## Monday, December 24, 2012

### MTS/ POSTMAN / GDS TO PA / POSTAL ASSISTANT MATERIAL

NUMBER SYSTEM-1

Numeral: In Hindu Arabic system, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number.

A group of digits, denoting a number is called a numeral.

TYPES OF NUMBERS

1. Natural Numbers: Counting numbers 1, 2, 3, 4, 5,….. are called natural numbers.

2. Whole Numbers: All counting numbers together with zero form the set of whole numbers. Thus,

(i) 0 is the only whole number which is not a natural number.

(ii) Every natural number is a whole number.

3. Integers: All natural numbers, 0 and negatives of counting numbers i.e., {…, -3,-2,-1, 0, 1, 2, 3,…..} together form the set of integers.

(i) Positive Integers: {1, 2, 3, 4, …..} is the set of all positive integers.

(ii) Negative Integers{- 1, – 2, – 3,…..} is the set of all negative integers.

(iii) Non-Positive and Non-Negative Integers: 0 is neither positive nor negative. So, {0, 1, 2, 3,….} represents the set of non-negative integers, while {0, -1,-2,-3,…..} represents the set of non-positive integers.

4.  Even Numbers: A number divisible by 2 is called an even number, e.g., 2, 4, 6, 8, 10, etc.

5.  Odd Numbers: A number not divisible by 2 is called an odd number. e.g., 1, 3, 5, 7, 9, 11, etc.

6.  Prime Numbers: A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.

Prime numbers up to 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,  53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Prime numbers Greater than 100: Let p be a given number greater than 100.

To find out whether it is prime or not, we use the following method :

Find a whole number nearly greater than the square root of p. Let k > *jp. Test whether p is divisible by any prime number less than k. If yes, then p is not prime. Otherwise, p is prime.

e.g,, We have to find whether 191 is a prime number or not. Now, 14 > V191.
Prime numbers less than 14 are 2, 3, 5, 7, 11, 13.

191 is not divisible by any of them. So, 191 is a prime number.

7. Composite Numbers: Numbers greater than 1 which are not prime, are known as composite numbers, e.g., 4, 6, 8, 9, 10, 12.
Note:
(i) 1 is neither prime nor composite.
(ii) 2 is the only even number which is prime.
(iii) There are 25 prime numbers between 1 and 100.

8.  Co-primes: Two numbers a and b are said to be co-primes, if their H.C.F. is 1. e.g., (2, 3), (4, 5), (7, 9), (8, 11), etc. are co-primes
Basic Formulas:
1. (a + b)2 = a2 + b2 + 2ab
2. (a – b)2 = a2 + b2 – 2ab
3. (a + b)2 – (a – b)2 = 4ab
4. (a + b)2 + (a – b)2 = 2 (a2 + b2)
5.  (a2 - b2) = (a + b) (a – b)
6.  (a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ca)
7.  (a3 + b3) = (a +b) (a2 – ab + b2)
8. (a3 – b3) = (a – b) (a2 + ab + b2)
9. (a3 + b3 + c3 -3abc) = (a + b + c) (a2 + b2 + c2 – ab – bc – ca)
10. If a + b + c = 0, then a3 + b3 + c3 = 3abc.

Natural numbers (Positive integers) :                  1, 2, 3, 4,....

Whole numbers (Non-negative integers) :          0, 1, 2, 3,....

Negative integers:  − 1, − 2, − 3,....

Integers: ...., −2, − 1, 0,1, 2,.....

Even numbers: ...., − 2, 0, 2, 4,.... (2n)

Odd numbers: ....., −3, −1,1, 3,... (2n + 1)

Prime numbers (exactly 2 factors) : 2, 3, 5, 7, 11,....

Composite numbers (more than 2 factors) : 4, 6, 8, 9,10,....

Perfect numbers (Sum of all the factors is twice the number) : 6, 28, 496, ..

Co-primes (relative primes) (Two numbers whose HCF is 1) : 2 & 3, 8 & 9,..

Twin primes (Two prime numbers whose difference is 2) : 3 & 5, 5 & 7,.

Rational numbers (qp form, p & q are integers, q ≠ 0 ) : 32, 23 , 2, 0.5,..

Irrational numbers (which cannot be represented in the form of a fraction)
:2, 35, e, π , 0.231764735...)

Decimal Numbers: 0.2, 1.25, 0.3333….

Terminating Decimal Numbers (which terminates): 0.23, 2.374, ….

Non Terminating Decimal Numbers (Which doesn’t terminate): 0.33…., 0.121212…, 0.2317…

Pure Recurring Decimals (All the figures after decimal point repeats) : 0.33…., 0.121212…

Mixed Recurring Decimals (Some figures after decimal repeats): 0.245555…, 0. 2343434…

Pure recurring decimal to fraction conversion

Ex. 0.ababab ….. = 99ab

Mixed recurring decimal to fraction conversion

Ex. 0.abcbcbc … = 990aabc−

→ 1 is the neither prime, nor composite.
→ 2 is the only even prime.
→ If x & y are two integers, then (x + y) ! is divisible by x !. y!
→ The product of ‘n’ consecutive numbers is divisible by n!.
→ (xn + yn) is divisible by (x + y), when n is an odd number.
→ (xn – yn) is divisible by (x + y)(x – y), when n is an even number.
→ (xn – yn) is divisible by (x – y), when n is an odd number.

9. Some Important points:

→ Every number ‘N’ can be written as N = ap × bq × cr …. . (a, b, c,…. are prime numbers.)
→ If p, q, r ……. are even, ‘N’ is a perfect square.
→ If p, q, r are multiples of 3, ‘N’ is a perfect cube.
→ Number of factors of N = (p+ 1) (q + 1) (r + 1) …..
→ Sum of the factors of N = ((a^(p+1)-1)/(a-1))((b^(q+1)-1)/(b-1)).........
→ Number of co – primes of ‘N’ , which are less than N = N (1 – 1/a) (1 – 1/b)….
→ Sum of these co-primes = N/2 × N (1 – 1/a) (1 – 1/b)….
→ Numbers of ways of writing ‘N’ as a product of 2 co-primes = 2 n – 1 , n is the number of different prime numbers in ‘N’
→ If n is a prime number, (n –1)! +1 is divisible by n.
→ If n is a natural number and p is a prime number, then (np –n) is divisible by p
→ The last digit of the powers of 2,3,7,8 repeats after every 4th power.
→ The last digit of any power of 0,1, 5,6 is always 0,1,5, 6 respectively.
→ The last digit of the powers of 4 and 9 repeats after every 2nd power.
→ The last two digits of any number is the remainder obtained by dividing that number by 100.
1. Decimal Fractions:
Fractions in which denominators are powers of 10 are known as decimal fractions.
 Thus, 1 = 1 tenth = .1; 1 = 1 hundredth = .01; 10 100

 99 = 99 hundredths = .99; 7 = 7 thousandths = .007, etc.; 100 1000
1. Conversion of a Decimal into Vulgar Fraction:
Put 1 in the denominator under the decimal point and annex with it as many zeros as is the number of digits after the decimal point. Now, remove the decimal point and reduce the fraction to its lowest terms.
 Thus, 0.25 = 25 = 1 ;       2.008 = 2008 = 251 . 100 4 1000 125
1. Annexing Zeros and Removing Decimal Signs:
Annexing zeros to the extreme right of a decimal fraction does not change its value. Thus, 0.8 = 0.80 = 0.800, etc.
If numerator and denominator of a fraction contain the same number of decimal places, then we remove the decimal sign.
 Thus, 1.84 = 184 = 8 . 2.99 299 13
1. Operations on Decimal Fractions:
1. Addition and Subtraction of Decimal Fractions: The given numbers are so placed under each other that the decimal points lie in one column. The numbers so arranged can now be added or subtracted in the usual way.
2. Multiplication of a Decimal Fraction By a Power of 10: Shift the decimal point to the right by as many places as is the power of 10.
Thus, 5.9632 x 100 = 596.32;   0.073 x 10000 = 730.
1. Multiplication of Decimal Fractions: Multiply the given numbers considering them without decimal point. Now, in the product, the decimal point is marked off to obtain as many places of decimal as is the sum of the number of decimal places in the given numbers.
Suppose we have to find the product (.2 x 0.02 x .002).
Now, 2 x 2 x 2 = 8. Sum of decimal places = (1 + 2 + 3) = 6.
.2 x .02 x .002 = .000008
1. Dividing a Decimal Fraction By a Counting Number: Divide the given number without considering the decimal point, by the given counting number. Now, in the quotient, put the decimal point to give as many places of decimal as there are in the dividend.
Suppose we have to find the quotient (0.0204 Ã· 17). Now, 204 Ã· 17 = 12.
Dividend contains 4 places of decimal. So, 0.0204 Ã· 17 = 0.0012
1. Dividing a Decimal Fraction By a Decimal Fraction: Multiply both the dividend and the divisor by a suitable power of 10 to make divisor a whole number.
Now, proceed as above.
 Thus, 0.00066 = 0.00066 x 100 = 0.066 = .006 0.11 0.11 x 100 11
1. Comparison of Fractions:
Suppose some fractions are to be arranged in ascending or descending order of magnitude, then convert each one of the given fractions in the decimal form, and arrange them accordingly.
 Let us to arrange the fractions 3 , 6 and 7 in descending order. 5 7 9

 Now, 3 = 0.6, 6 = 0.857, 7 = 0.777... 5 7 9

 Since, 0.857 > 0.777... > 0.6. So, 6 > 7 > 3 . 7 9 5
1. Recurring Decimal:
If in a decimal fraction, a figure or a set of figures is repeated continuously, then such a number is called a recurring decimal.

n a recurring decimal, if a single figure is repeated, then it is expressed by putting a dot on it. If a set of figures is repeated, it is expressed by putting a bar on the set.
 Thus, 1 = 0.333... = 0.3; 22 = 3.142857142857.... = 3.142857. 3 7

Pure Recurring Decimal: A decimal fraction, in which all the figures after the decimal point are repeated, is called a pure recurring decimal.

Converting a Pure Recurring Decimal into Vulgar Fraction: Write the repeated figures only once in the numerator and take as many nines in the denominator as is the number of repeating figures.

 Thus, 0.5 = 5 ; 0.53 = 53 ; 0.067 = 67 , etc. 9 99 999
Mixed Recurring Decimal: A decimal fraction in which some figures do not repeat and some of them are repeated, is called a mixed recurring decimal.
Eg. 0.1733333.. = 0.173.

Converting a Mixed Recurring Decimal Into Vulgar Fraction: In the numerator, take the difference between the number formed by all the digits after decimal point (taking repeated digits only once) and that formed by the digits which are not repeated. In the denominator, take the number formed by as many nines as there are repeating digits followed by as many zeros as is the number of non-repeating digits.

 Thus, 0.16 = 16 - 1 = 15 = 1 ;   0.2273 = 2273 - 22 = 2251 . 90 90 6 9900 9900
1. Some Basic Formulae:
1. (a + b)(a - b) = (a2 + b2)
2. (a + b)2 = (a2 + b2 + 2ab)
3. (a - b)2 = (a2 + b2 - 2ab)
4. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
5. (a3 + b3) = (a + b)(a2 - ab + b2)
6. (a3 - b3) = (a - b)(a2 + ab + b2)
7. (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)
8. When a + b + c = 0, then a3 + b3 + c3 = 3abc.
1. Laws of Indices:
1. am x an = am + n
2.
 am = am - n an
1. (am)n = amn
2. (ab)n = anbn
3.
 a n = an b bn
1. a0 = 1
Surds:
Let a be rational number and n be a positive integer such that a(1/n) = a
Then, a is called a surd of order n.
1. Laws of Surds:
1. a = a(1/n)
2. ab = a x b
1. (a)m = am
Square of numbers in 100

> The last two places will be the square of
the last two digits (keep if any carry) _ _ _ X X.

> The first three places will be the number plus
the last two digits plus any carry: X X X _ _.

Here is an Example:

> let the number be 108:

2. Square the last two digits (no carry): 8 × 8 = 64: _ _ _ 64
3. Add the last two digits to the number: 108 + 08= 116:
so 1 1 6 _ _
4. So 108 × 108 = 11664.

Sqaure of Numbers in 200 to 299

Steps to find square of numbers in 200's

> Choose a number in the 200s (start with numbers under 210, then try for larger ones).
>The first digit of the square is 4: 4 _ _ _ _
> The next two digits will be 4 times the last 2 digits: _ X X _ _
> The last two places will be the square of the last digit: _ _ _ X X

here we take an Example:

> let the number be 207:

> The first digit is 4
so 4 _ _ _ _

> The next two digits are 4 times the last digit:
4 × 7 = 28
so _ 2 8 _ _

Square the last digit: 7× 7 = 49
so _ _ _ 49

So finally we get 206 × 206 = 42849.

and For larger numbers work right to left:

> Square the last two digits (keep the carry): _ _ _ X X
> 4 times the last two digits + carry: _ X X _ _
> Square the first digit + carry: X _ _ _ _

example

>If the number to be squared is 225:

> Square last two digits (keep carry):
25x25 = 625 (keep 6): _ _ _ 2 5

> 4 times the last two digits + carry:
4x25 = 100; 100+6 = 106 (keep 1): _ 0 6 _ _

> Square the first digit + carry:

2x2 = 4; 4+1 = 5: 5 _ _ _ _
> So 225 × 225 = 50625.

Square a 2 Digit Number

for this example 37:

> Look for the nearest 10 boundary
> In this case up 3 from 37 to 40.
> Since you went UP 3 to 40 go DOWN 3 from 37 to 34.
> Now mentally multiply 34x40
> The way I do it is 34x10=340;
> Double it mentally to 680
> Double it again mentally to 1360
> This 1360 is the FIRST interim answer.
> 37 is "3" away from the 10 boundary 40.
> Square this "3" distance from 10 boundary.
> 3x3=9 which is the SECOND interim answer.
> Answer: 1360 + 9 = 1369

Square of 3 digit number

LET THE NUMBER BE ABC

SQ (ABC) is calculated like this

STEP 1. Last digit = last digit of SQ(C)

STEP 2. Second Last Digit = 2*B*C + any carryover from STEP 1.

STEP 3. Third Last Digit 2*A*C+ Sq(B) + any carryover from STEP2.

STEP 4. Fourth last digit is 2*A*B + any carryover from STEP 3.

STEP 5 . In the beginning of result will be Sq(A) + any carryover
from Step 4.

EXAMPLE:

SQ (431)

STEP 1. Last digit = last digit of SQ(1) =1

STEP 2. Second Last Digit = 2*3*1 + any carryover from STEP1.= 6

STEP 3. Third Last Digit 2*4*1+ Sq(3) + any carryover from STEP
2.= 2*4*1 +9= 17. so 7 and 1 carryover

STEP 4. Fourth last digit is 2*4*3 + any carryover (which is 1) . =
24+1=25. So 5 and carry over 2.

STEP 5 . In the beginning of result will be Sq(4) + any carryover
from Step 4. So 16+2 =18.

So the result will be 185761.
Squaring the numbers:
1.      Squaring the numbers ending with 5.
352 =
Ø  Separate the 5 from the digits in front. in this case there is only a 3 in front of the 5. Add 1 to 3 get 4 (3+1= 4)
Ø  Multiply these numbers together: 3 x 4 = 12
Ø  Write the square of 5 (25) after 12. We will get 1225.
135 2  = ??
Ø  Take 13, add 1 to it we will get 14.
Ø  Then 13 x 14 = 182
Ø  Add the square of 5 next to it. We will get 18225.
SQUARING THE NUMBERS NEAR TO 50:
1.      462 =
Ø  Forty six squared means 46 x 46. Rounding upwards, 50 x 50 = 2500.
Ø  Take 50 and 2500 as our reference points.
50   462
- 4
46 = 50-4, so 4 is a minus number.
Ø  So we take 4 from the 25 hundreds.
Ø  (25-4)  x 100= 2100
Ø  To get the rest of the answer, we square the number in the minus. ( 4 2= 16)
Ø  Add 2100 and 16 we will get 2116 is the answer.
562 =
Ø  Fifty six squared means 56 x 56. Rounding upwards, 50 x 50 = 2500.
Ø  Take 50 and 2500 as our reference points.
50   562
+6
56 = 50+4, so 6 is a positive number.
Ø  So we add 6 to 25 hundreds.
Ø  (25+6)  x 100= 3100
Ø  To get the rest of the answer, we square the number in the minus. ( 62= 36)
Ø  Add 3100 and 36 we will get 3136 is the answer.
SQUARING NUMBERS NEAR TO 500:
Ø  This is similar to our strategy for squaring numbers near 50.
Ø  Five hundred times 500 is 250000, we take 500 and 250000 as our reference number.
1.      5062 =
500   5062
+6
5002 = 250000
Ø   Five hundred and six is greater than 500,
Ø  Square of 500 is 250000
Ø  The number 6 is added to the thousands
Ø  (250+6) x1000 = 256000
Ø  Square 6 is 36.
Ø  256000+36 =256036 is the answer.
Square the number ends with 1:
1.      312 =
Ø  First, subtract 1 from the number. The number now ends in zero and should be easy to square. (302  = 3 x 3 x 10 x 10) = 900
Ø  Add 30 and its next number 31 (30+31). We will get 61)
Ø  Add (900 + 61) = 961.
2.      3512=?
Ø  3502 = 122500
Ø  350 +351 = 701
Ø  122500 +701 = 123201
We can also use the method for squaring numbers ending in 1 for those ending in 6.
3.      862 =
Ø  852 =7225
Ø  85+86 = 171
Ø  7225+171 = 7396
Squaring numbers ending with 9
1.      292 =
Ø  Add 1 to the number. The number now ends in zero and is easy to square.
Ø  302 = 900
Ø  Now add 30 with given number29 (30+29 =59)
Ø  Then sub (900 - 59 = 841)

2.      3492 =
Ø  3502 = 122500
Ø  350+349 = 699
Ø  Sub (122500 - 699 = 121801)

Square a 2 Digit Number Ending in 5
For this example we will use 25
• Take the "tens" part of the number (the 2 and add 1)=3
• Multiply the original "tens" part of the number by the new number (2x3)
• Take the result (2x3=6) and put 25 behind it. Result the answer 625.
Try a few more 75 squared ... = 7x8=56 ... put 25 behind it is 5625.
55 squared = 5x6=30 ... put 25 behind it ... is 3025. Another easy one! Practice it on paper first!

SINGLE STEP:
35 square
(3x[3+1]) / (5x5) = 12 / 25 = 1225
Square 2 Digit Number: UP-DOWN Method

Square a 2 Digit Number, for this example 37:
• Look for the nearest 10 boundary
• In this case up 3 from 37 to 40.
• Since you went UP 3 to 40 go DOWN 3 from 37 to 34.
• Now mentally multiply 34x40
• The way I do it is 34x10=340;
• Double it mentally to 680
• Double it again mentally to 1360
• This 1360 is the FIRST interim answer.
• 37 is "3" away from the 10 boundary 40.
• Square this "3" distance from 10 boundary.
• 3x3=9 which is the SECOND interim answer.
• Answer: 1360 + 9 = 1369

The basic rule for mental addition:
Ø  If you wanted to add 47, you would add 50 and subtract 3,
Ø  To add 38 to a number, add 40 and subtract 2,
If the unit’s digit is high, round off to the next ten and then subtract the difference. If the units’ digit is low, add the tens then the units.
Ø  With two digit mental addition you add the tens digit of each number first, then the units. If the unit’s digit is high, round off the number upwards and then subtract the difference. If you are adding47, add 50, and then subtract 3.
Ø  To add 35, 67, and 43 together you would begin with 34, add 70 to get 105, subtract 3 to get 102, add 10 to get 142 then the 3 to get your answer of 145.
355+752+694 =?
Ø  355+700 = 1055
Ø  1055+50+2 = 1107
Ø  1107+700-6 = 1807-6 = 1801
OR
Ø  You may prefer to add from left to right; adding the hundreds first, then the tens and then the units.
8461
+5678
Ø  We begin with the thousands column.8+5 = 13, since we are dealing with thousands, our answer is 13 thousand.
Ø  Observe that the numbers in the hundreds column conveniently add to 10, so that gives us another thousand. Then answer is 14000.
Ø  Then add 61 to 14000, we getting 14061.
Ø  An easy rule is: when adding a column of numbers add pairs of digits to make tens first, then add the other digits.
SUBTRACTION:
To subtract mentally, try and round off the number you are subtracting and then correct the answer.
To subtract 9, take 10 and add 1: to subtract 8, take 10 and add 2; to subtract 7, take 10 and add 3,
1.  Eg: 56-9 =
-1
(To take 9 from 56 in your head, the easiest and fastest method is to subtract 10, (46) and add1 we get 47.)
2.      54-38 = 16
+2
Ø  44-40, plus 2 makes 16

3.      436-87 =
+13
Ø  Take 100 to get 336. Add 13 and we will get 349 easy.
Ø  SUBTRACTING ONE NUMBER BELOW A HUNDREDS VALUE FROM ANOTHER WHICH IS JUST ABOVE THE SAME HUNDREDS NUMBERS.
Three digit SUBTRACTIONS:
1.      461 -275 =
25
161+25 = 160+20+5+1 = 186
2.      834 – 286 =
14
534+14 = 530+10+4+4 = 540+8 = 548

Subtraction method one:
1.      7254-3897 =

6   1    4
7   2   5  4
3  3   9   7
3    3    5   7
Ø  Subtract 7 from 4. We can’t, so we borrow 1 from the tens column.
Ø  Cross out the 5 and write 4.
Ø  Don’t say 7 from 14, we have to say 7 from 10 and add 4 we getting 3+4 = 7 ( the first digit of the answer)
Ø  Nine from 4 won’t go, so borrow again. Nine from 10 is 1, plus 4, the next digit answer.
Ø  Eight from 1 won’t go, so borrow again. Eight from 10 is 2, plus 1 is 3, three from 6 is 3, the final digit of the answer.
Subtraction method two:
7  2  5  4
3   8  9  7
3   3  5  7
Ø  Subtract 7 from 4. We can’t, so we borrow 1 from the tens column. Put a 1 in front of the 4 to make 14 and write a small 1 alongside the 9 in the tens column.  Don’t say 7 from 14, but 7 from 10, add 4 on top gives 7, the first digit of the number.
Ø  Ten ( 9+1) from 5 won’t go so borrow again in a similar fashion. Ten from 15 is 5 or 10 is zero, plus 5 is 5.
Ø  Nine from 2 won’t go, so borrow again. Nine from 10 is 1 plus 2 is 3.
Subtraction from a power of 10:
The rule is : SUBTRACT THE UNITS DIGIT FROM 10, THEN EACH SUCCESSIVE DIGIT FROM 9, THEN SUBTRACT 1 FROM THE DIGIT ON THE LEFT.
1.      1000
-574

Ø  10-4=6,
Ø  9 - 7 = 2,
Ø  9 - 5 = 4,
Ø  1 - 1 = 0
Subtracting smaller numbers:
If the number we are subtracting has fewer digits than the one you are subtracting from, then add zeros before the number (at least, mentally) to make the calculation:
For instance:
23 000 – 46 =
23 000
0 046
22 954
Use the same principle as subtraction method 2.
Shortcut for subtraction
Ø  What is the easiest way to take 90 from a number?
Take 100 and give back 10
Ø  What is the easiest way to take 80 from a number?
Take 100 and give back 20
Ø  What is the easiest way to take 70 from a number?
Take 100 and give back 30

100    98x135 = 13300-70 = 13230

-2      35             30
Ø  How do we take 70 from 13,330?
Ø  Take away 100 and give back 30