Aptitude Basics

Aptitude basics

 

Numbers

 Divisibility

1. A number is divisible by 2 if it is an even number.
2. A number is divisible by 3 if the sum of the digits is divisible by 3.
3. A number is divisible by 4 if the number formed by the last two digits is divisible by 4.
4. A number is divisible by 5 if the units digit is either 5 or 0.
5. A number is divisible by 6 if the number is divisible by both 2 and 3.
6. A number is divisible by 8 if the number formed by the last three digits is divisible by 8.
7. A number is divisible by 9 if the sum of the digits is divisible by 9.
8. A number is divisible by 10 if the units digit is 0.
9. A number is divisible by 11 if the difference of the sum of its digits at odd
places and the sum of its digits at even places, is divisible by 11.

Important formulas

i. ( a + b )( a - b ) = ( a ² - b² )
ii. ( a + b ) ² = ( a² + b² + 2 ab )
iii. ( a - b )² = ( a² + b² - 2 ab )
iv. ( a + b + c ) ² = a²+ b² + c² + 2 ( ab + bc + ca )
v. ( a³ + b³ ) = ( a + b )( a² - ab + b² )
vi. ( a³ - b³ ) = ( a - b )( a² + ab + b²)
vii. Sum of natural numbers from 1 to n
     
                   
viii. Sum of squares of first n natural numbers is =
                    
ix. Sum of cubes of first n natural numbers is A
                                 
x. HCF= (HCF of the numerators)/(LCM of the denominators)
xi. LCM= (LCM of the numerators)/HCF of the denominators
xii. Product of two numbers = Product of their H.C.F. and L.C.M

Note: When a number N is raised to any integral power m, the digit in the unit’s
place of the resulting value can be determined without actually evaluating the
power. The digits when raised to powers will give values in which the digits in
the unit’s place follow a cylindrical pattern. Following is the pattern to calculate
the digit in the unit’s place of any derived power.
                                      

HCF models:-

If N is a composite number such that N = a^p . b^q . c^r ….. 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 = (a^p+1 – 1 / a – 1) . (b^q+1 – 1 / b – 1) . (c^r+1 – 1 / c – 1)…..
(e) The number of co-primes of N (< N), Ø(N) = N(1 – 1/a) (1 – 1/b) (1 – 1/c) ….
(f) Sum of the numbers in (e) = N/2 . Ø(N)
(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.

 Averages, Mixtures and Alligation

• Average= (x₁+x₂+x₃+......x_n)/n
   where x₁,x₂,x₃, are quantities.
•  
• Weighted Average= (n₁x₁+n₂x₂+n₃x₃+.....n_kx_k) / n₁+n₂+n₃+......n_k
   where x₁,x₂,x₃.... x_k are the quality factors
  n₁,n₂,n₃,........n_k are the quantity factors Eg: If the average height of boys= 172cms and that of girls=154 cms, then find average height of the class with 18 boys and 12 girls?
• Here n₁ and n₂ are no. of boys and girls (Quantity factor)
  x₁ and x₂ are average heights (Quality factor)
• 
  
• Mixtures
  For mixtures, Average, x'= (n₁x₁+n₂x_{2)/ (}n₁+n₂)
                                                         
• Alligation:
  

=> n₁/n₂ = (x₂-x')/x'-x₁

Time, Speed and Distance

Distance= Speed*time
For a non-uniform motion Average speed= Total distance travelled/Total time taken
When the body travels at 'u' m/s for t1 seconds and 'v' m/s for t2 seconds, then
Average speed= (ut1+vt2)/(t1+t2)
When the body travels l distance at 'u' m/s and 'm' distance at 'v' m/s; Average speed = (mu+lv)/(l+m)

Relative Speed: Speed of a moving body w.r.t. another moving body is called relative speed.
Speed of A w.r.t. B
(i) When they are moving in same direction; Relative speed of A= A-B
(ii) When they are moving in opposite direction; Relative speed of A= A+B
Key points on Trains
When a train is crossing a pole distance travelled by the train= length of train
When a train of length l is crossing a bridge of length b; the distance travelled by train=l+b
When a train of length l is crossing a platform of length p; then distance travelled by train=l+p
When a train of length l1 is crossing/ overtaking another train l2; then distance travelled = l1+l2

 Percentages 

Percentage= (Sum of quantities)/(Number of quantities)
Percentage increase by x%= ((x+100)/100)*Initial
Percentage decrease by x%= ((100-x)/100)*Initial

Some common percentage conversions

Profit and Loss

Profit=SP-CP
Loss=CP-SP
Profit %= ((SP-CP)/CP) *100
Loss %= ((CP-SP)/ CP) *100
Discount= MP-SP
Discount %= ((MP-SP)/ MP) * 100
where SP= Selling Price, CP= Cost Price, MP= Marked Price

Simple and Compound Interest

Important formula and equations
Principal: The money borrowed or lent out for a certain period is called the principal or the sum.
Interest: Extra money paid for using other's money is called interest.
Simple Interest (SI): If the interest on a sum borrowed for certain period is reckoned uniformly, then it is called simple interest.
Let Principal= P, Rate= R% per annum (p.a) and Time= T years. Then
(i) Simple Interest= (P*R*T)/100
(ii) P = (100*SI)/(R*T); R= (100*SI)/(P*T) and T= (100*SI)/(P*R)
Key notes on Simple Interest
A sum of money becomes n times itself in T years at simple interest, then the rate of interest is
Rate= 100(n-l)%
               T
If a sum of money becomes n times in T years at SI then it will be m times of itself in ..... years
Required time= (m-l)*T years
                          (n-l)
If SI on a sum of money is 1/xth of the principal and the time T is equal to the rate percent R, then
Rate= Time=
A certain sum is at SI at a certain rate for T years. And if it had been put at R1 % higher rate, then it would fetch Rs.x more, then the
Principal= x*100
                 T*R1
The annual payment that will discharge a debt of Rs.P due in T years at the arte of interest R% per annum is Annual payment = 100P               
100T+RT(T-1)
                   2 
                                                                           
Let the rate of interest for first 1 years is r1% per annum, for the next t2 years is r2 % per annum and for the period beyond that is r3 %. Suppose all together the simple interest for t3 years is Rs.I. Then Principal=100*I                         
                         t1r1+t2r2+(t3-t1-t2)r3
The simple interest on a certain sum of money at r1 % per annum for t1 years is Rs.m. The interest on the same sum for t2 years at r2 % per annum is n.
Then the sum= (m-n)*100
                         r1t1-r2t2
Key notes on Compound interest
Compound Interest: (Amount - Principal)
Amount= P* (1+R/100)ⁿ
When the interest is compounded K times a year, Amount= P( 1 + R / K*100)^kt
When the interest is paid half yearly, say at r%per annum compound interest, then the amount after t years is given by:
P( 1 + R / 2*100)^2t
Similarly, if the interest is paid quarterly, say at r% per annum compound interest, then the amount due after t years is given by:
P( 1 + r / 4 * 100)^4t
Under the method of equated instalments, the value of each instalment is the same.
Equal Annual Instalment under
(a) Simple Interest, x = 2P(100 + nr)
                                     n[200 + (n - 1)r]
(b) Compound Interest, x = Pr / 100[1 – (100/100 + r) ⁿ ]
  

Progression

Important Equations and Formula
Sum of first n natural numbers= n(n+1)/2
Sum of the squares of first n natural numbers= (n(n+1)(2n+1))/6
Sum of the cubes of first n natural numbers= [n (n+1)/2]²
Sum of first n natural odd numbers= n²
Average = Sum of items/ Number of items
Arithmetic Progression (AP): An AP is of the form a, a+d, a+2d, a+3d,..... where a is called the 'first term' and d is called the 'common difference'.
nth term of an AP; tn=a+(n-1)d
Sum of the first n terms of an AP; Sn= n/2 [2a+(n-1)d] or Sn= n/2 (first term+last term)
Geometrical Progression (GP):
A GP is of the form a, ar, ar², ar³......... where a is called the 'first term' and r is called the 'common ratio'.
nth term of a GP; tn= ar^n-1
Sum of the first n terms in a GP; Sn= a(1-rⁿ)/1-r
Sum of infinite series of progression; S= a/(1-r)
Geometric mean of two number a and b is given as GM= sqrt(ab)
Harmonic Progression (HP)
If a₁,a₂,a₃,...................an are in AP, then 1/a₁, 1/a₂, 1/a₃, ........1/a_n, are in HP
nth term of this HP, tn =1/(1/a₁+(n-1)(a₁-a₂/a₁a₂) ) nth term of this HP from the end, tn=1/ (1/a₁-(n-1)(a₁-a₂/a₁a₂))
If a and b are two non-zero numbers and H is harmonic mean of a and b then a, H, b from HP and then H=2ab/(a+b)
Arithmetico-Geometric series
A series having terms a, (a+d)r, (a+2d)r²,...... etc is an Arithmetico-Geometric series where a is the first term, d is the common difference of the Arithmetic part of the series and r is the common ratio of the Geometric part of the series.
The nth term tn= [a+(n-1)d]r^n-1
The sum of the series to n terms is
Sn= a/1-r+ dr (1-r^n-1)/ (1-r)² - [a+(n-1)d]rn/ 1-r
The sum to infinity, S= a/ 1-r + dr (1-r^n-1)/(1-r)² ; r<1
Exponential Series
e^x = 1+x/1!+x²/2!+x³/3!+..........   (e is an irrational number)
coefficient of xn= 1/n!; Tn+1=x^x/n!
e^-x = 1-x/1!+x2/2!- x3/3!+..........
Logarithmic Series
log_e (1+x)= x-x²/2+x³/3+x⁴/4+........  (-1<x 1)
log_e (1-x)= -x-x²/2-x³/3- x⁴/4 -........  (-1x< 1)
log_e (1+x)/(1-x)= 2-(x+x³/3+x⁵/5+.........) (-1<x 1)