Expected DA wef Jan2014

Possibility of 11% increase in DA wef Jan2014

Maths - Train - Problems

Maths -Train - Problems

1. Train passing a telegraph post or a man standing

      Time = Train Length divided by Train Speed

2. Train passing the platform or crossing a bridge

      Time = Train Length + Object Length

                  divided by Train Speed

3. Two trains running at the same direction

     Time = Total length of two trains

                 divided by Difference in Speed of two trains

4. Two trains running at opposite directions

      Time = Total length of two trains

                  divided by Sum of their speeds

5. Train passing a man moving against its direcion

      Time  = Train Length

                   divided by Sum of their speeds

6. Train passing a man moving in its direction

      Time = Train Length

                  divided by Difference in their speeds 

  

         

  

Maths - Tests of Divisibility

A number is divisible by  2 : if its unit digit is 0,2,4,6, or 8

                                    by  3 : if the sum of its digits is divisible by 3

                                    by  4 : if the number formed by its last two digits
                                              is divisible by 4

                                    by  5 : if its unit digit is either 0 or 5

                                    by  6 : if it is divisble by 2 and 3

                                    by  8 : if the number formed by its last three digits
                                              is divisible by 8

                                    by  9 : if the sum of its digits is divisible by 9

                                   by 10 : if its unit digit is  0

                                   by 11 : if the differnce between the sum of its
                                               alternate digits is 0 or mutiple of 11.

                                   by 12 : if the number is divisible by 3 and 4

                                   by 25 : if the number formed by its last two digits
                                               is divisible by 25
 

Maths : Time, Speed & Distance

Maths :: Time, Speed & Distance

Points to remember:

1. To find Distance  =  Speed  x  Time

2. To find Speed     =   Distance divided by Time

3. To find Time       =   Distance divided by Speed

4.  1 km / per hr      =   5/18 m / per second

    [ i.e., to convert given speed @ km per hour 

             into speed @ metre per second,

             multiply it by 5/18 ]

5.  1 m / per sec      =  18/5 km / per hour

      [ i.e., to convert given speed @ metre per second

               into speed @ km per hour,

               multiply it by 18/5 ]

6. If 'a' is onward journey speed, and 'b' is return journey speed, 

    then the average speed in a travel between two stations, then

    the average speed for the whole journey would be

        2 multiplied by 'a and by 'b'

        divided by 'a' + 'b'

           i.e.,    2ab divided by (a + b)

7. If a boy walks from his house to school at the rate of 5 kmph, and 

    reaches the school 10 minutes earlier than the scheduled time, and 

    if he walks at the rate of 3 kmph , he reaches 10 minutes late, 

    then the distance of the school from his house  would be:

    Distance covered = Product of Two Speeds x Diff. in Time

                                    divided by Diff of Two Speeds x 60

  and so, the answer = 5 x 3 x (10 + 10)

                                  divided by  2 x 60 = 2.5 km

Maths : Time and Work

Maths :: Time and Work

Direct variation and Indirect variation

Examples of Direct variation:

1. Interest and principal

    More principal earns more interest

2. Salary (wages) and workers

    More workers means more salary

3. Articles and cost

    Purchase of more articles costs more money

4. Distance and time

    More distance to travel, more time to spend

Examples of Indirect variation

1. Work and time

     More number of workers, the lesser will be the time

                              i.e., less number of days

2. Speed and time

   (i) Higher the speed, the lower is the time required to

       cover a distance

  (ii) A more capable person takes lesser time to 

       complete a job

3. Number and share

    More persons means, less will be the share of

    each person

    

Algebra

Algebra

Points to remember:
 
1. The sum of the first n odd numbers               =   n^2

2. The sum of the first n even numbers             =   n(n+1)

3. The sum of the first n consecutive numbers  =   n(n+1) divided by 2

4. Sum of numbers 51 to 100  = ?
    [Hint here 51 = a, 100 = last number;
       Sum = number of terms divided by 2
                   multiplied by (a + last number)
      Hence sum = 50 divided by 2
                            mutliplied by (51 + 100)
                        = 25 x 151  =  3775]

5. The number of  hand-shake of n persons       =   n(n-1) divided by 2

6. n^th odd number                                            =   2n-1
    [e.g.100 th odd number is 2(100-1) = 199]

7. n^th term                                                        =   a + (n - 1) d

8. Distributive property :  a(b+c) = ab + ac

9. Cross-Product Rule:
     If a divided by b = c divided by d is true, a x d = b x c is also true.

10. Product of extremes = Product of means.

11. Product of any number  and zero = Zero

12. (Any number)^0                         = 1.

13. (a + b)^2    =  a^2 + b^2 + 2ab

14. (a - b)^2     =  a^2 + b^2 - 2ab

15. a^2 - b^2    =  (a + b) (a - b)

16. (a + b)^2 - (a - b)^2 divided by ab  = 4ab divided by ab = 4

17. a^3 + b^3 divided by a^2 - ab + b^2  = a + b

18. a^m  x   a^ n                =  a^(m+n)

19. a^m  divided by  a^n   =  a^(m-n)      

20. 2^10  = 1024

 

Maths - HCF AND LCM of numbers

HCF and LCM

Important points to remember :

1. HCF of two or more than two numbers is the greatest number that
    divides each of them exactly. It is also known as Greatest Common
    Measure or Greatest Common Divisor.

2. LCM (Least Common Multiple) : The least number which is exactly
    divisible by each one of the given numbers is called their LCM.

3. The product of two numbers is equal to the product of their
    HCF and LCM.

4. HCF of any consecutive numbers is 1.

5. HCF and LCM of fractions :

   (i)  HCF = HCF of Numerators divided by
                     LCM of Denominators

   (ii) LCM =  LCM of Numerators divided by
                       HCF of denominators

Model Questions:

(1) Two numbers are in the ratio of  15 :11. If their HCF is 13,
      find the numbers.
      [Solution: Let the numbers be 15 x and 11 x, and so x = 13 (HCF)
                      So the numbers are 15 x 13, and 11 x 13
                      Ans : 195  and  143.]

(2) Two numbers are in the ratio of 3 : 4, and their HCF is 4.
      Find their LCM.
      [Hint : Two numbers are 3 x 4(HCF) and 4 x 4 (HCF),
                  and so their LCM is 3 x 4 x 4 = 48 (Ans)]

(3) Two numbers are in the ratio of 2 : 3. If their LCM is 48,
      find the numbers.

(4) The HCF and LCM of two numbers are 11, and 693 respectively.
      If one number is 77, find the other.

(5) The product of two numbers is 1320. If their HCF is 6,
      find their LCM.

(6) The sum of two numbers is 216. If their HCF is 27, find the numbers.

(7) The sum of two numbers is 2000. If their LCM is 21879,
      find the numbers.

(8) The LCM and HCF of two numbers are 495 and 5 respectively.
      If their sum is10, find the numbers.

(9) The product of LCM and HCF of two numbers is 24. If their
      difference is 2, find the numbers.

(10) The HCF and LCM of two numbers are 84 and 21 respectively.
        If the numbers are in the ratio of 1 : 4, find the larger number.

(11) Find the greatest possible length which can measure exact;y
        the length of 4m 95cm, 9m16cm, and 16m65cm.

(12) The length , breadth, and height of a room are 8m25cm, 6m75cm,
        and 4m50cm respectively. Determine the longest rope which can
        measure the three dimensions of the room exactly.

(13 Two tankers contain 850 lt and 680 lt of petrol respectively. Find
        the maximum capacity of  a container, which can measure the
        petrol of either tank in exact number of times.

(14) A rectangular courtyard is 20m16cm long and 15m60cm broad.
        It is to be paved with square stones of same size. Find the least
        possible number of such tones.
        [Hint: HCF of 2016 and 1560 is 24,
                   No. of square stones required = length x breadth
                                                                    divided by HCF x HCF
                                                                   = 65 x 84
                                                                   = 5460 (Ans)]

(15) Find a rod of the greatest length which can measure exactly
        42m, 49m, 63m.

(16) Two bills of Rs. 6075 and Rs.8505 respectively are to be paid
        by cheques of the same amount. What will be the largest
        possible amount of each cheque ?

(17) There are four bells, which toll at intervels of 3,7,12, and 14
        seconds respectively. The four bells begin to toll at 12 noon.
        How often will they toll together in 14 minutes ?
        [Hint: LCM of 3,7,12,and 14  is 84
                  No. of times they toll together in 14 minutes, i.e.,
                                     in 14 x 60 seconds  = 14 x 60
                                                                         divided by LCM
                        i.e., 14 x 60  divided by 84  = 10 times (Ans)]

(18) 4 bells toll after an intervel of 8,9,12, and 15 seconds
        respectively. When they toll again ?

(19) The circumference of the wheels of a carriage are 3m25cm,
        and 5m. What is the least distance in which both the wheels
        make an exact number of revolutions ?

(20) A boy saves Rs 4.65 daily. Find the least no. of days
        in which he will be able to save an exact number of rupees.
        [Hint: The exact number of rupees will be a multiple of 100.
                   LCM of 100 and 465 = 5 x 20 x 93 = 9300
                   i.e., the boy saves 9300 paise
                   Hence, the no. of days required = 9300 divided by 465
                                                                      = 20 (Ans)]

(21) Electric posts occur at equal distance of 220m along a road
       and police constables are standing at equal distance of 300m
       along the same road. The first constable is standing at the foot
       of the first electric post. How far from it along the road is the
       next constable standing at the foot of an electric post ?
       [Hint: The required distance  = LCM of 220 and 300]

(22) In a walking competition three persons step off together.
       Their steps measure 85cm, 90cm, and 80cm respectively.
       At what distance from the starting point will they again
       step off together ?

(23) Three different containers contain  496, 403 and 713 litres
        of milk respectively. Find the biggest jug which can measure
        all the different quantities exactly.

(24) A rectangular  courtyard  3.78  mt long and 5.25 mt wide is
        to be paved exactly with square tiles, all of the same size.
        What is the largest size of the tile which could be used for
        the purpose ?

(25) Four differenct electronic devices make a beep after every
        30 minutes, 1 hour, 1 1/2 hour and 1 hour 45 minutes
        respectively. All the devices beeped together at 12 noon.
        (i) At what time will they all beep together again ?
        (ii) In 30 minutes, how many times do they beep together ?

(26) A, B, and C start at the same time in the same direction to
        run around a circular stadium. A completes a round in 252
        seconds, B in 308 seconds, and C in 198 seconds, all starting
        at the same point. After what time will they meet again at the
        starting point ?

(27) The traffic signal lights at three different road crossings change
        after every 48 seconds, 72 seconds, and 108 seconds respectively.
        If they all change simultaneously at 08:20 hours, then  at what
        time will they again change simultaneously ?
        [Solution: Interval of change  = LCM of 48,72, 108 seconds.
                  Hence, the lights will again change simultaneously
                  after every 432 seconds, i.e., 7 minutes 12 seconds.
                  So, the next simultaneous change will take place at
                  8:27:12 hrs.]

(28) Find the least number which when divided by 5,6,7, and 8
        leaves  a remainder of 3, but when divided by 9 leaves no
        remainder.
        [Solution : LCM of 5,6,7,8 = 840
          So, the required number is of the form 840k + 3
          Least value of k for which 840k +3 is divisible by 9
                         is  k = 2
          Hence the required number is  840 x 2  + 3 = 1683]

(29) The LCM of two numbers is 45 times their HCF. If one of
        the number is 125 and the sum of HCF and LCF is 1150,
        find the other number.

(30) Find the maximum number of students among whom 1001
        pens and 916 pencils can be distributed in a such a way
        that each student gets the same number of pens and same
        number of pencils.
    
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