Perfect Square Series
Perfect Cube Series
Ration Series
Geometric Series
Mixed Series
Perfect Square Series:
This Types of Series are based on square of a number which is in same order and one square number is missing in that given series.
Example 1: 441, 484, 529, 576, ?
Answer: 441 = 212, 484 = 222, 529 = 232, 576 = 242 ,625 = 252.
Example 2: 121, 144, 169, ?, 225
Answer: 121 = 112, 144 = 122, 169 = 132, 196 = 142, 225 = 152.
Example 3: ?, 2116, 2209, 2304, 2401, 2500
Answer: 2025 = 452, 2116 = 462, 2304 = 482, 2401 = 492, 2500 = 502
Example 4: 961, 1024, ?, 1156, 1225
Answer: 961 = 312, 1024= 322, 1089 = 332, 1156 = 342, 1225 = 352.
Example 5: 36, ?, 64, 81, 100, 121
Answer: 36 = 62, 49 = 72, 64 = 82, 81 = 92, 100 = 102, 121 = 112.
Perfect Cube Series:
This Types of Series are based on cube of a number which is in same order and one cube number is missing in that given series
Example 1: 1331, 1728, 2197, ?
Answer: 1331 = 113, 1728 = 123, 2197 = 133, 2744 = 143.
Example 2: 125, ?, 343, 512, 729, 1000
Answer: 125 = 53, 216 = 63, 343 = 73, 512 = 83, 729 = 93, 1000 = 103.
Example 3: 4096, 4913, 5832, ?, 8000
Answer: 4096 = 163, 4913 = 173, 5832 = 183, 6859 = 193, 8000 = 203.
Example 4: 1728, 1331, ?, 729, 512
Answer: 1728 = 123, 1331 = 113, 1000 = 103, 729 = 93, 512 = 83.
Ration Series:
This type of series are based on ration and change and order are in difference between the numbers is found out.all numbers are arranged in sequence order.
What is Proportion?
The idea of proportions is that two ratio are equal.
If a : b = c : d, we write a : b : : c : d,
Ex. 3 / 15 = 1 / 5
a and d called extremes, where as b and c called mean terms.
Proportion of quantities
the four quantities a, b, c, d said proportion then we can express it
a : b = c : d
Then a : b : : c : d <–> ( a x d ) = ( b x c )
product of means = product of extremes.
If there is given three quantities like a, b, c of same kind then then we can say it proportion of continued.
a : b = b : c the middle number b is called mean proportion. a and c are called extreme numbers.
So, b2 = ac. ( middle number )2 = ( First number x Last number ).
Few Examples for understanding
Example 1:
Rama distributes his pencil among his four friends Rakesh, Rahul, Ranjan, and Rohit in the ratio 1 / 2 : 1 / 3 : 1 / 4 : 1 / 5 . What is the minimum number of pencil that the person should have?
Answer :
Step 1: At First we need to do is LCM of 2,3,4 and 5 is 60.
Step 2: Then pencil are distributed in ratio among friends,
Rakesh = ( 1 / 2 x 60 ) = 30.
Rahul = ( 1 / 3 x 60 ) = 20.
Ranjan = ( 1 / 4 x 60 ) = 15.
Rohit = ( 1 / 5 x 60 ) = 12.
Step 3: Total number of pencils are ( 30 x + 20 x + 15 x + 12 x) = 77 x.
So minimum number of pencil is x = 1.
and Raman should have 77 pencils.
Example 2:
Two numbers are respectively 40% and 60% more than third number.Find the ration of two numbers ?
Answer :
Step 1: Let the third number is A
Then first number is 140% of A = 140 x A / 100 = 7A / 5 and second number is 160% of B = 160 x B / 100 = 8B / 5.
Step 2: now ratio of first and second number is 7A / 5 : 8B / 5 = 35A : 40B = 7 : 8.
Example 3 :
If 90 is divided into three parts proportional to 3, 5, 8, 9 then the smallest part is :
Answer :
ratio is = 3 : 5 : 8 : 2, sum of ratio terms is = 18.
So the smallest part is = ( 90 x 2 / 18 ) = 10.
Example 4:
If a : b = 3 : 7 and b : c = 5 : 9, find a : b : c .
Answer: A : B = 3 : 7, B : C = 5 : 9
= ( 5 x 7 / 5 ) : ( 9 x 7 / 5 ) = 7 : 63 / 5 .
= A : B : C = 3 : 7 : 63 / 5 = 15 : 35 : 63 .
Example 5: If A : B = 4 : 9 and B : C = 3 : 6 , then A : C is :
Answer : (A / B = 4 / 9, B / C = 3 / 6 )
= A / C = ( A /B x B / C) = ( 4 / 9 x 3 / 6 ) = 2 / 9 = A : C = 2 : 9.
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Example 1:
Share Rs.4200 among joy, sanjay and bijoy in the ration 2 : 4 : 6.Find the amount received by sanjay.
Answer :
Amount received by sanjay.
4 / 12 X 4200 = 1400= ( related ratio / sum of ratio ) x Total amount
So, the Amount received by sanjay is 1400.
Example 2:
Find the mean proportional between given two number that is 64 and 49.
Answer :
The mean proportion of two numbers is
Root of 64 and 49 is √8 x √ 7 = 8 x 7 = 56.
So, the mean proportional is 56.
Example 3:
The Salary of Three friend A, B, C are divided into ratio 5 : 6 : 8.If the increment has given of 10%, 20%, 25%, Find the new ratio of three friend salary ?
Answer :
Step 1: We assume ration as 5x, 6x, 8x
now the increment of new salary is A = 110 / 100, B = 120 / 100, C = 125 / 100.
Step 2: A,s new salary is 110 X 5x / 100 = 55x / 10.
B,s new salary is 120 X 6x / 100 = 36x / 5.
C,s new salary is 125 X 8x / 100 = 10.
Step 3:New ratio is 55x / 10 : 36x / 5 : 10.
Example 4 : Rs 1210 were divided among three person P, Q, R so that P : Q = 5 : 4 and Q : R = 9 : 10. Then R gets the amount.
Answer :
P : Q = 5 : 4, Q : R = 9 : 10 = ( 9 x 4 / 9 ) : ( 10 x 4 / 9 ) = 4 : 40 / 9.
So, A : B : C = 5 : 4 : 40 /9 = 45 : 36 : 40
Sum of ratio terms is = ( 45 + 36 + 40 ) =121.
R share of amount is Rs (1210 x 36 / 121) = Rs. 400.
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Example 1:
A money bag contains 50 p, 25 p, and 10 p coins in the ratio 5 : 9 : 4, and the total amounting to Rs.206.
Find the individual number of coins of each type.
Answer :
Step 1: Let the number of 50 p ,25 p, and 10 p coins be 5x, 9x, 4x respectively.
Then, 5x / 2 + 9x / 4 + 4x / 10 = 206
= 50x + 45x + 8x = 4120
= 103x = 4120
= x = 40.
Step 2: Number of 50 p coins is ( 5 x 40 = 200 ),
Number of 25 p coins is( 9 x 40 = 360 ),
Number of 10 p coins ( 4 x 40 = 160 ),
Example 2:
How many bags are required for filling 1824 kg of wheat if each bag filled with 152 kg of wheat ?
Answer :
Number of bags = 1824 / 152
= 12 .
Example 3:
On a self there are 4 books on Economics , 3 books on Management and 4 books on Statistics . In how many different ways can be the books be arranged so that the books on Economics are kept together ?
Answer :
Total ways = 8! x 4!
= ( 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 ) x ( 4 x 3 x 2 x 1 )
= 40320 x 24
= 967680 .
So , we can 967680 way be the books be arranged .
Example 4:
An urn contains 4 green , 5 blue , 2 red and 3 yellow marbles . If four marbles are drawn at random , what is the probability that two are blue and two are red ?
Answer :
Required probability = 5C2 x 2C2 / 14C4
= 10 / 1001 .
Example 5:
An urn contains 4 green , 5 blue , 2 red and 3 yellow marbles . If eight marbles are drawn at random , what is the probability that there are equal number of marbles of each color ?
Answer :
Required probability = 4C2 x 2C2 x 3C2 / 14C8
= 180 / 3003 = 60 / 1001 .
Example 6:
An urn contains 4 green , 5 blue , 2 red and 3 yellow marbles . If two marbles are drawn at random , what is the probability that both are red or at least one is red ?
Answer :
Required probability = 2C2 / 14C2 + [ 1 - 12C2 / 14C2]
= 1 / 91 + [ - 66 / 91 ]
= 1 / 91 + 25 / 91
= 26 / 91 .
Example 7:
An urn contains 4 green , 5 blue , 2 red and 3 yellow marbles . If three marbles are drawn at random , what is the probability that at least one is yellow ?
Answer :
Required probability = 1 – 11C3 / 14C3
= 1 – 165 / 364
= 199 / 364 .
Example 8:
An urn contains 4 green , 5 blue , 2 red and 3 yellow marbles . If three marbles are drawn at random , what is the probability that none is green ?
Answer :
Required probability = 10C3 / 14C3
= 30 / 91 .
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Example 1:
A liquid mixture contain alcohol and water in the ratio of 4 : 3. If 5 liters of water is added to the mixture the ration becomes 4 : 5.Find the quantities of alcohol in the given mixture ?
Answer :
Let the quantity of alcohol and water be 4x and 3x liters.
Then,
4x / 3x + 5 = 4 / 5
= 20x = 4(3x + 5)
= 8x = 20
= x = 2.5
Quantity of alcohol = ( 4 x 2.5 ) = 10 liters.
Example 2:
In a bottle mixture of 80 liters and the ratio of milk and water is 3 : 2. If this mixture ratio is to be 2 : 3. What the quantity of water to be further added ?
Answer :
Step 1: Quantity of Milk ( 80 x 3 / 5 ) = 48 liters, So Quantity of water in it ( 80 – 48 ) = 32 liters.
Step 2: New Ratio required 2 : 3, Let x water to be added, Then Milk : Water is = 48 : (32+x)
=48 / (32 + x).
Step 3: Now 48 / (32 + x) = 2 : 3
48 / (32 + x) = 2 / 3
2x = 144 – 64
x = 80/2
=40 liters.
Example 3 :
Rs 75,500/- are divided between A and B in the ratio 1 : 3 . what is the difference between thrice the share of A and twice the share of B ?
Answer :
Share of A = 75,500 x 1 / 1 + 3 = 75,500 x 1 / 4 = 18875 .
Share of B = 75,500 x 3 / 1 + 3 = 75,500 x 3 /4 = 56625 .
Difference between thrice the share of A and twice the share of B is
= 2B – 3A
= 2 x 56625 – 3 x 18875
= 113250 – 56625
= 56625 .
Example 4 :
The ratio between the number of men and women in a society is 31 : 23 , When 75 more women are added in the society, this ratio becomes 124 : 107 . How many more women should be added in the society in order to make the number of men and women be equal ?
Answer :
31x / 23x + 75 = 124 / 107
3317x = 2852x + 9300
465x = 9300
x = 20 .
So , the number of women in society after added
= 20 x 23 + 75
= 535 .
So , the number of men in society after added
= 31 x 20
= 620 .
Number of more women is = 620 – 535 = 85 .
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Geometric Series:
This type of series are based on ascending or descending order of numbers and each successive number is obtain by multiplying or dividing the previous number with a fixed number.
Examples 1: 5, 45, 405, 3645, ?
Answer: 5 x 9 = 45, 45 x 9 = 405, 405 x 9 = 3645, 3645 x 9 = 32805.
Examples 2: 73205, 6655, 605, 55, ?
Answer: 5 x 11 = 55, 55 x 11 = 605, 605 x 11 = 6655, 6655 x 11 = 73205.
Examples 3: 25, 100, ?, 1600, 6400
Answer: 25 x 4 = 100, 100 x 4 = 400, 400 x 4 = 1600, 1600 x 4 = 6400.
Examples 4: 9, 54, ?, 1944, 11664
Answer: 9 x 6 = 54, 54 x 6 = 324, 324 x 6 = 1944, 1944 x 6 = 11664.
Two stage Type Series:
A two tier Arithmetic series is one in which the
differences of successive numbers themselves
form an arithmetic series.
Examples 1: i. 3, 9, 18, 35, 58,——
ii. 6, 9, 17, 23,———-
Mixed Series:
This type of series are more then one different order are given in a series which arranged in alternatively in a single series or created according to any non-conventional rule.This mixed series Examples are describes in separately.
Examples 1:
111, 220, 438, ?, 1746
Answer:
from 111 to 220 we get using this 111 x 2 = 222 – 2 = 220,similarly we follow next steps
from 220 to 438 we get using this 220 x 2 = 440 – 2 = 438,
from 438 to ? we get using this 438 x 2 = 876 – 2 = 874,
from 874 to 1746 we get using this 874 x 2 = 1748 – 2 = 1746.
So the missing number is 874
Examples 2: 24, ?, 208, 622, 1864
Answer:
from 24 to ? we get using this 24 x 3 = 72 – 2 = 70, Similarly we follow next steps
from 70 to 208 we get using this 70 x 3 = 210 – 2 = 208,
from 208 to 622 we get using this 208 x 3 = 624 – 2= 622,
from 622 to 1864 we get using this 622 x 3 = 1866 – 2 = 1864.
So the missing number is 70
Examples 3: 11, 24, 50, 102, 206, ?
Answer:
11 x 2 = 22 +2 = 24,
24 x 2 = 48 + 2 = 50,
50 x 2 = 100 + 2 = 102,
102 x 2 = 204 + 2 = 206,
206 x 2 = 412 + 2 = 414.
So the missing number is 414.
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