Quant Boosters - Sibanand Pattnaik


  • QA/DILR Mentor | Be Legend


    Dudul n Gudul make round trips between Cuttack n Bhubaneswar respectively ..They first met at 10kms from Bhubaneswar, they reached their respective destinations n returned back n on their way , they meet for the 2nd time at 300 metres from Cuttack ..Find the distance between Cuttack n Bhubaneswar ?

    First time when Dudul n gudul meet , together they cover then entire distance between Cuttack n Bhubaneswar.. 2nd time when they meet they covered thrice the distance. So I can say 3 * 10000 - 300 = 29.7 kms is the distance between 2 cities

    Else s1/s2 = 10000/L - 10000
    And in second meet
    S1/S2 = (L + 300)/2L - 300
    So equating 2 equations we can find L.

    How many whole number solutions exist for the equation x + y + z = 48 such that x< y < z?
    a) 1225
    b) 192
    c) 200
    d) 872

    x + y + z = 48
    total whole number solutions = n+r-1 C r-1 = 48+3-1 C 3-1 = 50C2
    But 50C2 includes "ALL POSSIBLE WAYS IN WHICH 48 CAN BE DIVIDED INTO 3 GROUPS "
    So broadly there are 3 type of cases in which 48 can be divided into 3 groups ..
    CASE - 1 --> WHEN X , Y , Z are all equal i.e X = Y = Z = 16 .. so it can be done in only 1 way
    CASE - 2 ---> WHEN 2 GROUPS ARE SAME AND 1 GROUP is different .. so cases like
    0 0 48
    1 1 46
    2 2 44
    .......and so on till
    24 24 0

    so total 0 to 24 = (24 - 0 ) + 1 = 25 cases .. .. Just for your easy understanding i ve counted else you can directly do 2a + b = 48 so "a" can take values from 0 till 24 so 25 cases .

    And now each of these 25 cases would have been arranged in 3!/2! ways ..why ?? because " 2 are same and 1 is different" .. so total 25 * 3 = 75 cases

    BUT THESE 75 SOLUTIONS INCLUDE THAT "16 , 16 , 16" CASE ASWELL..so exclude that .. So finally 75 - 3 = 72 CASES ..

    CASE - 3 ---> WHEN ALL THE GROUPS ARE DIFFERENT
    This includes cases like 3 , 24 , 21 ; i.e where X , Y ,Z are different ..
    Now each these "ALL DIFFERENT ELEMENT " sets like ( 3 , 24 , 21 ) can be arranged among themselves in 3! i.e 6 ways ..But we need only ONE TYPE OUT OF THOSE 6 WAYs ..In simple words those 6 ways will include cases like

    X < Y > Z ; X < Y < Z ; X > Y < Z ; X < Z < Y ; X > Z > Y ;X > Y > Z
    BUT WE NEED ONLY "X > Y > Z" type cases so 1 out of 6 i.e 1/6 OF (ALL DIFFERENT CASES )
    So X > Y > Z CASES = (TOTAL - CASE 2 - CASE 1 )* 1/6
    = (50c2 - 72- 1)/6 = [192]

    What is the probability of selecting a number is selected [100,999] such that sum of digits of number is 14.

    a+b+c = 14
    1st digit cannot be 0.
    (a'+1)+b+c = 14
    a'+b+c= 13 total = 15C2=105
    a' cannot be 9 or more since (a'+1) would be 10 or more then
    -> (9+a')+b+c = 13
    a'+b+c= 4 -> 6C2
    b and c cannot be 10 or more.
    a'+(b+10)+c = 13
    a'+b+c= 3 -> 5C2 and same for c
    => 105-6C2-2*5C2 = 105-15-20 = 70
    => 70/900 = 7/90

    The volume of a mixture of milk and water was increased by 68.75% by adding pure milk to it. If the ratio of milk n water in the resultant solution is 3:1, find the ratio of milk n water in the original solution ?

    68.75 = 11/ 16
    let Vol 1= 16 then vol 2 = 27
    We know water proportion initially * Volume 1 = Water proportion finally *Volume 2
    => water proportion initially * 16 = (1/4) * 27
    =>water proportion initially = 27/64
    so water : milk = 27: 37

    How many pairs of factors of N=360 will be coprime to each other?

    360 = 36 * 10 = 2^3 * 3^2 * 5
    ORDERED PAIRS = (2 * 3 + 1)(2 * 2 + 1) (2 * 1 + 1) = 105
    UNORDERED PAIR = [(2 * 3 + 1)(2 * 2 + 1) (2 * 1 + 1) -1 ]/2 = 52
    As in this case variables are implicit so Its UNORDERED PAIRS .. so 52

    How many ways can 360 be expressed as Product of 2 coprimes?

    Let us suppose you have to express the number 360 as product of 2 co primes i,e a and b
    So 360 = 2^3 * 3^2 *5
    SO TOTAL 3 PRIMES Are there in 360 ..
    We have to express as product of 2 co-primes "a" and "b"
    So each of the primes i.e 2 , 3 and 5 has just 2 choices i.e either they can go "a" or to "b" .. so 2 * 2 * 2 = 8
    there these 8 include cases like (1. 360 ) and (360,1) i.e "repeat cases
    so 8/2 = 4
    the formula is (2^n)/2 = 2^(n-1)

    Real madrid and Manchester united play a football match in which REAL beats MANCHESTER 4 - 2 ..In how many Ways can the goals be scored provided MANCHESTER never had a lead over REAL during the match?

    if Manchester never had a lead means 1st goal was scored by REAL .. So score card at this juncture would ve been 1 - 0 ..The only way MANCHESTER could take lead is by scoring next 2 goals , i.e a pattern like RMMRRR.. Except this one case , MANCHESTER Can no way take lead as such ..
    So (5!)/3! *2! = 10
    NOW excluding this one case of RMMRRR ... We have 10 - 1 = 9

    How many integers P are there between 0 and 10^100 such that units digit of P^3 is 1 ?

    So , each number from 1 - 10 has a distinct unit digit for its cube ..
    occurrence of each digit in cube is 1/10 times
    So 0 - 10^100 it will be 10^100 * 1/10 = 10^99

    20 groups of five shooters each compete in an shooting championship. A shooter finishing in Nth position contributes N points to his team, and there are no ties. The team that wins will be the one that has the least score . Given that , the 1st position team's score is not the same as any other team, the number of winning scores that are possible is

    The group’ scores must sum to 1 + 2 + . . . + ..+99+100 =5050. The winning group can be ATMOST 1/20 *5050 = 252.5 and is at least 1 + 2 + 3 + 4 + 5 = 15. However, not all scores between 15 and 252 inclusively are possible because all teams must have integer scores and no team can tie the winning team.

    number of scores possible = 251 - 15 + 1 ...= 237

    X varies inversely as y and x varies directly as the square of z. If y decreases by 43.75% and z decreases by 75%, by what percent does x change?
    a. 88.88%
    b. 50%
    c. 45.45%
    d. 54.54%
    e. 44.44%

    Here my approach would be
    X varies inversely as Y so XY is Constant
    X varies directly with z^2 so x/z^2 is a constant
    so (XY)/Z^2 = constant
    X = (constant * Z^2 )/Y
    So multiplying factor of X = (1 * multiplying factor of Z^2 )multiplying factor of Y
    => multiplying factor of X = [1 * (1/4)^2] / (9/16)
    => multiplying factor of X = 1/9
    As mulpliying factor of x is 1/9 so change is "x" is 8/9 .. i.e 11.11 * 8 = 88.88 %

    Total positive integral solutions of 4x + 5y + 2z = 111?

    Answer by Bruce Wayyne
    y needs to be odd, else LHS would become even.
    say y = 2k-1 k is positive
    => 4x + 5 (2k-1) + 2z = 111
    2x + 5k +z = 58
    Let k = 1,
    2x+z= 53 so (1, 51)...(26, 1) so 26
    k=2
    2x+z=48 so (23, 2).....(1, 46) so 23
    k=3
    2x+z=43 so (1,41)....(21, 1) so 21
    k=4
    2x+z= 38 so (1, 36)....(18,2) so 18
    k=5
    2x+z= 33 so (1,31)....(16,1) so 16
    Hence the pattern is +3, +2, +3, +2....so on.
    so (26+21+16+11+6+1) + (23+18+ 13 + 8 + 3) = 146

    The volume of a mixture of milk and water was increased by 68.75% by adding pure milk to it..if the ratio of milk n water in the resultant solution is 3:1.. find the ratio of milk n water in the original solution ?

    68.75 = 11/ 16
    let Vol 1= 16 then vol 2 = 27
    We know water proportion initially * Volume 1 = Water proportion finally *Volume 2
    => water proportion initially * 16 = (1/4) * 27
    =>water proportion initially = 27/64
    so water : milk = 27: 37
    or
    milk : water = 37: 27

    Fresh grapes contain 84% water while raisins contain 20% water.How many kg of raisins can be prepared from 80 kg of fresh grapes?

    16% of Grapes = 80% of raisins
    => Raisins = 1/5 * Grapes
    Grapes = 80
    => Raisins = 1/5 * 80 = 16

    Ratios method:
    84% = 21 / 25
    if weight = 25 ; then water = 21 and rest = 4
    20% = 1/5
    if weight = 5 : then water = 1 and rest = 4
    So ratio of weight of Grapes : raisins = 5:1
    so i need 80 kg of Grapes that mean 5 becomes 16 times so 1 should also become 16 times ... so 16 kgs

    Ram bought a few mangoes and apples spending an amount of at most 2000.If each mango cost 4 and Apple 6 and Ram bought at least 1. Find the different possible amounts could have spent in purchasing fruits?

    Sum of 3 number is even so either all three are even or 2 odd and 1 even ..
    7^4 > 2002 .. so it must be less than this ... 5^4 is 3 digit number so we MUST TAKE 6^4 else how can you form a 4 digit number .. ??
    that way 6^4 is included , now only 2 cases are possible
    either the other 2 number ll be even or they have to be odd ... By common sense , we ignore 4 and 2 so it must be 3^4 and 5^4
    so 6+ 5+ 3 = 14


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