Number System Conversion Questions and Solutions | Binary, Octal, Decimal, Hex

If you want to clearly understand how to convert numbers from one system to another, you're in the right place. Number system conversion is very useful, especially in computer science. Computers use the binary number system (only 0s and 1s) to process data, while humans often prefer the hexadecimal system because it's easier to read and understand.

That’s why learning how to convert between systems like decimal, binary, octal, and hexadecimal is important. This topic is commonly asked in board exams, so practising these conversions can help you revise quickly and understand the topic better. You can also check your answers with the correct solutions to improve your learning and confidence.

What Does Number System Conversion Mean?

Number system conversion means changing a number from one type of number system to another by using certain methods or rules. For example, we can convert a number from the decimal system (base 10) to the binary system (base 2), and the other way around. Each number system has its own way of writing numbers, which helps us do calculations like addition, subtraction, and division.

There are many number systems, but the four most common ones are:

• Binary (Base 2): Uses only 0 and 1
• Octal (Base 8): Uses digits from 0 to 7
• Decimal (Base 10): Our everyday number system using digits 0 to 9
• Hexadecimal (Base 16): Uses digits 0–9 and letters A–F

Further reading: Number System .

To fully understand the concept, practice the number system conversion questions and solutions provided below.

A Number System Conversion Table provides a quick reference for converting values between different number systems such as Binary, Decimal, Octal, and Hexadecimal. It's a useful tool for students, programmers, and engineers working with digital systems and computer architecture.

Hexadecimal to Binary Conversion Table

A Hexadecimal to Binary Conversion Table helps in quickly converting each hexadecimal digit (0–F) into its 4-bit binary equivalent. This table is especially useful in computer science and digital electronics for translating data between numbering systems.

Hexadecimal	Binary
0	0000
1	0001
2	0010
3	0011
4	0100
5	0101
6	0110
7	0111
8	1000
9	1001
A	1010
B	1011
C	1100
D	1101
E	1110
F	1111

Number System Conversion Questions with Solutions

Conversion from Decimal Number System to Other Bases:

The steps below are used to convert a number from the decimal number system to the binary/octal/hexadecimal number system. The process of converting a number from the decimal system to another number system is illustrated below.

Step 1: Identify the base of the desired number system. For example, if we need to convert a specific number to the octal system, the base of the desired number is 8.

Step 2: Divide the given number by the base of the desired number system and write down the quotient and remainder in quotient-remainder form. Repeat this process until the quotient becomes less than the base.

Step 3: The equivalent number in the octal number system is obtained by writing the last quotient and all the remainders from bottom to top.

1. Convert the number 25 10 to the binary system.

Solution:

Given: 25 10 .

Now, we have to convert the decimal system to the binary number system . Hence, the base of the desired number is 2.

Step 1: Divide 25 by 2, we get: quotient = 12 and remainder = 1

Step 2: Divide 12 by 2, we get: quotient = 6 and remainder = 0

Step 3: Divide 6 by 2, we get: quotient = 3 and remainder = 0

Step 4: Divide 3 by 2, we get: quotient = 1 and remainder = 1

Now, write the quotient obtained in step 4 and the remainder from step 4 to step 1.

Hence, the binary equivalent of 25 10 is 11001 2 .

2. Convert the number 3456 10 to the binary system.

Solution:

Given: 3456 10 .

Now, we have to convert the decimal system to the binary number system. Hence, the base of the desired number is 2.

Step 1: Divide 3456 by 2.

⇒ Quotient = 1728 & Remainder = 0

Step 2: Divide 1728 by 2.

⇒ Quotient = 864 & Remainder = 0

Step 3: Divide 864 by 2.

⇒ Quotient = 432 & Remainder = 0

Step 4: Divide 432 by 2.

⇒ Quotient = 216 & Remainder = 0

Step 5: Divide 216 by 2.

⇒ Quotient = 108 & Remainder = 0

Step 6: Divide 108 by 2.

⇒ Quotient = 54 & Remainder = 0

Step7: Divide 54 by 2.

⇒ Quotient = 27 & Remainder = 0

Step 8: Divide 27 by 2.

⇒ Quotient = 13 & Remainder = 1

Step 9: Divide 13 by 2.

⇒ Quotient = 6 & Remainder = 1

Step 10: Divide 6 by 2.

⇒ Quotient = 3 & Remainder = 0

Step 11: Divide 3 by 2.

⇒ Quotient = 1 & Remainder = 1

To get the equivalent binary number for the decimal representation 3456 10 , write the quotient obtained in step 11 and the remainder from step 11 to step 1.

Therefore, 3456 10 = 110110000000 2 .

3. Convert 85 10 to the octal number system.

Solution:

Given: 85 10

Here, the required base number is 8 (i.e., octal number system). Hence, follow the below procedure to convert the decimal system to the octal system.

Step 1: Divide 85 by 8.

⇒ Quotient = 10 & Remainder = 5

Step 2: Divide 10 by 8.

⇒ Quotient = 1 & Remainder = 2

Since, the quotient “1” is less than “8”, we can stop the process.

Therefore, 85 10 = 125 8

4. Convert 500 10 to the hexadecimal number system.

Solution:

Given: 500 10

Now, we have to convert the decimal system to the hexadecimal number system .

So, divide the given number by 16.

Step 1: Divide 500 by 16.

⇒ Quotient = 31 & Remainder = 4 (4 can be represented as “4” in the hexadecimal number system)

Step 2: Divide 31 by 16.

⇒ Quotient = 1 & Remainder = 15 (15 can be represented as “F” in the hexadecimal number system)

As the quotient 1 is less than 16, stop the process.

Hence, 500 10 = 1F4 16 .

Other Base System to Decimal Conversion

The steps below are used to convert a number system from the binary or octal or hexadecimal system to the decimal system.

Step 1: Starting with the rightmost digit, multiply each digit of the provided number by the exponents of the base.

Step 2: Each step we take from right to left, the exponents should increase by one, starting with 0.

Step 3: Simplify and add each of the above-obtained products.

5. Convert the binary number 111001 2 to the decimal number system.

Solution:

Given binary number: 111001 2

Now, multiply each digit of the given binary number by the exponents of the base, starting with the right to left such that the exponents start with 0 and increase by 1.

Hence, the digits from right to left are written as follows:

1 = 1 × 2 0 = 1

0 = 0 × 2 1 = 0

0 = 0 × 2 2 = 0

1 = 1 × 2 3 = 8

1 = 1 × 2 4 = 16

1 = 1 × 2 5 = 32

Now, add all the product values obtained.

= 1 + 0 + 0 + 8 + 16 + 32

= 57

Hence, the decimal equivalent of 111001 2 is 57 10 .

I.e., 111001 2 = 57 10 .

Alternate Method:

111001 2 = (1 × 2 5 ) + (1 × 2 4 ) + (1 × 2 3 ) + (0 × 2 2 ) + (0 × 2 1 ) + (1 × 2 0 )

111001 2 = 32 + 16 + 8 + 0 + 0 + 1

111001 2 = 57 10 .

6. Convert the octal number 345 8 to the decimal number.

Solution:

Given octal number: 345 8

Since the base of the octal number system is 8, we have to multiply each digit of the given number with the exponents of the base.

Thus, the octal number 345 8 can be converted to the decimal system as follows:

345 8 = (3 × 8 2 ) + (4 × 8 1 ) + (5 × 8 0 )

345 8 = (3 × 64) + ( 4 × 8) + (5 × 1)

345 8 = 192 + 32 + 5

345 8 = 229 10

Hence, the decimal equivalent of the octal number 345 8 is 229 10 .

7. Convert 10110 2 to the decimal number system.

Solution:

Given binary number: 10110 2

Now, multiply each digit of 10110 with the exponents of the base, such that the exponents start with 0, and increase by 1 when moving from right to left.

So, 10110 2 = (1 × 2 4 ) + (0 × 2 3 ) + (1 × 2 2 ) + (1 × 2 1 ) + (0 × 2 0 )

10110 2 = 16 + 0 + 4 + 2 + 0

10110 2 = 22 10

Hence, the decimal number system 22 10 is equivalent to the binary number system 10110 2 .

8. Convert the hexadecimal number 3A4 to the decimal number system.

Solution:

The given hexadecimal number is 3A4.

As we know, the base of the hexadecimal number is 3A4.

Thus, 3A4 in the decimal number system is given as follows:

3A4 16 = (3 × 16 2 ) + (10 × 16 1 ) + (4 × 16 0 )

3A4 16 = (3 × 256) + (10 × 16) + (4 × 1)

3A4 16 = 768 + 160 + 4

3A4 16 = 932 10

Hence, the hexadecimal number 3A4 16 is equivalent to 932 10 .

Conversion from One Number System to Another Number System:

To convert a number from one number system (octal, binary, or hexadecimal) to another, it is first converted to the decimal system, and then the above methods are used to convert it to the desired number system.

9. Convert 1011 2 to octal number system.

Solution:

Given: 1011 2 .

To convert the binary number system to the octal number system, first we have to convert the binary system into decimal system, and then convert the decimal system to the octal number.

Conversion from Binary System to the Decimal System:

1011 2 = (1 × 2 3 ) + (0 × 2 2 ) + (1 × 2 1 ) + (1 × 2 0 )

1011 2 = (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1)

1011 2 = 8 + 0 + 2 + 1

1011 2 = 11 10

Conversion from Decimal Number System to the Octal Number System:

Here, we have to convert 11 10 to the octal system, and the required base is 8.

Hence,

Step 1: Divide 11 by 8.

⇒ Quotient = 1 & Remainder = 3

Since, quotient “1” is less than “8”, we cannot proceed further.

Therefore, the octal equivalent of 11 10 is 13 8

Hence, the octal number system equivalent to 1011 2 is 13 8 .

10. Convert 542 8 to the hexadecimal number system.

Solution: decimal= 354, hexa = 162

Given: 542 8 (i.e., octal number system)

Conversion from Octal to Decimal Number System:

The conversion from the octal number system to the decimal system is as follows:

542 8 = (5 × 8 2 ) + (4 × 8 1 ) + (2 × 8 0 )

542 8 = (5 × 64) + (4 × 8) + (2 × 1)

542 8 = 320 + 32 + 2

542 8 = 354 10

Conversion from Decimal to the Hexadecimal Number System:

Now, the number in the decimal system is 354 10 .

Thus, the process of converting the decimal number system to the hexadecimal system is as follows:

Step 1: Divide 354 by 16

⇒ Quotient = 22 & Remainder = 2 (2 can be represented as “2” in the hexadecimal system)

Step 2: Divide 22 by 16

⇒ Quotient = 1 & Remainder = 6 (6 can be represented as “6” in the hexadecimal system)

So, 354 10 = 162 16

Therefore, the octal number 542 8 equivalent to the hexadecimal number system is 162 16 .

Further Reading

• Perimeter and Area Questions
• Arithmetic Questions
• BODMAS Rule Questions
• Functions Questions
• Decimal to Binary Questions
• Conditional probability Questions

Practice Questions

Try to solve the following number system conversion questions:

Convert 78 10 to binary number system.

Convert 5BC to the decimal number system.

Convert 876 8 to the hexadecimal number system.

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