---
title: "Convert (110111)gray to binary"
subject: "Digital Logic"
chapter: 1
key: "semester_3/digital_logic:convert-gray-110111-binary"
source: https://www.bctengineer.com/q/semester_3/digital_logic/convert-gray-110111-binary
site: BCT Engineering
---

# Convert (110111)gray to binary

Digital Logic · Chapter 1: Introduction

- Asked in: 2080 Baisakh, 1.5 marks
- Video: [Lec-9: Binary to Gray Code Conversion & Vice-versa](https://www.youtube.com/watch?v=R7uuIACpdGQ) — Gate Smashers
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/convert-gray-110111-binary
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-1

## Question

Convert (110111)gray = (?)₂.

## Short answer

Copy MSB, then compute each binary bit as previous binary ⊕ current Gray: 1, 1⊕1=0, 0⊕0=0, 0⊕1=1, 1⊕1=0, 0⊕1=1 gives (100101)₂.

## Answer

Copy the MSB; each next binary bit = previous binary bit ⊕ current Gray bit.

| Gray bit | 1 | 1 | 0 | 1 | 1 | 1 |
|---|---|---|---|---|---|---|
| Operation | copy | 1⊕1 | 0⊕0 | 0⊕1 | 1⊕1 | 0⊕1 |
| Binary bit | 1 | 0 | 0 | 1 | 0 | 1 |

**Answer: $(110111)_{Gray} = (100101)_2$** (= 37 in decimal)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is Gray converted to binary?
   A. XOR adjacent Gray bits
   B. Invert all Gray bits
   C. Copy MSB; next binary bit = previous binary bit XOR current Gray bit
   D. AND previous binary bit with Gray bit
   Answer: C. B_i = B_(i+1) XOR G_i. (Key point: Rule)

2. In 110111, the second binary bit is?
   A. 1
   B. 2
   C. Same as Gray
   D. 0 (1 XOR 1)
   Answer: D. B1 = 1 and G = 1, so 0. (Key point: Second bit)

3. (110111) Gray in binary?
   A. 100101
   B. 110111
   C. 101010
   D. 100110
   Answer: A. Following the XOR chain. (Key point: Final answer)

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