Directions: In a certain number system there are only two notations to represent numbers: & and #.0 is represented by & and 1 by #. The sub…

According to the given information,

Number

0

1

2

3

4

5

Code

&

#

#&

##

#&&

#&#

Simplified Code

0

1

10

11

100

101

So, this entire code language is based on only 2 symbols & and #, i.e. 0 and 1.

Let’s work out the logic behind the arrangement of these digits.

On observation, we notice that starting from the rightmost digit, each digit is multiplied with corresponding power of 2, starting with 20, and then all these terms are added to obtain the number in decimal system.

[The rightmost term is multiplied with 20, second term from right is multiplied with 21, third term from right is multiplied with 22 and so on]

2 → 10 ⇒ (0 × 20) + (1 × 21) = (0 × 1) + (1 × 2) = 0 + 2 = 2

3 → 11 ⇒ (1 × 20) + (1 × 21) = (1 × 1) + (1 × 2) = 1 + 2 = 3

4 → 100 ⇒ (0 × 20) + (0 × 21) + (1 × 22) = (0 × 1) + (0 × 2) + (1 × 4) = 0 + 0 + 4 = 4

5 → 101 ⇒ (1 × 20) + (0 × 21) + (1 × 22) = (1 × 1) + (0 × 2) + (1 × 4) = 1 + 0 + 4 = 5

and so on.

Based on this logic,

#&&##& ↔ 100110

100110 ⇒ (0 × 20) + (1 × 21) + (1 × 22) + (0 × 23) + (0 × 24) + (1 × 25)

= (0 × 1) + (1 × 2) + (1 × 4) + (0 × 8) + (0 × 16) + (1 × 32)

= 0 + 2 + 4 + 0 + 0 + 32 = 38

###& ↔ 1110

1110 ⇒ (0 × 20) + (1 × 21) + (1 × 22) + (1 × 23)

= (0 × 1) + (1 × 2) + (1 × 4) + (1 × 8) = 0 + 2 + 4 + 8 = 14

Sum of the two numbers = 38 + 14 = 52

50% of 52 = 26.

Now, we need to convert 26 into the given code language.

26 = 16 + 8 + 2

∴ 26 = (1 × 24) + (1 × 23) + (0 × 22) + (1 × 21) + (0 × 20)

∴ 26 ⇒ 11010

Converting 0s and 1s to &s and #s respectively, we get

26 = ##&#&