100
  1 +
𝑅3 100
  
16. In general, interest is considered to be Simple unless otherwise stated.  
Time and Work 
1. If 1/n of a work is done by A in one day, then A will take n days to complete the full work. 
2. If  A can do a piece do a piece of work in X days and B can do the same work in Y days, then
both of them working together will do the same work in XY/(X+Y) days 
3. If A, B and C, while working alone, can complete a work in X, Y and Z days respectively,
then they will together complete the work in XYZ/(XY+YZ+ZX) days 
4. If A does 1/nth of a work in m hours, then to complete the full work A will take nxm hours. 
5. If A and B can together finish a piece of work in X days, B and C in Y days and C and A in Z
days, then
a) A, B and C working together will finish the job in (2XYZ/XY+YZ+ZX) days. b) A alone will finish the job in (2XYZ/XY+YZ- ZX) days.
c) B alone will finish the job in (2XYZ/ZX+XY- YZ) days.  d) C alone will finish the job in (2XYZ/ZX+YZ- XY) days. 
6. If A can finish a work in X days and B is k times efficient than A, then the time taken by both
A and B working together to complete the work is X/(1+k). 
7. If A and B working together can finish a work in X days and B is k times efficient than A,
then the time taken by A working alone to complete the work is (k+1)X  and B working
alone to complete the work is (k+1/k)X.  
Time and Distance 
1. 1 Kmph = (5/18) m/s 
2. 1 m/s = (18/5) Kmph 
3. Speed(S) = Distance(d)/Time(t) 
4. Average Speed = Total distance/Total Time = (d1+d2)/(t1+t2) 
5. When d1 = d2 , Average speed = 2S1S2/(S1+S2),  where S1 and S2 are the speeds for covering d1
and d2 respectively 
6. When t1 = t2 , Average speed = (S1+S2)/2,  where S1 and S2 are the speeds during t1 and t2
respectively 
7. Relative speed when moving in opposite direction is S1 +S2 
8. Relative speed when moving in same direction is S1 - S2 
9. A person goes certain distance (A to B) at a speed of S1 kmph and returns back (B to A) at a
speed of S2 kmph. If he takes T hours in all, the distance between A and  B is T(S1S2/S1+S2)

10. When two trains of lengths l1 and l2 respectively travelling at the speeds of s1 and s2
respectively cross each other in time t, then the equation is given as s1+s2 = (l1+l2)/t 
11. When a train of lengths l1 travelling at a speed s1 overtakes another train of length l2
travelling at speed s2 in time t, then the equation is given as s1 - s2 = (l1+l2)/t  
12. When a train of lengths l1 travelling at a speed s1  crosses a platform/bridge/tunnel of length
l2 in time t, then the equation is given as s1 = (l1+l2)/t 
13. When a train of lengths l travelling at a speed s  crosses a pole/pillar/flag post in time t, then
the equation is given as s = l/t 
14. If two persons A and B start at the same time from two points P and Q towards each other
and after crossing they take T1 and T2 hours in reaching Q and P respectively, then (A’s speed)/(B’s speed) =T2/ T1  
Mensuration
Circle: 
1. Diameter, D = 2r  
2. Area = r2 sq. units 
3. Circumference = 2r units 
Square: 
4. Area = a2 sq. units  
5. Perimeter = 4a units 
6. Diagonal, d =  2 a units 
Rectangle: 
7. Area = l x b sq. units 
8. Perimeter = 2(l+b) units 
9. Diagonal, d =  𝑙2 + 𝑏2 units 
Scalene Triangle: 
10. Area =  𝑠 𝑠 − π‘Ž  𝑠 − 𝑏 (𝑠 − 𝑐) sq. units; s = (a+b+c)/2 
11. Perimeter = (a+b+c) units 
Isosceles Triangle: 
12. Area = 𝑏 4
 4π‘Ž2 − 𝑏2 sq units 
13. Perimeter = 2a + b units
  b = base length; a = equal side length 
Equilateral Triangle: 
14. Area =  3 4
π‘Ž2 sq. units
15. Perimeter = 3a units 
a = side of the triangle 
Right-angled triangle: 
16.  Area = (½)b x h sq. units 
17. Perimeter = b + h + hypotenuse 
18. Hypotenuse =  𝑏2 + β„Ž2 units 
Cuboid: 
19. Volume = (Cross section area  height)  =  l  b  h cubic units 
20. Lateral Surface Area (LSA) = 2[(l+b)h] sq. units 
21. Total surface area (TSA) = 2(lb+bh+hl) sq. units 
22. Length of the diagonals = 2 22 h bl   units 
Cube: 
23. Volume = a3 cubic units 
24. LSA  =  4 a2  sq. units 
25. TSA =  6a2 sq. units
26. Length of diagonal = a3 units  
Sphere: 
27. Volume  = (4/3) r3 cubic units 
28.  Surface Area = 4r2 sq. units 
29. If R and r are the external and internal radii of a spherical shell, then its Volume = 4/3[R3-r3]
cubic units 
Hemisphere: 
30. Volume = (2/3)r3 cubic units 
31. TSA = 3r2 sq. units 
Cylinder: 
32. Volume = r2h cubic units 
33. Curved surface Area (CSA) (excludes the areas of the top and bottom circular regions) = 2rh sq. units 
34. TSA = Curved Surface Area + Areas of the top and bottom circular regions =       2rh + 2r2=2r[r+h] sq. units
Cone: 
35. Volume = (1/3)r2h cubic Units
36. Slant Height of cone,    l = 2 2 hr  units 
37. CSA = rl sq. units 
38. TSA = r(r+l) sq. units 
Trigonometry  
1. Right Triangle Definition 
Assume that: 
0 < πœƒ < πœ‹ 2
 or 0° < πœƒ < 90°      
sinπœƒ =
π‘œπ‘π‘ β„Žπ‘¦π‘
  cosecπœƒ =
β„Žπ‘¦π‘ π‘œπ‘π‘
cosπœƒ =
π‘Žπ‘‘π‘— β„Žπ‘¦π‘
  secπœƒ =
β„Žπ‘¦π‘ π‘Žπ‘‘π‘—
cosπœƒ= π‘œπ‘π‘ π‘Žπ‘‘π‘—
  cotπœƒ =
π‘Žπ‘‘π‘— π‘œπ‘π‘ 
2. Tangent cotangent Identities 
tanπœƒ =
π‘ π‘–π‘›πœƒ π‘π‘œπ‘ πœƒ
   cotπœƒ = π‘π‘œπ‘ πœƒ π‘ π‘–π‘›πœƒ
3. Reciprocal Identities 
sinπœƒ =
1 π‘π‘œπ‘ π‘’π‘ πœƒ
   cosecπœƒ =
1 π‘ π‘–π‘›πœƒ
cosπœƒ =
1 π‘ π‘’π‘πœƒ
   secπœƒ =
1 π‘π‘œπ‘ πœƒ
tanπœƒ =
1 π‘π‘œπ‘‘πœƒ
   cotπœƒ =
1 π‘‘π‘Žπ‘› πœƒ  
4. Pythagorean Identities 
sin2πœƒ + cos2πœƒ = 1
tan2πœƒ + 1 = sec2πœƒ
1 + cot2πœƒ = cosec2πœƒ 
5. Even and Odd Formulas
sin (-πœƒ) = - sinπœƒ  cosec (-πœƒ) = - cosec πœƒ
cos(-πœƒ) =  cosπœƒ  sec(-πœƒ) = secπœƒ
tan (-πœƒ) = - tanπœƒ     cot (-πœƒ) = - cotπœƒ 
6. Double Angle Formulas 
sin(2πœƒ) = 2 sinπœƒ cosπœƒ
cos (2πœƒ) = cos2πœƒ - sin2πœƒ
                = 2 cos2πœƒ - 1
                 = 1 – 2 sin2πœƒ
tan (2πœƒ) =
2π‘‘π‘Žπ‘› πœƒ 1−π‘‘π‘Žπ‘› 2πœƒ  
7. Half Angle Formulas
sinπœƒ = ±  1−cos⁡ (2πœƒ) 2
cosπœƒ = ±  1+cos⁡ (2πœƒ) 2
tanπœƒ = ± 
1−cos⁡ (2πœƒ) 1+cos⁡ (2πœƒ) 
8. Sum and Difference Formulas
sin (A ± B) = sinA cosB ± cosA sinB
cos (A ± B) = cosA cosB ∓ sinA sinB
tan (A ± B) =
tan 𝐴 ±tan 𝐡 1 ∓ tan 𝐴 tan 𝐡
9. Product to Sum Formulas 
sinA sinB = 1 2
 [cos (A - B) – cos (A + B)]
cosA cosB = 1 2
 [cos (A - B) + cos (A + B)]
sinA cosB = 1 2
 [sin (A + B) + sin (A - B)]
cosA sinB = 1 2
 [sin (A + B) - sin (A - B)] 
10. Sum to Product Formulas 
sinA + sinB = 2 sin ( 𝐴+𝐡 2
) cos (
𝐴−𝐡 2
)
sinA - sinB = 2 cos ( 𝐴+𝐡 2
) sin ( 𝐴−𝐡 2
)
cosA + cosB = 2 cos ( 𝐴+𝐡 2
) cos ( 𝐴−𝐡 2
)
cosA - cosB = - 2 sin ( 𝐴+𝐡 2
) sin ( 𝐴−𝐡 2
) 
11. Co function  Formulas 
Sin ( πœ‹ 2
− πœƒ) = cosπœƒ         cos ( πœ‹ 2
− πœƒ) = sin πœƒ
cosec ( πœ‹ 2
− πœƒ) = secπœƒ      sec ( πœ‹ 2
− πœƒ) = cosπ‘’π‘πœƒ
tan ( πœ‹ 2
− πœƒ) = cotπœƒ          cot ( πœ‹ 2
− πœƒ) = tanπœƒ  
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