aptitude

click on the topics below to go to the particular topic.

1) shortcuts on addition,subtraction,multiplication,division
2)tricks for finding the unit digit of a number with exponent
3) Averages
4)Ratios and proportions

    INTRODUCTION TO APTITUDE:     

    The quantitative aptitude test measures the numerical ability and accuracy in mathematical calculations. The  questions range from purely numeric calculations  to  problems of arithmetic reasoning, graph and table reading, percentage analysis, categorization and quantitative analysis The quantitative aptitude  test measures  the numerical ability and accuracy in mathematical calculations. The questions range from purely numeric calculations to problems of arithmetic reasoning, graph and table reading, percentage analysis, categorization and quantitative analysis. If you are good in aptitude  and   logical reasoning you can easily manage GATE ,campus placements ,GRE ,bank exams and many more.

REASONING:                                                  


      Logical reasoning is the process which uses arguments, statements,   premises and  axioms  to  define  weather  a statement  is true or false, resulting in a logical  or illogical reasoning. In today’s logical reasoning three different types of reasoning can be distinguished, known as deductive reasoning, inductive reasoning and abductive reasoning based on respectively deduction, induction and abduction. These logical reasoning questions are some what confusing and  tricky.From next post onwards we are going to see problems on aptitude and resoning.

shortcuts on addition,subtraction,division,multiplication

 In this post, i am going to tell about shortcuts on addition,subtraction,multiplication and division.

SHORTCUTS IN ADDITION:

Addition of numbers close to multiples of ten (e.g. 19, 29, 89, 99 etc.)
116 + 39
= 116 + (40 - 1)
= 116 + 40 - 1  ---> add 40 then subtract 1        = 156 - 1
= 155

116 + 97
= 116 + (100 - 3)
= 116 + 100 - 3   ---> add 100 then subtract 3
= 216 - 3
= 213

Addition of decimals
12.5 + 6.25
= (12 + 0.5) + (6 + 0.25)
= 12 + 6 + 0.5 + 0.25   ---> add the integers then the decimals
= 18 + 0.5 + 0.25
= 18.75

SHORTCUTS IN SUBTRACTION:

Subtraction by numbers close to 100, 200, 300, 400, etc.
250 - 96
= 250 - (100 - 4)
= 250 - 100 + 4    ---> subtract 100 then add 4  = 150 + 4
= 154

250 - 196                          = 250 - (200 - 4)

= 250 - 200 + 4    ---> subtract 200 then add 4
= 50 + 4
= 54

216 - 61
= 216 - (100 - 39)
= 216 - 100 + 39
= 116 + (40 - 1)  ---> now the operation is addition, which is much easier
= 156 - 1
= 155

Subtraction of decimals
47 - 9.9
= 47 - (9 + 0.9) ---> "double subtraction"
= 47 - 9 - 0.9   ---> subtract the integer first then the decimal
= 38 - 0.9
= 37.1

18.3 - 0.8
= 18 + 0.3 - 0.8
= (18 - 0.8) + 0.3  ---> subtract 0.8 from 18 first
= 17.2 + 0.3
= 17.5

SHORTCUTS IN MULTIPLICATION:


12 x 15
= 12 x 5 x 3
= 60 x 3
= 180

Multiplication by distribution
12 x 17
= (12 x 10) + (12 x 7) ---> 12 is multiplied to both 10 & 7
= 120 + 84
= 204

Multiplication by "giving and taking"      
12 x 47
= 12 x (50 - 3)
= (12 x 50) - (12 x 3)
= 600 - 36
= 564

Multiplication by 5 --> take the half(0.5) then multiply by 10
428 x 5
= (428 x 1/2) x 10 = 428 x 0.5 x 10
= 214 x 10
= 2140

Multiplication by 10  ---> just move the decimal point one place to the right
14 x 10
= 140   ---> added one zero

Multiplication by 50 ---> take the half(0.5) then multiply by 100
18 x 50
= (18/2) x 100 = 18 x 0.5 x 100
= 9 x 100
= 900

Multiplication by 100 ---> move the decimal point two places to the right
42 x 100
= 4200  ---> added two zeroes

Multiplication by 500 ---> take the half(0.5) then multiply by 1000
21 x 500
= 21/2 x 1000
= 10.5 x 1000
= 10500

Multiplication by 25 ---> use the analogy $1 = 4 x 25 cents
25 x 14
= (25 x 10) + (25 x 4) ---> 250 + 100 ---> $2.50 + $1
= 350

Multiplication by 25 ---> divide by 4 then multiply by 100
36 x 25
= (36/4) x 100
= 9 x 100
= 900 

Multiplication by 11 if sum of digits is less than 10
72 x 11
= 7_2  ---> the middle term = 7 + 2 = 9
= place the middle term 9 between 7 & 2
= 792

Multiplication by 11 if sum of digits is greater than 10
87 x 11
= 8_7  ---> the middle term = 8 + 7 = 15
because the middle term is greater than 10, use 5 then
add 1 to the first term 8, which leads to the answer of
= 957

Multiplication of 37 by the 3, 6, 9 until 27 series of numbers --> the "triple effect"           
solve 37 x 3               
multiply 7 by 3 = 21, knowing the last digit (1), just combine two more 1's giving the triple digit answer 111           
               
solve 37 x 9               
multiply 7 by 9 = 63, knowing the last digit (3), just combine two more 3's giving the triple digit answer 333                               
solve 37 x 21               
multiply 7 by 21 = 147, knowing the last digit (7), just combine two more 7's giving the triple digit answer 777                           

Multiplication of the "dozen teens" group of numbers --
(i.e. 12, 13, 14, 15, 16, 17, 18, 19) by ANY of the numbers within the group:
solve 14 x 17               
4 x 7 = 28;  remember 8, carry 2               
14 + 7 = 21               
add 21 to whats is carried (2)               
giving the result 23               
form the answer by combinig 23 to what is remembered (8)               
giving the answer 238   

Multiplication of numbers ending in 5 with difference of 10
45 x 35 
first term = [(4 + 1) x 3] = 15; (4 is the first digit of 45 and 3 is the first digit of 35 --> add 1 to the higher first digit which is 4 in this case, then multiply the result by 3, giving 15)
last term = 75
combining the first term and last term,
= 1575

75 x 85
first term = (8 + 1) x 7 = 63
last term = 75
combining first and last terms,
= 6375

15 x 25
= 375

Multiplication of numbers ending in 5 with the same first terms (square of a number)
25 x 25
first term = (2 + 1) x 2 = 6
last term = 25
answer = 625  ---> square of 25

75 x 75
first term = (7 + 1) x 7 = 56
last term = 25
answer = 5625 ---> 75 squared



SHORTCUTS IN DIVISION:

Division by parts ---> imagine dividing $874 between two persons
874/2
= 800/2 + 74/2
= 400 + 37
= 437

Division using the factors of the divisor: "double division"
70/14
= (70/7)/2 ---> 7 and 2 are the factors of 14
= 10/2
= 5

Division by using fractions:
132/2
= (100/2 + 32/2) ---> break down into two fractions
= (50 + 16)
= 66

Division by 5 ---> divide by 100 then multiply by 20
1400/5
= (1400/100) x 20
= 14 x 20
= 280

Division by 10  ---> move the decimal point one place to the left
0.5/10
= 0.05  ---> 5% is 50% divided by ten

Division by 50 ---> divide by 100 then multiply by 2
2100/50
= (2100/100) x 2
= 21 x 2
= 42

700/50
= (700/100) x 2
= 7 x 2
= 14

Division by 100 ---> move the decimal point two places to the left
25/100
= 0.25

Division by 500 ---> divide by 100 then multiply by 0.2
17/500
= (17/100) x 0.2
= 0.17 x 0.2
= 0.034

Division by 25 ---> divide by 100 then multiply by 4
500/25
= (500/100) x 4
= 5 x 4
= 20

750/25
= (750/100) x 4
= 7.5 x 2 x 2
= 30
                  thats all for now .next post i will give more tips on aptitude.

          

tricks for finding the units digit of a number with exponent.

  In this  post i am going to give the tricks  to find the  units digit of a given number with exponent.For this you have to know about the cyclicity of a given number.
     Let’s take an example to understand this:

Example 1 :  find the unit digit of 3⁵⁶.
Solution :  Now it’s a big term so we cannot find the last digit by doing 3 x 3 x 3 x 3 x 3……. 56 times so we use the concept of cyclicity
Step 1 : 3¹ = 3
3² = 9
3³ = 27
3⁴ = 81
3⁵ = 243
So now pay attention to the last digits we saw that the last digit repeats itself after a cycle of 4 and the cycle is 3 ,9,7,1,this is called the cyclicity of any number ,therefore when we need to find the unit digit of any number like 3ⁿ we just need to find the number on which the cycle of last digit ends .  And in the next step we will divide the power with the cyclicity.

• if the remainder will be 1 then the unit digit will be 3
• if the remainder will be 2 then the unit digit will be 9
• if the remainder will be 3 then the unit digit will be 7
• if the remainder will be 0 then the unit digit will be 1

This is all about the cyclicity
Why the power is divided by number 4.
We will divide the power with 4 because cycle repeat itself after 4 values, and also we need to find the remainder which tells us the required values to complete the next cycle.
Now the main question was that how much is the last digit of 3⁵⁴
So we know the cycle repeats itself after 4 so we will divide the 54 with 4 ,so on dividing 54 by 4 the remainder becomes 2 .Now as we discussed above if the remainder is 2 the last digit would be 9, so in the end the unit digit of 3⁵⁴ is 9.
(Type 2 where power of 2 and 3 digits number is to be considered)
Example 2 : What will be the unit digit of 24⁴⁵ or 347⁴⁵
Solution :
Lets take some example to understand it very clearly
We know that unit digit of 3 x 3 = 9
And the unit digit of 453 x 543 = 9
The main purpose of the above expression is that the unit digit of any multiplication depends upon the unit digit of numbers , whatever is the number big or small the unit digit always depends upon the multiplication of the last digit .
So the last digit of 24⁴⁵ can be found by 4⁴⁵
So the Cyclicity of 4 is 2 because the cycle of last digit repeats after two values
4¹ = 4
4² =16
4³ = 64
So when we divide 45 with 2 then we will get the remainder as 1 and the last digit will be 4
Now come to the case number second unit digit of 347⁴⁵
The unit digit of this number can be find by the same method
The Cyclicity of 7 is  4
7¹  = 7
7² = 49
7³ = 343
7⁴ = 2401
So on dividing 45 with 4 , 1 will be the remainder and the last digit would be 7
Type 3 ( where _p^qr is to be considered )
What will be the last digit of ₁₂²³⁴⁵
To find the last digit of this type of number we  will start the question from the base the base is 12. It means we will see the cyclicity of 2 because the last digit is depends upon the unit digit of 12. Lets do it step vise step
Before the steps we will write the last digits of
2¹ = 2
2² = 4
2³ = 8
2⁴ = 6
2⁵ = 2
Step 1: Now we know that cyclicity of last digit of 12 i.e 2 is 4 , hence the divide the power of 12 i.e 23⁴⁵  with 4
Step 2:  Now the remainder 23⁴⁵ /4 will determine the last digit.
Step 3:  The remainder will be 3 because we can write remainder of 23 /4 = 3 or -1 and -1 ⁴⁵ / 4 will give us remainder as -1 or 3
Hence in the end the last digit of ₁₂²³⁴⁵ is nothing but 12³   = 8
Practice question:
1) Find the unit digit of  ₃₂²⁵⁴⁵ ? 
Answer = 2
2)



        754
 5238           =         unit digit  is 4
                                                                                                                                                             
the explanation for the above  is  the cyclicity of the 8 is 4 and 754/4 the remainder is 2 .the unit digit of  8^2 is 4.

3)
                    899
                456
            738        
      783
2356                               =     unit digit is  6.
the cyclicity of 6 is same of the number .so,unit digit anything power of  6 is 6.



i think you all got the clarity about the finding the unit digit of  any  number  with exponents.

Ratios and proportions

Introduction:

• The ratio of numbers A and B can be expressed as the ratio of A to B.
• The numbers A and B are sometimes called terms with A being the antecedent and B being the consequent.
• The proportion expressing the equality of the ratios A:B and C:D is written A:B = C:D or A:B::C:D. 

• A, B, C and D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means. The equality of three or more proportions is called a continued proportion.

•  Product of extremes = Product of Means
  To solve proportion, we use above principal, A single term in the proportion is called proportional.

  “a” is the 1st proportional.
  “b” is the 2nd proportional.
  “c” is the 3rd proportional.
  “d” is the 4th proportional.1

• EX:Find the third proportional in 3:2=x:8 ?
• sol;product of extremes = product of means

                      2x = 16     

         Points to remember:

•  1/a:1/b =b:a
• 1/a:1/b:1/c=bc:ac:ab

Exercise problems:
1)the two terms are in the ratio  5:7.the sum of the two terms are 108. find the two numbers?
 a)
      let the two numbers are 5x,7x.
sum of the two numbers are 5x+7x=108
                                              12x=108
                                                 x=9
                       1  part   = 9
                      5   part's = 5*9 = 45
                      7 part.s=  7*9  =63

2)the length and breadth of rectangle are in the ratio of 9:5. the length exceeds the breadth by 280 meters.find the perimeter of the rectangle?
a)           l:b    =    9:5
            l-b=9-5=4
    in the question the 4  part is given as 280.
                       1 part =  70
                        length=9*70=630 meters
                       breadth=5*70=350 meters
                       perimeter=2(l+b)=  2(630+350)
                                                 =1960

3) If a sum of Rs.11400 is divide among A,B and C in the ratio of 3:4:5. find the share of  C?
 a)  
      A:B:C=  3:4:5
    let  the share of A,B and C are 3x,4x,5x,respectively
             3x+4x+5x     =  11400
                        12x    = 11400
                          x     =  950
the share of c is 5*950=4750.

4)The two numbers are in the  ratio 17:18 .when 8 is added to both the numbers the ratio becomes 19:20.
Find the smaller number?
A)
        let the numbers be   A:B=  17:18
     when 8 is added the new ratio becomes A:B=19:20      

the change in the ratio is 2 .So     2 parts =8
                                                 1 part  = 4
                                  the smallest number is 17*4=68.

5) If a sum of Rs. 2460 is divided  among A,B and C such that A gets 7/6 of what B gets and B gets 4/5 of what C gets .find the share of B?
A)
        A=7/6 B
        B=4/5 c
     
        A:B      =  7 : 6
            B:C  =      4 : 5
      A : B: C = 14:12:15.
the trick is very simple. take the L.C.M of 6 and 4 is 12.
    write the 12 in place of B
multiply 7 with 2 because to get 12 ,6 is multiplied with 2 and multiply 5 with 3.
the A:B:C   =  14:12:15
       41x=2460
          x= 60
the share of B is 12x=12*60=720

Averages

In this post, i am going to tell about the shortcuts in averages.

Points to remember in Averages:

Average =  ( Sum of observations ) . Numbers of observations

• An average, or more accurately an arithmetic mean is, in general terms, the sum of n different data divided by n.

Average =  Total of data No. of data

Total = Average × No. of data

•  If the value of each item in a group is increased/decreased by the same value x, then the average of the group also increases/decreases by x. For instance, if the income of each person in a group increases by 15, the average income of the group also increases by Rs. 15.

This is valid only when the value of each item increases/decreases by the same amount.

•  If the average age of group of people is x years, then their average age after n years will be (x + n) and their average age n years ago would have been (x – n) years.
This is because with each passing year, each person’s age increases by 1 and vice versa.

•  If the value of each item in a group is multiplied/divided by the same value x (where x ≠ 0 in the case of division), then the average of the group also gets multiplied/divided by x.

•  The average of a group always lies between the smallest value and the longest value in that group.

• If the terms in Arithmetic progression and the number of terms is odd,then the average will be equal to middle term.
• EX: 

the average of 4 ,6,8 is    6.

• If the terms are in Arithmetic progression and the number of term are even ,the average is equal to average of  middle term.
• Ex:

the average of 4,6,8,10 is (6+8)/2     =7

• the average of  N odd numbers is N.
• the average of N even numbers is N+1.

exercise problems:

1)Ex. 1:  The average of 11 results is 50. If the average of first six results is 49 and that of last six is 52, find the sixth result.

Sol. :  The total of 11 results = 11 × 50 = 550
The total of first 6 results = 6 × 49 = 294
The total of last 6 results = 6 × 52 = 312
The 6th result is common on both;
∴ Sixth result = 294 + 312 – 500 = 56

Ex. 2 :  The average age of a family of 6 members is 22 years. If the age of the youngest member be 7 years, then what was the average age of the family at the birth of the youngest member?
 

Sol. :  Total ages of all members = 6 × 22 = 132 years.
7 years ago, total sum of ages = 132 – (6 × 7) = 90 years.
But at that time there were 5 members in the family.
∴ Average at that time = 90 ÷ 5 = 18 years.

Ex. 3:  A man bought 13 shirts of Rs. 50 each, 15 pants of Rs. 60 each and 12 pairs of shoes at Rs. 65 a pair. Find the average value of each article.
Sol. : By the use of averge formula,  

Average =	13 × 50 + 15 × 60 + 12 × 65	= Rs. 58.25
	13 + 15 + 12

Ex. 4 :  The average score of a cricketer in two matches is 27 and in three other matches is 32. Then find the average score in all the five matches.

Sol. : By the use of average formula,

Average in 5 matches =	2 × 27 + 3 × 32	=	54 + 96	= 30.
	2 + 3		5

Ex. 5 :  The average of 11 results is 30, that of the first five is 25 and that of the last five is 28. Find the value of the 6th number.

Sol. : By the use of average formula,

6th number = Total of 11 results – (Total of first five results + Total of last five results)
= 11 × 30 – (5 × 25 + 5 × 28)
= 330 – 265 = 65

Ex. 6: A batsman in his 17th innings makes a score of 85, and thereby increases his average by 3. What is this average after 17 innings?
Sol. :  Let the average after 16th innings be x, then

16x + 85 = 17 (x + 3) = Total score after 17th innings.

∴     x = 85 – 51 = 34

∴    average after 17 innings = x + 3 = 34 + 3 = 37.

Ex. 7: A cricketer has completed 10 innings and his average is 21.5 runs. How many runs must he make in his next innings so as to raise his average to 24?
Sol. :  Total of 10 innings = 21.5 × 10 = 215

Suppose he needs a score of x in 11th innings; then

average in 11 innings =	215 + x	= 24
	11

or, x = 264 – 215 = 49

Ex8: the average of 9 consecutive even numbers is 68.what is the product of the smallest and the largest of number

Sol:

                n=9

             the numbers are 

              60   62  64   66  68  70   72  74 76 

             product of smallest and largest =60*76       =4560.

EX 9:the  average of set of 24 numbers is 32.if each number is tripled then what will be the resultant age?

from the above points to remember 

                    

                                      =32*3

                                      =96

EX:In a examination students average marks were 71 per parer.if they had obtain 15 more marks in English ,9 more marks in science ,its average marks per paper will be 74.find the number of papers in the exam.

sol:

          let the number of papers is n

             

          the  number of paper in exam=(71n+15+9)/n=74

                                                       n  =     8.