Quant Boosters - Vikas Saini - Set 2



  • A takes time =30/3 = 10.
    B takes time = 32/4 = 8.
    Suppose speed of escalator is a.
    10(a+3) = 8(a+4)
    a = 1.
    Total steps = 10(1+3) = 40.



  • Q21) A man can walk up a moving up escalator in 30 seconds. The same can walk down this moving up escalator in 90 seconds. How much time he will take when escalator is stationary ?



  • Man’s speed is a and escalator speed is b.
    (a+b)30 = (a-b)90
    a = 2b.
    Total steps = 90b.
    Time taken by him = 90b / 2b = 45 seconds.



  • Q22) A and B speed ratio is 2:3. The steps taken by A & B are 25 & 30 at climbing escalator. When A is walking up and B is walking down then after how many steps B will take to meet A ?



  • Time taken by A = 25/2 = 12.5 seconds.
    Time taken by B = 30/3 = 10 seconds.
    Speed of escalator is a.
    (2+a) 12.5 = (3+a)10
    25 + 12.5a = 30 +10a
    a = 2.
    Total steps = (3+2)10 = 50.
    Steps taken by B = 50 x 3 / (3+2) = 30 steps.



  • Q23) A & B walk up an escalator (moving stairway), the escalator moves at a constant speed. A takes 3 steps for every 2 steps of B. A gets to the top of the escalator having taken 25 steps, while B takes only 20 steps to reach the top. If the escalator were turned off. How many steps would they have to take up ?



  • Time taken by A = 25/3.
    Time taken by B = 20/2 = 10.
    Let speed of escalator = a.
    (a+2)10 = (a+3)25/3
    30a + 60 = 25a + 75.
    a = 3.
    Total steps = (3+2)10 = 50.



  • Q24) On an upward moving escalator A, B, C take 10,8,5 steps respectively to reach the top. A takes 30 step to come down on same moving up escalator. Find the time ratio by B & C.



  • Let speed of A = 1step/sec.
    Speed of escalator = a.
    10(1+a) = 30(1-a)
    10 + 10a = 30 – 30a
    a = 1/2
    Total steps = 10 ( 1+1/2) = 15.
    B takes time = 15-8 = 7.
    C takes time = 15-5 = 10.
    Time ratio = 7:10



  • Q25) 677 has exactly 5 digits when converted into base 'n' from the decimal system. What is the minimum possible value of n.



  • Let's start from base 3.
    smallest 5 digit number in base 3 = (10000)_3 = 3^4.
    Largest 5 digit number in base 3 = (22222)_3 = 2(11111)_3 = 3^5 - 1.
    in base 3 not possible.
    Now let's take base 4.
    smallest number = (10000)_4 = 4^4 = 256.
    Largest number = (33333)_4 = 4^5 - 1 = 1023.
    256 < 677 < 1023.
    Hence base 4.



  • Q26) How many 3 digit numbers are there in base 9 system.



  • Hundred place = 1 to 8 = 8 digit.
    Ten's place = 0 to 8 = 9 digit.
    Unit's place = 0 to 8 = 9 digit.
    Total = 8 x 9 x 9 = 648 digits.



  • Q27) How many base 10 numbers are three digit numbers in base 7 and four digit numbers in base 6.



  • Smallest 3 digit number in base 7 = (100)_7 = 7^2 = (49)_10.
    Largest 3 digit number in base 7 = (666)_7 = 7^3 - 1 = (342)_10.
    [49,342] numbers are 3 digits numbers in base 7.
    Smallest 4 digit number in base 6 = 6^3 = 216.
    Largest 4 digit number in base 6 = 6^4 - 1 = 1295.
    [216,1295] numbers are 4 digit numbers in base 6.
    [216,342] common numbers.
    Total decimal numbers = 342 - 216 + 1 = 127.



  • Q28) N in base 10 = aaa in base 3. Here a is a single digit number, then N is a multiple of.
    (a) 9
    (b) 3a
    (c) 13
    (d) 26.



  • (aaa)_3 = a (111)_3
    a (3^2 + 3 + 1)
    (13a)_10 = (N)_10.
    Hence 13.



  • Q29) How many 3 digits are there in decimal system which have exactly 3 digits when expressed in both base 7 & base 13.



  • Smallest 3 digit number in base 7 = (100)_7 = 7^2 = 49.
    Largest 3 digit number in base 7 = (666)_7 = 7^3 - 1 = 342.
    [100,342] are the number which are exactly 3 digit number in base 7.
    Smallest 3 digit number in base 13 = (100)_13 = 13^2 = 169.
    Largest 3 digit numbers in base 13 = (CCC)_13 = 12^3 - 1 = 1727.
    [169,999] are the three digit numbers which are exactly 3 digit numbers in base 13.
    [ 169,342] is common in both.
    Total = 342 - 169 + 1 = 174 numbers.



  • Q30) In base x, 206 + 216 = 424. What is value of x ?


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