Quant Boosters - Shashank Prabhu, CAT 100 Percentiler - Set 2

[shashank_prabhu]

99! mod 101 gives 1 as the remainder from Wilson's theorem.
95! * 96 *97 * 98 * 99 mod 101 = 1
95! * -5 * -4 * -3 * -2 mod 101 = 1
95! * 120 mod 101 = 1
95! * 19 mod 101 = 1
Let 95! mod 101 be k
19k mod 101 = 1
19k = 101a + 1
k = (101a + 1)/19
a=3, k=16

[shashank_prabhu]

Q28) Let a secret three digit number be cba. If the sum of cab + bac + bca + abc + acb = 2536, what is c + b + a ?

[shashank_prabhu]

222(a+b+c)=2536+(cba)
(a+b+c) > 11
Also, 2536 mod 222 is 94 and so, (cba) mod 222 needs to be 128. Possible values are 128, 350, 572, 794. Only 572 satisfies and so, c+b+a=14.

[shashank_prabhu]

Q29) If 15x + 20y = 375, then find the minimum value of (x^2 + y^2), where x and y are natural numbers

[shashank_prabhu]

Looking for Cauchy Schwarz inequality here:
(x^2+y^2)(15^2+20^2) >= (15x+20y)^2
(x^2+y^2) >= 375^2/625
(x^2+y^2) >= 225

[shashank_prabhu]

Q30) At what price the shopkeeper sell his article so that if he sells the article at the marked price he makes a profit of 25% and if he increased selling price by 250 he gets a 25 % profit on the selling price find the cost price?

[shashank_prabhu]

Marked price: 1.25x
Increased selling price: 1.25x+250
Profit: 0.25x+250
Profit on selling price:
(0.25x+250)/(1.25x+250)=25/100
25x+25000=31.25x+6250
6.25x=18750
x=3000.

[Vikrant Garg]

@shashank_prabhu can you please explain this ?

[zabeer]

@Vikrant-Garg

Original quantity - 3L
After first replacement - 3 - 1 = 2 L wine and 0 + 2 = 2 L water (wine : water = 1 : 1)
Now when we take 1 L from the container it will have 0.5 L of wine and 0.5 L of water. So after second replacement we have 2 - 0.5 = 1.5 L wine and 2 - 0.5 + 2 = 3.5 L of water (wine : water = 3 : 7)
When we take 1L from the container it will have 0.30L of wine and 0.70L of water. So after third replacement, we have 1.5 - 0.25 = 1.20 L wine and 3.5 - 0.7 + 2 = 4.8 L of water (wine : water = 1 : 4)
and so on

So after
1st step = 2
2nd step = 1.5
3rd step = 1.2
and so on..

Got the pattern ?
it's 6/(n + 2), where n is the number of steps so far.

So after 19th step, concentration of wine would be 6/21 = 2/7

Alternatively,
Milk quantity is being reduced as 3L -> 2L -> 1.5L -> 1.2L -> ...
3 x 2/3 -> 2
2 x 3/4 -> 1.5
1.5 x 4/5 -> 1.2 and so on.
So we need to find out 3 x 2/3 * 3/4 * 4/5 * 5/6 * .... * 20/21 (19 steps)
Every thing cancels out and we will get 3 * 2/21 = 2/7

[Vikrant Garg]

@zabeer thanks for the explanation.