---
title: "Conversions: (21)₈, (671)₁₀ to BCD, (110101.011)₂"
subject: "Digital Logic"
chapter: 1
key: "semester_3/digital_logic:number-conversions-21-671"
source: https://www.bctengineer.com/q/semester_3/digital_logic/number-conversions-21-671
site: BCT Engineering
---

# Conversions: (21)₈, (671)₁₀ to BCD, (110101.011)₂

Digital Logic · Chapter 1: Introduction

- Asked in: 2081 Bhadra, 6 marks
- Video: [Binary to Octal and Octal to Binary Conversion | Number Conversion](https://www.youtube.com/watch?v=04QUJNuw6WE) — Learn Coding
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/number-conversions-21-671
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-1

## Question

Convert the following number system: i) (21)₈ = (?)₂ ii) (671)₁₀ = (?)BCD iii) (110101.011)₂ = (?)₁₀

## Short answer

(i) Octal 21 = 010 001 binary = 10001₂. (ii) 6→0110,7→0111,1→0001 gives BCD 0110 0111 0001. (iii) 2⁵+2⁴+2²+2⁰+2⁻²+2⁻³ = 53.375.

## Answer

#### (i) $(21)_8$ to binary
Write each octal digit in 3 bits: 2 = 010, 1 = 001.

$(21)_8 = (010\ 001)_2$ = **$(10001)_2$** (check: $2\times8 + 1 = 17 = 16 + 1$).

#### (ii) $(671)_{10}$ to BCD
Code each decimal digit in 4 bits:

| Digit | 6 | 7 | 1 |
|---|---|---|---|
| BCD | 0110 | 0111 | 0001 |

**Answer: $(671)_{10} = (0110\ 0111\ 0001)_{BCD}$**

#### (iii) $(110101.011)_2$ to decimal

$$
\begin{aligned}
&= 1\cdot2^5 + 1\cdot2^4 + 0\cdot2^3 + 1\cdot2^2 + 0\cdot2^1 + 1\cdot2^0 + 0\cdot2^{-1} + 1\cdot2^{-2} + 1\cdot2^{-3} \\
&= 32 + 16 + 4 + 1 + 0.25 + 0.125 \\
&= 53.375
\end{aligned}
$$

**Answer: $(110101.011)_2 = (53.375)_{10}$**

## Check yourself

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

1. What is (21) octal in binary?
   A. 10001
   B. 10010
   C. 01010
   D. 100001
   Answer: A. 2 = 010 and 1 = 001, leading zero dropped. (Key point: Octal to binary)

2. What is (671) decimal in BCD?
   A. 1010011111
   B. 0110 0111 0001
   C. 0110 0111 0010
   D. 0111 0110 0001
   Answer: B. 6, 7, 1 are coded separately. (Key point: Decimal to BCD)

3. What is (110101.011) binary in decimal?
   A. 53.75
   B. 52.375
   C. 53.375
   D. 26.375
   Answer: C. 32 + 16 + 4 + 1 + 0.25 + 0.125. (Key point: Binary to decimal)

4. What are the weights of fractional bits in binary?
   A. 2^1, 2^2, 2^3
   B. 10^-1, 10^-2, 10^-3
   C. 8^-1, 8^-2, 8^-3
   D. 2^-1, 2^-2, 2^-3
   Answer: D. Each place right of the point halves the weight. (Key point: Fraction weights)

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