Quant Boosters  Sibanand Pattnaik  Set 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

Q21) 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

Q22) 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 %

Q23) 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 = 2k1 k is positive
=> 4x + 5 (2k1) + 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

Q24) 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

Q25) HCF of 3 numbers is 6 and their sum is 120. How many such triplets exist ?

6x + 6y + 6z = 120
=> x + y + z = 20
19c2 casesHCF > 1
only possible case for HCF = Either 2 and 5 ..
case 1 (when HCF =2)
2x + 2y + 2z = 20
=> x + y + z = 10
9c2Case 2 , (When HCF = 5)
5x + 5y + 5z = 20
=> x + y + z = 4
so 3c2Therefore , required cases = 19c2  9c2  3c2 = 132 (ORDERED )
Now remove the 2 same 1 cases where HCF is 1
cases , 1 , 1 ,18 ; 3, 3 , 14 ; 7,7,6; 9,9,2 ... each of these arranged in 3!/2! ways so , 12 ways ..
so (132  12)/6 + 4 = 24
(for un ordered) ..

Q26) 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

Q27) If 1/a + 1/b + 1/c + 1/d = 2, where a , b , c , d are distinct natural numbers, what is the value of a + b + c + d ?

If sum of Factors of N excluding N is equal to N then N is called a perfect number ..
for Eg 28 = 1 + 2 + 4 + 7 + 14
496 = 1 +2 +4 +8 + 16 + 31 + 62 + 124 + 248
PERFECT NUMBERS show another property
The sum of the reciprocal of the factors of a perfect number INCLUDING THE NUMBER ITSELF = 2
Again i repeat "INCLUDING THE NUMBER ITSELF"
for eg : 28 is a perfect number
1+ 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 2

Q28) Find the largest value of a for which 100 + a^3 becomes perfectly divisible by 10 + a

by Factor theorem, if 100 + a^3 is divisible by 10 + a then substitute a = 10 in 100 + a^3 = 100  1000 = 900
Now 10 + a must be a factor of 900
so let K * (10 + a) = 900
Now to obtain the greatest value of "a", we have to put K as 1
so 10 + a = 900
=> a = 890

Q29) How many ways can 22 identical balls be distributed in 3 identical boxes ?

It is unordered distribution of a+b+c = 22
so total cases = 24c2
2 same n 1 different cases .. will be .. 0 0 22 ; 1 1 20 ....till 11 11 0 .
so total 0  11 that is 12 cases and each of them ll be arranged in 3!/2! i.e 3 ways
why ?? because they are of type A A B ..
so( 24c2  36 ) / 6 + 12 = 52

Q30) X and Y have some chocolates with them which they wish to sell . The cost of each chocolate is equal to the number of total chocolates with both of them together initially. Together they sell all the chocolates and after that they start distributing the money collected in this particular fashion. First X takes a 10 rupee note, then Y takes a 10 rupee note and so on. In end it's turn of Y who didn't get any more 10 rupees. How much rupees Y get in his last turn?