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Number Systems: Question 7

Syllabus 1.1

Structured 6 marks

A smart parking meter records each parking bay's session code and displays it in different number systems depending on which maintenance app is used to read it.

(a) Convert the hexadecimal value 2F to a denary value. Show your working. [2]

(b) Convert the denary value 214 to a hexadecimal value. Show your working. [2]

(c) Convert the 8-bit binary value 11001001 directly to a denary value. Show your working. [2]

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Worked solution

Part (a): Hexadecimal to denary (2F)

A 2-digit hexadecimal number has a 16s column and a 1s column, just as a 2-digit denary number has a 10s column and a 1s column.

  • The first digit, 2, is in the 16s column: 2 x 16 = 32
  • The second digit, F, is worth 15 in denary, and is in the 1s column: 15 x 1 = 15

Adding the columns: 32 + 15 = 47.

So 2F in hexadecimal = 47 in denary.

Part (b): Denary to hexadecimal (214)

Divide repeatedly by 16 and read off the quotient and remainder.

  • 214 DIV 16 = 13 remainder 6 (since 16 x 13 = 208, and 214 - 208 = 6)

The quotient, 13, becomes the first hexadecimal digit. Since 13 is greater than 9, it is written as the hexadecimal digit D. The remainder, 6, becomes the second hexadecimal digit.

So 214 in denary = D6 in hexadecimal.

Check: D6 = (13 x 16) + 6 = 208 + 6 = 214. Correct.

Part (c): Binary to denary (11001001)

Use the binary place values 128, 64, 32, 16, 8, 4, 2, 1, and add together the place values where the bit is 1.

Bit pattern: 1 1 0 0 1 0 0 1

  • bit 7 (128): 1, so include 128
  • bit 6 (64): 1, so include 64
  • bit 5 (32): 0
  • bit 4 (16): 0
  • bit 3 (8): 1, so include 8
  • bit 2 (4): 0
  • bit 1 (2): 0
  • bit 0 (1): 1, so include 1

Adding the included place values: 128 + 64 + 8 + 1 = 201.

So 11001001 in binary = 201 in denary.

Final answers

  • (a) 2F = 47
  • (b) 214 = D6
  • (c) 11001001 = 201