Question

In a number system the product of $ 44 $ and $ 11 $ is $ 3414 $ . The number $ 3111 $ of this system when converted to the decimal number system becomes
A. $ 406 $
B. $ 1086 $
C. $ 213 $
D. $ 691 $

Answer 1

Hint: This given question deals with the decimal number system. Decimal number system consists of digits from $ 0 - 9 $ including $ 0 $ and $ 9 $ . The base of this system is $ 10 $ as the total number of digits available in the decimal number system is $ 10 $ . All the other remaining digits can easily be expressed with the help of these $ 10 $ -digit numbers. Here in this question, we have to convert the number $ 3111 $ into decimal numbers.

Complete step-by-step answer:
The binary numbers with n number of digits: $ {d_{n - 1}}...{d_3}{d_2}{d_1}{d_0} $
The conversion of binary numbers to decimal numbers can be easily obtained by the sum of the product of the binary digits ( $ {d_n} $ ) and their power of $ 2\left( {{2^n}} \right) $ :
Decimal number = $ {d_0} \times {2^0} + {d_1} \times {2^1} + {d_2} \times {2^2} + {d_3} \times {2^3} + .... $
So, given is the product of $ 44 $ and $ 11 $ is $ 3414 $ . We have to find out what the number $ 3111 $ be when converted into a decimal number system.
We know that when $ 44 $ is multiplied by $ 11 $ , we get the product as $ 484 $ .
Let us assume that if base is $ x $ then, $ 3411 $
 $
   \Rightarrow 3{x^3} + 4{x^2} + 1{x^1} + 4{x^0} = 484 \\
   \Rightarrow 3{x^3} + 4{x^2} + x + 4 = 484 \\
   \Rightarrow 3{x^3} + 4{x^2} + x = 484 - 4 \\
   \Rightarrow 3{x^3} + 4{x^2} + x = 480 \\
  $
The above equation if satisfied only and only if $ x = 5 $ . So, we can conclude that the base is $ 5 $ .
Then, in decimal number system, the number $ 3111 $ can be written as:
 $
   \Rightarrow 3 \times {5^3} + 1 \times {5^2} + 1 \times {5^1} + 1 \times {5^0} \\
   \Rightarrow 3 \times 125 + 1 \times 25 + 1 \times 5 + 1 \\
   \Rightarrow 375 + 25 + 5 + 1 \\
   \Rightarrow 406 \\
  $
Hence, the number $ 3111 $ when converted into a decimal number system becomes $ 406 $ .
So, the correct answer is “Option A”.

Note: One of the most common mistakes which students make is they mix up the conversion part. The conversion part is the most essential part in these types of questions. Students should have clear concepts of the decimal number system as well as number system to solve these types of problems.