### Clues and helpful tips on typical mathematical puzzles

Percentages

1. If the commodity price increases by x% then the consumption has to be reduced by x/(100+x) to maintain the amount spent
2. If the commodity price decreases by x% then the consumption has to be increased by x/(100-x) to maintain the amount spent
3. if a’s income is x% more than b on b’s income then b’s income is x/(100+x)% less than a on a’s income
4. if a’s income is x% less than b on b’s income then b’s income is x/(100-x)% more than a on a’s income
5. When it is stated that a is x% more than b then it means on the base of b. thus

a = b+x%b

Simple & compound interest

1. Simple interest = PNR/100 where p = principal; N=period (years) and R = rate

Amount = P+I

1. compound interest = A = P (1+R/100)^N

This in compound interest the formula will give the final amount. Principal will have to be deducted from it to get the interest portion. In case the interest is half yearly then rate should be halfed and period should be doubled. Similarly for quarterly. Then the interest rate for different years is different then P (1+R1/100) (1+R2/100) (1+R3/100)

1. population. The formula for compound interest can also be used for populations. But where the populations is decreasing then the sign will change to ‘-‘ instead of ‘+’

Averages

1. Mode is the number that occurs most no of times in given sample
2. Median is the middle number of the given sample. Where the number of items in given sample is odd then (n+1)/2th number and if the number is even it is simple average between n/2 and n+2/2th numbers
3. Arithmetic mean : it is sum of all numbers / no of numbers
4. weighted average mean: it is sum of product of numbers and their respective weights / sum of weights
5. Geometric mean: between two numbers it is x = Sqrt (ab). If geometric mean of one group of numbers (a) is X and that of group (b) is Y then geometric mean of both the groups will be (X+Y)/ (a+b)
6. Harmonic mean : between two numbers = number of numbers / (sum of reciprocal of numbers) ie. 2ab/(a+b). This also gives the average speed when same length distances are covered in different speeds
7. GM^2 = AM*HM

Ratio proportion and variation

1. comparing two quantities as ratios:
1. both the quantities should be of same kind
2. both should have the same measurement per unit
3. ratio is a pure number i.e. it does not have any measurement. It just denotes how many times one quantity is of one of other
2. compounding : if two different ratios (say a:b and c:d) are expressed in different units then if we require to combine these two ratios then it will be AC:BD
3. if a/b=c/d=e/f then the ratio is equal to a+c+e / b+d+f

Mixtures and allegation

1. Alligation rule

Quantity of cheap = Price of dear – average price

Quantity of dear Average price – price of cheap

2. if a vessel contains ‘a’ litres of liquid A and if ‘b’ litres are withdrawn and replaced by liquid B then if ‘b’ litres of the mixture is again withdrawn and replaced by liquid B. the operation is repeated ‘n’ number of times then

Liquid a left in vessel = ((a-b)/a)^n

Initial liquid in vessel

Profit or loss

If the a and b are two successive discounts that have been given then effective discount rate will (a+b-(ab/100))

Time speed and distance

Km/hr * (5/18) = m/s

Km = 5/8 mile

1. while traveling if a person changes his speed in m:n ratio then the time taken will also change in n:m ratio

2. if the A to B is traveled in T1 time and a speed and B to A if T2 time and b speed then the average speed is give by

(2ab) / (a+b) ………. Harmonic mean

Distance is given by (T1+T2) (2ab/(a+b))

Or (T1-T2) (2ab/(a-b))

Or (a-b) (T1T2 / (T1+T2))

3. if two persons start towards each other from different points and arrive at two points in a hrs and b hrs respectively after having met then ratio of their speed is given by SQRT (b) / SQRT (a) = a’s speed / b’s speed

Work

1. if A can do a work in x days then 1/xth work is done in one day

2. if A is X times better workman than B then A will take 1/xth time of that taken by B

3. if A and B do work in X and Y days then they will complete the same work in XY / (X+Y) days and in one day (X+Y)/ XY days work will be done

4. if A and B can do a piece of work in X days and if A alone will be able to complete the work in a days more than X and b can in b days more than X then X2 = ab

5. if a pipe can fill a vessel in x hrs then 1/xth part of the vessel is filled in one hour

6. if A pipe is X times bigger than B then A will take 1/X times lesser time than B

7. if A and B fill the pipe in m and n hours respectively then both will fill the pipe in MN / (m+n) hours and (m+n) / mn th part of vessel will be filled in one hour

8. if one inlet pipe fills the vessels in M hrs and other pipe empties the vessel in N hrs then the vessel will be filled in MN / (N-M) hrs. and (N-M)/MN the part will be filled

9. if an inlet pipes taken X minutes to fill the cistern and has taken a minutes longer then the leak will empty the cistern in a*(1+a/x) minutes

10. A and B can fill the cistern in X hrs and A alone will fill the same in a minutes more than X and b can fill it in b minuted more than X then X= sqrt (ab)

Clocks and Calendars

a) A dial of the clock is divided into 60 parts each called minute spaces

b) The hour hand goes 5 minute spaces in one hour and minutes hand goes 60 minute spaces in one hour. Thus the minute hand gains 55 minute spaces over the hour hand in one hour

c) When two hands are in 90 degree they are 15 minute spaces apart. This occurs twice in an hour.

d) When the two hands are in opposite directions they are 30 minute hands apart this occurs once in an hour

e) Two hands are in straight line when they coincide or are in opposite directions

f) The angle between the two hands = 6(x-11/12m)

X= hour hand convert into minute spaces i.e.* 5 of the earlier clock

M = the later part of the time i.e. minutes

g) The years that are divisible by 400 are the only ones that are leap year.

When mentioned rRST then S will be the top vertices

Numbers

a) ODD +/- ODD = EVEN

b) ODD +/- EVEN = ODD

c) EVEN +/- EVEN = EVEN

d) ODD * ODD = ODD

e) ODD * EVEN = EVEN

f) EVEN * EVEN = EVEN

g) HCF of two numbers is the number that divides both the numbers exactly

h) LCM of two numbers is the number that is divided by both the numbers exactly

i) HCF*LCM= product of both the numbers

j) HCF of fraction is HCF of the numerators / LCM of denominators

k) LCF of fractions is LCM of numerators and HCF of denominators

l) if three numbers a,b,c are divided by N in such manner that r is the remainder each time then smallest value of N is LCM of (a,b,c)+r

m) if three numbers a,b,c divide N is such manner that remainders are p,q,r then if (a-p) = (b-q) = (c-r) then the smallest value of N is LCM of (a,b,c) – (a-p)

Indices

a) m * An=A(m+n)

b) m / An=A(m-n)

c) (A­m)n=A(m*n)

d) Nth root of A = A1/n

e) 1/A = A-n

f) AnBn=(AB)n

g) (A+B)2=A2+B2+2ab

h) (A-B)2=A2+B2—2ab

i) (A+B)2-(A-B)2=4AB

j) (A+B)2+ (A-B)2=2(A+B)2

Inequalities

a) If a>b then a+m>b+m

b) If a>b then am>bm for m>0 and am

c) If a>b then 1/a<1/b

Logarithms

a) Log n A = logm A/logmn

Progressions

a) Arithematic progressions

Sum = (n/2)*[2A+(n-1)d]

= (n/2) * (a+l)

Nth Term = A+(N-1)D

N = number of terms

D is the common differences

A is the first term

L is the last term

b) Geometric progression

Arn-1=Nth Term

Sum = A(1-rn)/(1-r)

Geometric mean = (ab)1/2

c) Harmonic mean = it is the arithmetic mean of reciprocals of numbers

Sum and nth number of harmonic mean is reciprocal of arithmetic mean

Harmonic mean of two numbers is 2ab/(a+b)

Permutation and combination

a) Fundamental principal of addition: if one thing can be done m number of ways and other thing can be done in n number of ways independent of other. Then either of them can be done in (m+n) ways

b) Fundamental principal of multiplication: if one thing can be done m number of ways and other thing can be done in n number of ways independent of other. Then either of them can be done in (m*n) ways

c) Permutation : permutation of n objects taken r at a time is the arrangement in a straight line of r objects taken at a time denoted by N!/(N-R)!

d) The number of permutation of n objects taken all at a time = n!

e) The number of permutations of n objects taken all at a time when p of them are like, q are like = n!/p!q!

f) Combination is the selection of r objects in n objects. Denoted as N!/(n-r)!r!

g) Number of permutations of n objects taken all at a time in circle (n-1)!

h) When the repetition of allowed then permutation nr

Volumes

a) Triangle :

Area = ½ base * height - universal

Area of equilateral triangle = sqrt(3)/4 side2

b) Rectangle : length * breadth

c) Square : side 2

Diagonal = side * sqrt(2)

Area = ½ product of diagonal

d) Parallelogram : base * height

e) Rhombus: ½ product of diagonals

f) Circle : Area PiR2 Circumference 2PiR

g) Cuboid : is the rectangular solid having 6 faces with all the faces as rectangles

Volumes : l*b*h

Area for 4 walls : 2 (i+b)* h

Total surface area of cuboid : 2 (lb + bh + lh)

Body diagonals of cuboid : sqrt(l2+b2+h2)

h) Cube

Volume = a3

Total surface area of Cube 6 * a2

i) Cylinder

Volume Pi R2H

Curved surface= 2PiRH

Total surface = 2PiR(R+H)

j) Cone

Volume = 1/3 PiR2h

Curved surface area = PiRL where L=Sqrt (R2+H2)

Total surface area : PiR(R+L)

k) Sphere

Volume : 4/3 PiR3

Surface : 4 Pi R2

Triangle

a) Sum of angles is 180

b) Exterior angle is equal to sum of interior angle non adjacent to it i.e. angles other than the complementary angle of the exterior angle

c) Sum of any two sides is more than the third side

d) Equilateral triangle is the triangle with all the sides as same

Area = 3/4side2 Height 3/2side

Perimeter = 3Side

e) Right angle triangle

45 -90-45 triangle Hypotenuse = 2 * Side

30-60-90 triangle 30 side = ½ hypotenuse

60 side = (3/2) hypotenuse

f) If the angles of two triangles are same then they are similar then all the attributes that they have will have same proportion – heights, sides etc.

Rectangle

a) Diagonals are equal and bisect each other

b) Diagonal = (a2+b2)

c) Of all the given rectangles of same area or perimeter square will have the maximum area

Parallelogram

a) Diagonals bisect each other

b) Opposite angles are same

c) Each diagonal divides the parallelogram in triangles of same area

Trapezium

a) Only one pair of opposite side are parallel to each other

b) Area = ½ * (sum of parallel sides) * height

c) Isosceles trapezium is the one that is inscribed in a circle. The oblique sides are equal. The opposite angles made by oblique sides with the parallel side are equal.

Circle

a) Tangents drawn from an external side are equal

Some points that can be helpful

1. 2N-1 or 2m+1 is always odd

2. If √N is an integer then N is always an integer

3. N^3-n = N(n-1)(n+1)

4. Think negative also as the maximum traps are account of positive to negative changes – best input nos 2-,2,3,-2,0.5,-0.5

5. Distance between two points on coordinate is given by the formula ((A-x)^2+(B-Y)^2)) where x.y and a,b are the pair of coordinates

6.

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