Binary & Quinary Number Systems – Class 8 Math | Math Mantra Nepal

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CDC Class 8 Mathematics | Unit: Number Systems

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Review of Number Systems

In mathematics and computer science, numbers are represented using different bases. In our daily lives, we use the Decimal System (Base 10) with digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$. In Class 8 CDC Curriculum, we study two other vital bases: Binary System (Base 2) and Quinary System (Base 5).

Binary Number System (Base 2)

Uses only two digits: 0 and 1.
Base value is 2 (written as $(\dots)_2$).
Extensively used in computer circuits, microprocessors, and electronic price coding.

Quinary Number System (Base 5)

Uses five digits: 0, 1, 2, 3, and 4.
Base value is 5 (written as $(\dots)_5$).
Historically derived from counting digits using 5 fingers of one hand.

Binary Switch / Light Bulb Analogy

Digital computers represent binary numbers using electrical switches (or Light Bulbs):
• An ON Bulb (Switched ON) represents digit 1.
• An OFF Bulb (Switched OFF) represents digit 0.

Board Work Example: Convert Decimal $27_{10}$ to Binary

Successive Division Method (by 2):

Divisor	Dividend	Remainder
2	27	1 ↑ (LSB)
2	13	1
2	6	0
2	3	1
2	1	1 ↑ (MSB)
–	0	–

Reading Remainders from Bottom to Top:

$27_{10} = 11011_2$

Bulb Light Representation for $11011_2$

1 (ON)

0 (OFF)

1 (ON)

5 Light Bulbs: ON – ON – OFF – ON – ON

Solved Model Questions (Set of 10)

Solved step-by-step line-by-line vertically for conversions among Decimal, Binary, and Quinary systems, including practical real-life shop coding models.

Question 1 (Decimal to Binary)

Convert the Decimal number $35_{10}$ into Binary System.

Solution:

Given, Decimal Number = $35$

Dividing continuously by 2 and recording remainders:

2 | 35 — Remainder: 1

2 | 17 — Remainder: 1

2 | 8 — Remainder: 0

2 | 4 — Remainder: 0

2 | 2 — Remainder: 0

2 | 1 — Remainder: 1

Writing remainders from bottom to top: $35_{10} = 100011_2$.

Question 2 (Binary to Decimal)

Convert the Binary number $110101_2$ into Decimal System.

Solution:

Given, Binary Number = $110101_2$

Expanding with powers of 2:

$$ 110101_2 = (1 \times 2^5) + (1 \times 2^4) + (0 \times 2^3) + (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) $$

$$ \text{or, } 110101_2 = (1 \times 32) + (1 \times 16) + (0 \times 8) + (1 \times 4) + (0 \times 2) + (1 \times 1) $$

$$ \text{or, } 110101_2 = 32 + 16 + 0 + 4 + 0 + 1 $$

$$ \therefore 110101_2 = 53_{10} $$

Question 3 (Decimal to Quinary)

Convert the Decimal number $148_{10}$ into Quinary System (Base 5).

Solution:

Dividing continuously by 5 and recording remainders:

5 | 148 — Remainder: 3

5 | 29 — Remainder: 4

5 | 5 — Remainder: 0

5 | 1 — Remainder: 1

Writing remainders from bottom to top: $148_{10} = 1043_5$.

Question 4 (Quinary to Decimal)

Convert the Quinary number $4231_5$ into Decimal System.

Solution:

Expanding with powers of 5:

$$ 4231_5 = (4 \times 5^3) + (2 \times 5^2) + (3 \times 5^1) + (1 \times 5^0) $$

$$ \text{or, } 4231_5 = (4 \times 125) + (2 \times 25) + (3 \times 5) + (1 \times 1) $$

$$ \text{or, } 4231_5 = 500 + 50 + 15 + 1 $$

$$ \therefore 4231_5 = 566_{10} $$

Question 5 (Binary to Quinary)

Convert the Binary number $11100_2$ into Quinary System.

Solution:

Step 1: Convert Binary $11100_2$ to Decimal

$$ 11100_2 = (1 \times 2^4) + (1 \times 2^3) + (1 \times 2^2) + (0 \times 2^1) + (0 \times 2^0) $$

$$ = 16 + 8 + 4 + 0 + 0 = 28_{10} $$

Step 2: Convert Decimal $28_{10}$ to Quinary (by dividing by 5)

5 | 28 — Remainder: 3

5 | 5 — Remainder: 0

5 | 1 — Remainder: 1

Therefore, $11100_2 = 103_5$.

Question 6 (Quinary to Binary)

Convert the Quinary number $134_5$ into Binary System.

Solution:

Step 1: Convert Quinary $134_5$ to Decimal

$$ 134_5 = (1 \times 5^2) + (3 \times 5^1) + (4 \times 5^0) $$

$$ = 25 + 15 + 4 = 44_{10} $$

Step 2: Convert Decimal $44_{10}$ to Binary (by dividing by 2)

2 | 44 — Remainder: 0

2 | 22 — Remainder: 0

2 | 11 — Remainder: 1

2 | 5 — Remainder: 1

2 | 2 — Remainder: 0

2 | 1 — Remainder: 1

Therefore, $134_5 = 101100_2$.

Question 7 (Decimal 40 to Binary & Light Switch Pattern)

Convert the Decimal number $40_{10}$ into Binary System. Represent the resulting binary code using a sequence of Light Bulbs (ON/OFF status).

Solution:

Given, Decimal Number = $40$

Successive division by 2:

2 | 40 — Remainder: 0

2 | 20 — Remainder: 0

2 | 10 — Remainder: 0

2 | 5 — Remainder: 1

2 | 2 — Remainder: 0

2 | 1 — Remainder: 1

Binary Value: $40_{10} = 101000_2$

Light Bulb Switch Status for $101000_2$:

1 (ON)

0 (OFF)

1 (ON)

0 (OFF)

Hence, $40_{10} = 101000_2$ and the bulb pattern is: ON – OFF – ON – OFF – OFF – OFF.

Question 8 (Sita’s Shop – Binary Price Coding)

Sita runs a fancy shop and labels price tags using binary codes containing only digits 0 and 1.
(a) If an item has a price code $101_2$, what is its price in Rupees?
(b) How will Sita write the price code for an item costing $Rs.\,65$?

Solution:

Part (a): Price for Code $101_2$

$$ 101_2 = (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) $$

$$ \text{or, } 101_2 = (1 \times 4) + (0 \times 2) + (1 \times 1) = 4 + 0 + 1 = 5 $$

$$ \therefore \text{Price in Rupees} = Rs.\,5 $$

Part (b): Code for Price $Rs.\,65$

Dividing 65 continuously by 2:

2 | 65 — Remainder: 1

2 | 32 — Remainder: 0

2 | 16 — Remainder: 0

2 | 8 — Remainder: 0

2 | 4 — Remainder: 0

2 | 2 — Remainder: 0

2 | 1 — Remainder: 1

$$ \therefore 65_{10} = 1000001_2 $$

Therefore, price for code $101_2$ is $Rs.\,5$ and Sita writes $1000001_2$ for $Rs.\,65$.

Question 9 (Ram’s Shoe Shop – Quinary Price System)

Ram owns a shoe store and uses a price coding system consisting of digits from 0 to 4.
(a) On which number system is Ram’s price coding based?
(b) Which code will he use to label a pair of shoes priced at $Rs.\,800$?
(c) If a pair of shoes has a price code $202_5$, what is its actual price in Rupees?

Solution:

Part (a): Identification of Number System

Since the digits used are 0, 1, 2, 3, and 4 (5 digits), Ram’s price coding is based on the Quinary Number System (Base 5).

Part (b): Code for Shoes Priced at $Rs.\,800$

Dividing 800 continuously by 5:

5 | 800 — Remainder: 0

5 | 160 — Remainder: 0

5 | 32 — Remainder: 2

5 | 6 — Remainder: 1

5 | 1 — Remainder: 1

$$ \therefore 800_{10} = 11200_5 $$

Part (c): Actual Price for Code $202_5$

$$ 202_5 = (2 \times 5^2) + (0 \times 5^1) + (2 \times 5^0) $$

$$ \text{or, } 202_5 = (2 \times 25) + (0 \times 5) + (2 \times 1) = 50 + 0 + 2 = 52 $$

$$ \therefore \text{Actual Price} = Rs.\,52 $$

Hence, System = Quinary (Base 5), Code for $Rs.\,800 = 11200_5$, and Price for code $202_5 = Rs.\,52$.

Question 10 (System Comparison)

Which number is greater: $10110_2$ or $42_5$? Show by converting both into Decimal System.

Solution:

Converting $10110_2$ to Decimal:

$$ 10110_2 = (1 \times 16) + (0 \times 8) + (1 \times 4) + (1 \times 2) + (0 \times 1) = 16 + 4 + 2 = 22_{10} $$

Converting $42_5$ to Decimal:

$$ 42_5 = (4 \times 5) + (2 \times 1) = 20 + 2 = 22_{10} $$

Since both equal $22_{10}$, the two numbers are equal ($10110_2 = 42_5$).

Class 8 BLE Model Practice Questions (Set of 10)

Specially designed according to CDC Nepal Class 8 BLE Exam pattern.

1. Convert the Decimal number $45_{10}$ into Binary System.

2. Convert the Binary number $111011_2$ into Decimal System.

3. Convert the Decimal number $234_{10}$ into Quinary System (Base 5).

4. Convert the Quinary number $3104_5$ into Decimal System.

5. Convert the Binary number $101101_2$ directly into Quinary System.

6. Convert the Quinary number $241_5$ into Binary System.

7. Convert Decimal number $52_{10}$ into Binary and write its Light Bulb ON/OFF status.

8. Sita encodes prices using digits 0 and 1 only. (a) What price does code $1101_2$ represent? (b) Write the code for a dress costing $Rs.\,75$.

9. Ram labels shoe prices using digits 0, 1, 2, 3, and 4. (a) Name his number system. (b) Find the actual price for code $314_5$. (c) Write the code for shoes priced at $Rs.\,120$.

10. Which digits are used in Base 5 (Quinary system)? Write the largest 3-digit number in Quinary System.

Answer Key (10 BLE Practice Questions)

Q1: $101101_2$

Q2: $59_{10}$

Q3: $1414_5$

Q4: $404_{10}$

Q5: $140_5$ (as $101101_2 = 45_{10}$)

Q6: $111001_2$ (as $241_5 = 71_{10}$)

Q7: $110100_2$ (Bulbs: ON-ON-OFF-ON-OFF-OFF)

Q8: (a) $Rs.\,13$, (b) $1001011_2$

Q9: (a) Quinary (Base 5), (b) $Rs.\,84$, (c) $440_5$

Q10: Digits = 0, 1, 2, 3, 4. Largest 3-digit number = $444_5$

Number Systems Concept Quiz (10 Questions)

Score: 0 / 10

Test your knowledge of bases, digits, and system conversions!

1. Which digits are used in Binary Number System?

2. What is the base of Quinary Number System?

3. In a light bulb representation of binary numbers, an ON bulb represents:

4. Which of the following is NOT a valid number in Quinary System?

5. What is the decimal equivalent of binary $101_2$?

6. Sita labels shop price tags using binary digits (0 and 1). What is the price in Rupees for code $1010_2$?

7. What is the decimal value of quinary $10_5$?

8. Decimal number $8_{10}$ in Binary System is:

9. Ram uses digits 0, 1, 2, 3, and 4 to code shoe prices. Which number system is his store using?

10. To convert a Decimal number into Quinary System, we repeatedly divide by:

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Math Mantra Nepal

Binary and Quinary Number System – BLE

Math Mantra Nepal Edu

Review of Number Systems

Binary Number System (Base 2)

Quinary Number System (Base 5)

Binary Switch / Light Bulb Analogy

Board Work Example: Convert Decimal \(27_{10}\) to Binary

Solved Model Questions (Set of 10)

Solution:

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Class 8 BLE Model Practice Questions (Set of 10)

Answer Key (10 BLE Practice Questions)

Number Systems Concept Quiz (10 Questions)