---
title: "(1110)gray→BCD and (1430)₁₀→Excess-3"
subject: "Digital Logic"
chapter: 1
key: "semester_3/digital_logic:gray-to-bcd-decimal-to-excess3"
source: https://www.bctengineer.com/q/semester_3/digital_logic/gray-to-bcd-decimal-to-excess3
site: BCT Engineering
---

# (1110)gray→BCD and (1430)₁₀→Excess-3

Digital Logic · Chapter 1: Introduction

- Asked in: 2073 Shrawan, 3+2 marks
- Video: [How To Convert Gray Code to Binary and Binary to Gray Code](https://www.youtube.com/watch?v=cbmh1DPPQyI) — The Organic Chemistry Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/gray-to-bcd-decimal-to-excess3
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-1

## Question

Perform the following code conversions: i) (1110)gray = (?)BCD ii) (1430)₁₀ = (?)Excess-3

## Short answer

(i) Gray 1110 → binary 1011 (decimal 11) → BCD 0001 0001. (ii) Decimal 1430: add 3 to each digit → 4,7,6,3 → Excess-3 0100 0111 0110 0011.

## Answer

#### (i) $(1110)_{Gray}$ to BCD
**Step 1: Gray to binary.** Copy the MSB; each next binary bit = previous binary bit ⊕ current Gray bit.

| Gray bit | Operation | Binary bit |
|---|---|---|
| 1 (MSB) | copy | 1 |
| 1 | 1 ⊕ 1 | 0 |
| 1 | 0 ⊕ 1 | 1 |
| 0 | 1 ⊕ 0 | 1 |

$(1110)_{Gray} = (1011)_2 = 8 + 2 + 1 = (11)_{10}$

**Step 2: decimal to BCD.** Code each digit: 1 → 0001, 1 → 0001.

**Answer: $(1110)_{Gray} = (0001\ 0001)_{BCD}$**

#### (ii) $(1430)_{10}$ to Excess-3
Add 3 to each digit and write it in 4 bits:

| Digit | Digit + 3 | Excess-3 |
|---|---|---|
| 1 | 4 | 0100 |
| 4 | 7 | 0111 |
| 3 | 6 | 0110 |
| 0 | 3 | 0011 |

**Answer: $(1430)_{10} = (0100\ 0111\ 0110\ 0011)_{XS-3}$**

## Check yourself

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

1. What is the rule for Gray to binary conversion?
   A. Copy MSB; each next binary bit = previous binary bit XOR current Gray bit
   B. Each binary bit = Gray bit AND previous Gray bit
   C. Copy all bits unchanged
   D. Each binary bit = XOR of adjacent Gray bits
   Answer: A. B_i = B_(i+1) XOR G_i. (Key point: Gray to binary rule)

2. What is (1110) Gray in decimal?
   A. 14
   B. 11
   C. 7
   D. 13
   Answer: B. Binary 1011 = 8 + 2 + 1 = 11. (Key point: Gray result)

3. What is that decimal 11 in BCD?
   A. 1011
   B. 0001 1011
   C. 0001 0001
   D. 0010 0001
   Answer: C. Each digit is coded separately. (Key point: BCD result)

4. What is (1430) decimal in Excess-3?
   A. 0001 0100 0011 0000
   B. 0100 0111 0110 0000
   C. 0100 0111 0011 0110
   D. 0100 0111 0110 0011
   Answer: D. Digits 1,4,3,0 become 4,7,6,3. (Key point: XS-3 of 1430)

## Related in Digital Logic

- [Gray→BCD, decimal→Excess-3 and 2's complement addition](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Agray-bcd-excess3-twos-addition) — Digital Logic, chapter 1
- [BCD code and 3-bit binary-to-Gray converter](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Abcd-code-gray-converter) — Digital Logic, chapter 1
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- [39 to Gray/Excess-3 and 2's complement addition](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aconvert-39-gray-excess3-addition) — Digital Logic, chapter 1
- [Digital/analog signals, Gray and Excess-3 codes](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Adigital-analog-gray-excess3) — Digital Logic, chapter 1
- [Convert (1100 0011)Excess-3 to BCD](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aconvert-excess3-11000011-bcd) — Digital Logic, chapter 1

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