Aptitude and Reasoning Questions

1 million distinct 3 digit initials are needed.

Let the number of required alphabets in the language be ‘n’.

Therefore, using ‘n’ alphabets we can form n * n * n = n3 distinct 3 digit initials.

Note distinct initials is different from initials where the digits are different.

For instance, AAA and BBB are acceptable combinations in the case of distinct initials while they are not permitted when the digits of the initials need to be different.

This n3 different initials = 1 million 

i.e. n3=106  (1 million = 106)

  => n = 102 = 100

Hence, the language needs to have a minimum of 100 alphabets to achieve the objective.

Let the speed of the boat upstream be p kmph and that of downstream be q kmph

Time for upstream = 8 hrs 48 min = 845hrs

Time for downstream = 4 hrs

Distance in both the cases is same.

=> p x  845= q x 4

=> 44p/5 = 4q

=> q = 11p/5

 Now, the required ratio of Speed of boat : Speed of water current

= q+p2:q-p2

=> (11p/5 + p)/2  : (11p/5 - p)/2

=> 8 : 3

C.P. of 50 kg rice = Rs. (36 x 20 + 14 x 36) = Rs. (720 + 504) = Rs.1224.  

S.P. of 56 kg rice = Rs. (50 x 30) = Rs.1500.

Profit = 1500 - 1224 = Rs.276 

Profit % = ProfitC.P×100  = 2761224×100 = 22.54%.

The figure may be labelled as shown

Rectangles :

The simplest rectangles are CVSR, VETS, RSWM and STKW i.e 4 in number.

The rectangles composed of two components each are CETR, VEKW, RTKM and CVWM i.e 4 in number.

The rectangles composed of three components each are ACRP, PRMO, EGHT and THIK i.e 4 in number.

The rectangles composed of four components each are CEKM, AVSP, PSWO,VGHS and SHIW i.e 5 in number.

The rectangles composed of five components each are AETP, PTKO, CGHR and RHIM i.e 4 in number.

The rectangles composed of six components each are ACMO and EGIK i.e 2 in number.

The rectangles composed of eight components each are AGHP, PHIO, AVWO and VGIW i.e 4 in number.

The rectangles composed of ten components each are AEKO and CGIM i.e 2 in number.

AGIO is the only rectangle having sixteen components

Total number of rectangles in the given figure = 4 + 4 + 4 + 5 + 4 + 2 + 4 + 2 + 1 = 30.

Hexagons :

The hexagons in the given figure are CDEKLM, CEUKMQ, CFHJMQ, BEUKNP and BFHJNP. So, there are 5 hexagons in the given figure.

B runs around the track in 10 min.

 i.e ,    Speed of B = 10 min per round

 Therefore,  A beats B by 1 round

  Time taken  by A to complete 4 rounds 

  = Time taken by B to complete 3 rounds  

 = 30 min 

Therefore, A's speed = 30/4 min per round = 7.5 min per round

 Hence, if the race is only of one round A's time over the course = 7 min 30 sec

We have to arrange 6 books.

The number of permutations of n objects is n! = n. (n – 1) . (n – 2) ... 2.1

Here n = 6 and therefore, number of permutations is 6.5.4.3.2.1 = 720

Here,P(A)=4/9=p/(p+q)

P(odd against event A)=q/p=5/4

Let width = x

Then, height = 6x and length = 42x 

42x * 6x * x = 16128

x = 4