---
title: "Convert (5A.B)₁₆ to Excess-3"
subject: "Digital Logic"
chapter: 1
key: "semester_3/digital_logic:convert-5a-b-excess3"
source: https://www.bctengineer.com/q/semester_3/digital_logic/convert-5a-b-excess3
site: BCT Engineering
---

# Convert (5A.B)₁₆ to Excess-3

Digital Logic · Chapter 1: Introduction

- Asked in: 2078 Kartik, 2 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/convert-5a-b-excess3
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-1

## Question

Convert (5A.B)₁₆ = (?)Excess-3.

## Short answer

First convert (5A.B)₁₆ to decimal (90.6875)₁₀. Then add 3 to each decimal digit’s 4-bit BCD to obtain excess-3: 1100 0011.1001 1011 1010 1000.

## Answer

Excess-3 code is defined for decimal digits, so first convert the hexadecimal number to decimal, then add 3 (0011) to the BCD code of each decimal digit.

**Step 1: Hex to decimal**

$$
\begin{aligned} (5A.B)_{16} &= 5 \times 16^1 + 10 \times 16^0 + 11 \times 16^{-1} \\ &= 80 + 10 + 0.6875 \\ &= (90.6875)_{10} \end{aligned}
$$

**Step 2: Each digit + 3, written in 4 bits**

| Digit | 9 | 0 | . | 6 | 8 | 7 | 5 |
|---|---|---|---|---|---|---|---|
| Digit + 3 | 12 | 3 | . | 9 | 11 | 10 | 8 |
| XS-3 | 1100 | 0011 | . | 1001 | 1011 | 1010 | 1000 |

**Answer:** $(5A.B)_{16} = (90.6875)_{10} = (1100\ 0011.1001\ 1011\ 1010\ 1000)_{XS\text{-}3}$

## Check yourself

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

1. Why must (5A.B)16 be converted to decimal first?
   A. XS-3 works only on hex digits
   B. Hex has no binary form
   C. XS-3 needs octal input
   D. XS-3 is defined for decimal digits
   Answer: D. XS-3 codes each decimal digit as digit + 3. (Key point: Method)

2. What is (5A.B)16 in decimal?
   A. 90.11
   B. 91.6875
   C. 80.6875
   D. 90.6875
   Answer: D. 80 + 10 + 11/16 = 90.6875. (Key point: Hex to decimal)

3. What is the XS-3 code of decimal digit 9?
   A. 1001
   B. 1010
   C. 1100
   D. 1000
   Answer: C. 9 + 3 = 12 = 1100. (Key point: Digit coding)

4. What is the XS-3 of the first two digits 90?
   A. 1001 0000
   B. 1100 0011
   C. 0011 1100
   D. 1100 0000
   Answer: B. 9 gives 1100 and 0 gives 0011. (Key point: Final result)

## Related in Digital Logic

- [Convert (93)₁₀ to Excess-3](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aconvert-93-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
- [Convert (1001 1010)excess-3 to BCD](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aconvert-excess3-10011010-bcd) — Digital Logic, chapter 1
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