## Thursday, 24 November 2011

### Numbers -17 (XAT-2011) (3 Marks)

The micromanometer in a certain factory can measure the pressure inside the gas chamber from 1 unit to 999999 units. Lately this instrument has not been working properly. The problem with the instrument is that it always skips the digit 5 and moves directly from digit 4 to 6. What is the actual pressure inside the gas chamber if the micromanometer displays 003016?
(1)2201          (2)2202          (3)2600          (4)2960          (5)None of these options
Solution follows here:
Solution:
This can be done by digit-by-digit analysis using concepts of Permutations and Combinations. But the best way out is using number theory as given below:
Here we need to skip number ‘5’ while counting. This means, we need to consider only 9 digits 0, 1, 2, 3, 4, 6, 7, 8, 9 of the base-10 system. The digits to be used in the ‘New system’ are 0, 1, 2, 3, 4, 6, 7, 8, 9. This implies the ‘New system’ is in some-way similar to Base-9 system, which also considers nine single digit numbers (0 to 8). So, we compare the Base-10, Base-9 and the ‘New system’ here:

 Base-10 Base-9 New-system Base-10 Base-9 New-system 0 0 0 9 10 10 1 1 1 10 11 11 2 2 2 11 12 12 3 3 3 12 13 13 4 4 4 13 14 14 5 5 6 14 15 16 6 6 7 15 16 17 7 7 8 16 17 18 8 8 9 17 18 19

Observe that the numbers 0, 1, 2, 3, 4 are same in Base-9 system and the new system. Similarly, numbers 10, 11, 12, 13, 14 are same respectively in Base-9 and the new systems.
But numbers 5,6,7,8 in Base-9 system are respectively equal to 6,7,8,9 in the new system. Similarly, 15,16,17,18 in Base-9 system are respectively equal to 16,17,18,19 in the new system.
If we go by this logic, 3016 in the new system is equal to 3015 in Base-9 system. Now we need to find (3015)9 equivalent in Base-10 system.
(3015)9 = 5*90 + 1*91 + 0*92 + 3*93 = 5 + 9 + 3*729 = 2201