Shortcut for Square - A Simple Method

We have already discussed a method for finding the squares of the given number (check that method Here). Now we shall discuss another simplest method to calculate square of the given number. First of all one thing you should keep in mind, for you r convinience we've mentioned each and every step in detail. But You should do all these calculations MENTALLY. I mean without using pen or pencil. So practice well to do these problems orally. Now lets go to the topic, Finding the square of a number ending in 5 is very simple. If the last digit of the number is 5, then the last two digits of the square will be 25.What ever is the earlier part of the number multiply it with one more than itself and that will be the first part of the answer. (The second part of the answer will be 25 only).

35² = 1225  Here 3 x 4 = 12 so the answer is 1225

45²   = 2025

55²   = 3025

75²   = 5625

95²   = 9025

125²   = 15625

175²   = 30625

235²  = 55225

195²   = 38025

245²   = 60025

Well friends, now we know the squares of numbers 25, 30, 35, 40, 45, 50, 55 etc. If we want to find the squares of any other number, we find it using these squares which we already know.

For 262 we will go from 252, for 322 we go from 30 2 and so on.

One way is writing 262 = (25+1)2. But we need not even calculate (a+b)2 by adopting the following method.

262  = 252 + 26th odd number. i.e., 625 + 51 = 676

(a+b)2   = a2 + 2ab + b2

262   = (25 + 1 )2

           = 252  + 2 x 25 x 1 + 12

        = 625 + 50 + 1

       = 625 + 51 

        = 676

So, the answer is same. Now we shall look at an easiest method which will help you to calculate Squares Mentally. 

12   = 1 = 1

2 2  = 4 = 1+3

32   = 9 = 1+ 3 + 5

42   = 16 = 1 + 3 + 5 + 7

52   = 25 = 1 + 3 + 5 + 7 + 9

I mean, to get n2, you should add the FIRST n odd numbers. If we want 172, it will be the sum of the FIRST 17 odd numbers. 

n^th odd number is equal to (2n - 1)

Suppose you want to find out 62 knowing what 52 is, we can move from 52 to 62.

62 will be the sum of 1st 6 odd numbers. But the sum of the first 6 odd numbers can be written as " Sum of the first 5 odd numbers" + "Sixth odd number". Since we already know that the sum of the FIRST 5 odd numbers is 52 , i.e., 25, we need to add the sixth odd number, i mean (2x 6-1) = 11 to 25 to give us 62 = 36.

Similarly

312 = 900 + 31st Odd number = 900 + 61 = 961

362 = 1225 + 36th odd number = 1225 + 71 

                = 1296 (remember that 352 = 1225)

412 =  1600 + 81 = 1681

462 =  2025 + 91 = 2116

1262 = 15625 + 251 = 15876

1962 =  38025 + 391 = 38416

2162 =  46225 + 461 = 46656      Read More ☛

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00:59 - By Unknown 0

 Finding the Square of the given Number

In this post we shall discuss about the Shortcut of finding the Square of the Given Number. This is very easy and Simple method. All the you need to learn this method is little concentration and little patience. Remember friends, To make you understand the process clearly, we are posting the process in Step by Step process. But you have to do all these steps Mentally and you should Only write the Answer on Paper... 

Lets start with an example. Assume that you have to find the Square of the number 38.

So, the general process is 38²  multiplying 38 with itself. But this process take much time and effort. So, instead of calculating 38 X 38, just use the formula a² + 2ab + b²

But here you should apply a little trick. If you apply the above formula as it is... you cant get the answer... Just do as mentioned below.

Now, follow the below mentioned steps

Thats it... The answer is 1444.

Now lets have a look at another example. Lets see how to find out the square of 56. I mean 56².

Now treat 6 as a, and 5 as b and apply the a² + 2ab + b² formula as explained above.

• First of all, find the a ² i.e., 6 ² = 36. 

• Now find 2ab i.e., 2 X 6 X 5 = 60

• Now find the b ² i.e.,  5 ²  =  25, now  add this 25 to 6 = 31. And put this before the numbers 3 6. So it will become 3 1 3 6
• This is our answer 56 ² = 3136

Advantages of this Method :

• We are dividing the given number into two small numbers and then performing calculations on those smaller numbers. So that you can do calculations Mentally in very easy way. 
• Initially you may think that this method is little confusing. Just do 3 or 4 problems, so that you can find the advantage of this method.

If you have any Comments, please use the comments box below. Please tell your friends about us if you like this Post.. Good Day and Happy Reading :)

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00:58 - By Unknown 0

 Aptitude Shortcuts - Average

Aptitude Questions with Answers - Average

Before going to solve the below problems, first read the basics of Average here. 

1.  What is the average of first 21 multiples of 7 ?

1.  49
2. 147
3. 77
4. Cant be determined
5. None of these

2.  In a class, the average age of 30 boys is 13 years and the average of 20 girls is 12 years. what is the average age of the whole class?

1. 12.5 Yrs
2. 14.2 Yrs
3. 12.3 Yrs
4. 12.6 Yrs
5. None of these

3.  The average of seven numbers is 18. The average of first three numbers is 14 and the average of last three numbers is 19. What is the middle number?

1. 42
2. 57
3. 27
4. Cant be determined
5. None of these

4.  The average age of a class of 32 students is 16 yrs. if the teacher's age is also included, the average increases by one year. Find the age of the teacher

1. 49
2. 46
3. 48
4. Cant say
5. NOne of these

5.  The average weight of a class of 20 boys was calculated to be 58.4 kgs and it was later found that one weight was misread as 56 kg instead of 65 kg. What is the correct weight?

1. 67.4kg
2. 58.85kg
3. 57.75kg
4. 49.4kg
5. None of these

6.  The average weight of 8 persons increases by 1.5kg when a person weighting 65kg is replaced by a new person. What could be the weight of the new person?

1. 53kg
2. 58.85kg
3. 75kg
4. 77kg
5. None of these

7.  The average temperature for Monday, Tuesday and Wednsday is 36.3 degrees C. The average temperature for Tuesday, Wednesday and Thursday is 36.7 degrees C. if Monday's temperature recorded as 39 degrees c, find the Thursday's temperature ?

1. 40.2 degrees C
2. 39.2 degrees C
3. 41.4 degrees C
4. 40.4 degrees C
5. None of these

8.  A man spends Rs. 1600 per month on an average for the first three months, Rs 1550 for next four months and Rs. 1800 per month for the last five months and saves Rs . 5200 a year. What is his average monthly income?

1. Rs. 2050
2. Rs. 210
3. Rs. 1950
4. Rs. 2200
5. None of these

9.  There were 45 students in a hostel, if the numbers of students increased by 7, the expenses of the mess were increased by Rs. 39 per day while the average expenditure per head diminished by Re.1. What is the original expenditure of the mess?

1. Rs. 624
2. Rs. 562
3. Rs. 585
4. Rs. 598
5. None of these

10.  The average age of father and his two sons is 27 Years. Five years ago, the average age of the two sons was 12 Years. If the difference between the ages of the two sons is four years, what is the present age of the father?

1. 42
2. 48
3. 44
4. 47
5. None of these

Answers :

1.  Required average  = 7(1+2+....+21) / 21

                 =  (7/21)X ( (21x22) / 2 )  (because sum of first 21 natural numbers)

                  =  77

2.  Total age of 50 students 

                             =  (30X 13+20 X 12) = 630

                          Average =  630/50 = 12.6 Years

3.  The total of seven numbers = 7X18 = 126

          The total of first 3 and last 3 numbers is = 3 X 14+3 X 19 = 99

          So, the middle number is (126 - 99 ) = 27

4.  Total age of students is 32X16 = 512 Years

     Total age inclusive of teacher = 33X (16+1) = 561

      So, Teacher's age is 561-512 = 49 Yrs

                               There is a shortcut for these type of problems :

                                                    Teacher's age is 16+(33X1) = 49 Years :)

5.  Actual total weight is (20X 58.4 - 56 + 65) = 1177 Kgs

          Actual average weight is 1177/20 = 58.85kgs

6.  Let the average weight of 8 persons be 'x'

      years and the age of the new person be 'y'   years

                         so, (8x - 65 + y) / 8  = x+1.5

                          so, y = 77kgs                   Shortcut : 65 + 8X1.5 = 77kgs

7.  MOn + Tue + wed Temperature  =  3X36.3 = 108.9

        Tue + wed Temperature = 108.9 - 39 = 69.9

         Tue + wed + Thu temperature = 3X36.7 = 110.1

           So, Thursday's temperature = 110.1 - 69.9 = 40.2 degrees C

8.  Total expenditure for the first 3 months = 3X1600 = 4800

      Total expenditure for 4 months = 4X1550 = 6200

           Total expenditure for 5 months = 5X1800 = 9000

           Total expenditure and saving (which is income for one year)

               = 4800 + 6200 + 9000 + 5200 = Rs. 25200

              So, Average monthly income = 25200/12  =  Rs. 2100

9.  Let the original expenditure be Rs.x

     Original average expenditure = X/45

            New average expenditure = (x+39)/52

            So (x/45) - ((x+39) / 52) = 1  so x = 585

               so, original expenditure is Rs 585

10.  The total present age of father and two sons is 3S27 = 81 yrs

           The total present age of sons is (12+5) X 2 = 34Years

           so, present age of father is 81 - 34 = 47 yrs

                                                               

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Math Trick Speed Math Math Shortcuts Aptitude Shortcut Tricks for Bank Exams ssc

00:56 - By Unknown 0

 Average

• Average of Given Items :
  
  

Average =		Sum of the given Items
		Number of those Items




• Average Speed :
  
  
  Suppose a Person covers a certain distance at X kmph and an equal distance at Ykmph.
  
  Then, the average speed for the whole journey will be (2XY)/(X+Y) Kmph

✒ Lets do some problems on Average (Click Here)

00:54 - By Unknown 0

 Mathematical Operations

Friends, today we shall discuss another important model of Reasoning Section Mathematical Operations. Of course, even these questions are asked under reasoning section these are mainly dependent on mathematical operations. All calculations will be performed using BODMAS Rule. We can divide these problems in three types.Those are,

1. Symbol Substitution
2. Mathematical Logic
3. Interchange of Signs & Numbers

Now lets discuss these types with detailed examples.

1st Type : Symbol Substitution

In this type, to find the value of the given expression you should replace the symbols by mathematical operations. Then apply the BODMAS Rule (i.e., Brackets, of, Division, Multiplication, Addition, Subtraction).

For the detailed understanding, lets have a look at an example

Example : 42% (9-2) + 6 x 3 - 4

Solution :
42 % (9-2) + 6 x 3 - 4
= 42 % 7 + 6 x 3 - 4
= 6 + 6 x 3 - 4
= 6 + 18 - 4
= 24 - 4 = 20


2nd Type : Mathematical Logic

In this type, they just give you some logical statements followed by some mathematical operations. You should solve the given mathematical operations with the help of given logical statements.

Example : If % stands for greater than, x stands for addition, + stands for division, - stands for equal to, > stands for multiplication, = stands for less than, < stands for minus, then which of the following alternatives is correct ?

1. 3 + 2 < 4 % 6 > 3 x 2 
2. 3 x 2 < 4 % 6 + 3 < 2
3. 3 > 2 < 4 - 6 x 3 x 2 
4. 3 x 2 x 4 = 6 + 3 < 2 

Usually for these type of questions people tend to make direct decisions with the help of given statements. But confusion arises when you tried to  coordinate given mathematical symbols with the logical statements mentioned above. It is because lack of coordination between eyes and brain. So, to avoid confusion you just should make a note of given logical statements in simple mathematical equations format. So that it will be easier for your eye to pick the correct operation.

by using above logical statements, you can use a rough sketch like this,

now you can calculate the values of given operations mentally by picking the right hand side symbols instead of left hand side. Or you just can replace the symbols by taking the reference of above rough sketch.

Solution : Using the symbols correctly as given in the above operations, for the option (2) you will get, 3+2-4>6%3-2. Using BODMAS rule in this inequality, we will get 5-4>2-2 or, 1>0, which is correct.

3rd Type : Interchange of Signs & Numbers

In this type they just give you some random set of numbers and mathematical symbols. And then ask you to choose the correct of symbols to fit the given equation.

Example : Select the correct set of symbols which will fit in the given equation 5  0  3  5  = 20

1. x, x, x
2. =, +, x
3. x, +, x
4. +, -, x

Solution : Using the set of symbols given in options, you will get the Option 2 as the correct answer.

Using the set of symbols given in option 2, you will get the equation,

5-0+3x5 = 20

Using BODMAS rule you will get,

5-0+15=20 or 20-0 = 20 which is correct

Now lets have a look at some examples,

1. '<' means subtraction, '>' means Addition, '+' means Multiplication and '$' means Division. Then what will be the value of 27 > 81 $ 9 < 6 ?

Solution :

27+81%9-6 = 27+9-6 = 36-6 = 30

2. If % means +, - means x, x means -, + means %, then what will be the value of 15 - 2 % 900 + 90 x 100 ?

Solution :

15x2+900%90-100 = 30 + 10 - 100 = -60

3. Q means Addition, J means Multiplication, T means Subtraction, K means Division then what is the value of 30K2Q3J6T5 ?

Solution :

3%2+3x6-5  =  15 + 18 - 5 = 28

4. P means x, R means +, T means %, S means -, then what will be the value of 18T3P9S8R6 ?

Solution :

By substituting the given symbols, you will get the answer 52

5. P means Multiplication, T means Subtraction, M means Addition, B means Division then what will be the value of 28B7P8T6M4 ?

Solution :

28 % 7 x8 - 6 + 4 

= 4 x 8 - 6 + 4 

= 32 - 6 + 4 

= 36 - 6 = 30

Thats all for now friends. In our next post we shall discuss some more problems of Mathematical Operations with higher difficulty level. Happy Reading :)

03:11 - By Unknown 0

 BODMAS

• In The simplifications of numerical questions, the order of mathematical operations to be followed is given in the abbreviated form : "BODMAS", where

• B stands for 'Bracket'
• O stands for 'Of'
• D stands for 'Division'
• M stands for 'Multiplication'
• A stands for 'Addition' and
• S stands for 'Subtraction'.
• Order to be followed by in case of brackets is as follows :
• (   )  : Small Brackets,
• {  } : Curly Brackets   and
• [  ]  : Square Brackets

00:53 - By Unknown 0

 Problems on LCM and HCF

☛ Read Basics of LCM and HCF Here, Before starting Practice 


1. Find the least number which when increased by 4 is exactly divisible by 8, 16, 24, 30 and 32 ?
a)   480
b)   484
c)   476
d)   472
e)   None of these

2. What is the greatest number of five digits which when 3769 is added to it will be exactly divisible by 5, 6 , 10, 12, 15 and 18 ?
a)   4309
b)   99459
c)   100539
d)   99911
e)   None of These

3. Find the minimum number of square tiles required to pave the floor of a room of 2m 50cm long and 1m 50cm broad ?
a)    50
b)   750
c)   45
d)   15
e)   None of these

4. Five bells toll together at the intervals of 5, 6, 8, 12 and 20 seconds respectively. Find the number of times they toll together in one hour's time (Inclusive of the toll at the beginning)
a)   120
b)   31
c)   30
d)   5
e)   None of These

5. A milk man has three different kinds of milk 493liters, 551 liters and 435 liters. Find the minimum number of equal size containers required to store all the milk without mixing.
a)   29
b)   51
c)   58
d)   49
e)    None of these

6. The circumference of the front and back wheels of a vehicle are 6 3/14 m and 8 1/18 m respectively. At any given moment, a chalk mark is put on the point of contact of each wheel with the ground. Find the distance traveled by the vehicle so that both the chalk marks are again on the ground at the same time
a)   217.5 m
b)   435 m
c)    412m
d)   419m
e)   None of these

7. The LCM of two numbers is 28 times of their HCF. The sum of their LCM and HCF is 1740. If one of the numbers is 420, the other number is
a)   150
b)   225
c)   180
d)   240
e)   None of these

8. Two persons A and B walk around a circular track whose radius is 1.4 km. A walks at a speed of 176 meters per minute while B walks at a speed of 110 meters per minute. if they both start at the same time, from the same point and walk in the same direction, at what interval of time would they both be at the same starting point again? (in Hours)
a)   6 2/3
b)   2 1/3
c)   5 1/4
d)   3 2/3
e) None of these

9. Find the least number which when divided by 8, 9, 15, 24, 32 and 36 leaves remainders 3, 4, 10, 19, 27 and 31 respectively?
a)   2880
b)   2885
c)   2974
d)   2875
e)   None of these

10. Find the greatest number which when divide 357, 192 and 252 leaves same remainder in each case
a)   45
b)   1
c)   15
d)   Cant be determinde
e)    None of these




Solutions :


1.  LCM of 8, 16, 24, 30 and 32 is 480
     So, Required number is 480 - 4  =  476

2.  LCM of 5, 6, 10, 12 and 18 is 540
      On dividing  (99999 + 3769)  by 540, the remainder is  88
      So, the required number is 99999 - 88 = 99911

3.  HCF of 250 cm and 150 cm is 50 cm, which is the side of the tile

     So, the required number of tiles  =  (250 X 150) / (50X50)  =  15

4.  Time after which all the bells tol together is the LCM of 5, 6, 8, 12 and 20.
       i.e., 120 seconds = 20   minutes
      The number of times they toll together in one hour =  60/2  = 30 + 1 (beginning tone)
         So, the answer is 31

5.  As minimum number of containers are required, the size of the container should be maximum and the size is also equal. so size of the container will be HCF of 493, 551 and 435 i.e., 29
     So, required number of containers is =  (493+551+435) / 29  =  51

6.  The required distance is the LCM of 6 3/14  and 8 1/18
      LCM of 6 3/14 and 8 1/18 =  LCM (87/14, 145/18)
      =>  LCM(87,145) / HCF (14,18)  =  435/2  =  217.5m
7.  LCM =  28 HCF
      LCM + HCF = 1740
       =>  28  HCF + HCF = 1740
        HCF =  1740/29  =  60 and LCM = 28X60
      if a and b are two numbers, then
       LCM of (a & b) X HCF of (a & b) = aXb
      28X60X60 = x X 420  = >  x = 240
         So, the other number is 240


8.   Circumference of the track is 2πr
        = 2 X (22/7) X 1400m = 8800 m


         Time taken by A to complete one round
           =  (8800m) / (176m/min)  =  (8800X60)/(176) =  3000 sec

        Time taken by B to complete one round

          = (8800) / (110m/min)  =  (8800X60) / (110)  =  4800 sec

         Time they meet together at the starting point is LCM of 3000 and 4800 sec

           i.e., 24000 sec   =  6 2/3 hours

            So, they meet at the starting point after 6 2/3 hours

9.   We can observe that the difference between the numbers and their remainders is same 

         i.e., 8-3  =  9-4  =  15-10  =  24-19  =  32-27  =  36-31  =  5

        So, required answer is 

                                                LCM (8, 9, 15, 24, 32, 36) - 5

                    = > 2880 - 5 = 2875

10.   Required answer is 

               HCF (357-192, 192-252, 357-252)

                 = > HCF (165, 60, 105) = 15

00:52 - By Unknown 0