Binary number system in Intro to Computing - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ trace

intermediate

Trace the binary addition

What is the result of adding the binary numbers 1011 and 1101? Show the final binary sum.

A 10110

B 11000

C 10000

D 11110

💡 Hint

Add each bit from right to left, carrying over when sum exceeds 1.

Adding 1011 (11 decimal) and 1101 (13 decimal) in binary:

• 1 + 1 = 10 (write 0, carry 1)
• 1 + 0 + carry 1 = 10 (write 0, carry 1)
• 0 + 1 + carry 1 = 10 (write 0, carry 1)
• 1 + 1 + carry 1 = 11 (write 1, carry 1)
• Carry 1 is written as the leftmost bit

Final sum: 11000 (24 decimal).

🧠 Conceptual

intermediate

Understanding binary place values

Which decimal number does the binary number 100101 represent?

A 41

B 29

C 37

D 45

💡 Hint

Calculate the sum of powers of 2 where bits are 1.

Binary 100101 means:

• 1 × 2⁵ = 32
• 0 × 2⁴ = 0
• 0 × 2³ = 0
• 1 × 2² = 4
• 0 × 2¹ = 0
• 1 × 2⁰ = 1

Sum = 32 + 4 + 1 = 37 decimal.

❓ Comparison

advanced

Compare binary and decimal representations

Which of the following binary numbers is greater than decimal 20 but less than decimal 30?

A 11001

B 11100

C 10101

D 10011

💡 Hint

Convert each binary to decimal and check the range.

Convert each binary to decimal:

• 10101 = 21
• 11100 = 28
• 11001 = 25
• 10011 = 19

Numbers greater than 20 and less than 30 are 21, 25, and 28. Among options, 21 (10101), 25 (11001), and 28 (11100) fit the condition.

❓ identification

advanced

Identify the error in binary to decimal conversion

A student converts binary 1110 to decimal and writes the result as 14. What is the mistake in their conversion?

A They correctly converted; 1110 is 14 decimal.

B They forgot to multiply the leftmost bit by 8.

C They added the place values incorrectly; 1110 is 15 decimal.

D They reversed the bits; 1110 is 7 decimal.

💡 Hint

Calculate the decimal value of 1110 carefully.

Binary 1110 means:

• 1 × 2³ = 8
• 1 × 2² = 4
• 1 × 2¹ = 2
• 0 × 2⁰ = 0

Sum = 8 + 4 + 2 + 0 = 14 decimal, so the student's conversion is correct.

🚀 Application

expert

Convert and add binary numbers with carry

What is the decimal value of the binary sum of 1111 and 1011? Show the final decimal result.

A 32

B 30

C 28

D 26

💡 Hint

Add the binary numbers bit by bit, then convert the result to decimal.

Adding 1111 (15 decimal) and 1011 (11 decimal) in binary:

• 1 + 1 = 10 (write 0, carry 1)
• 1 + 1 + carry 1 = 11 (write 1, carry 1)
• 1 + 0 + carry 1 = 10 (write 0, carry 1)
• 1 + 1 + carry 1 = 11 (write 1, carry 1)
• Carry 1 is written as the leftmost bit

Final binary sum: 11010

Convert 11010 to decimal: 16 + 8 + 0 + 2 + 0 = 26 decimal.

Correction: The sum binary is 11010 which is 26 decimal, so option D is correct.

Practice

1. What digits are used in the binary number system?

easy

A. Digits 1 to 10

B. Digits 0 to 9

C. Digits 0 to 7

D. Only 0 and 1

Solution

1. Step 1: Understand the binary system basics

   The binary number system uses only two digits to represent all numbers.

2. Step 2: Identify the digits used

   These digits are 0 and 1, representing off and on states in computers.

3. Final Answer:

   Only 0 and 1 -> Option D

4. Quick Check:

   Binary digits = 0 and 1 [OK]

Hint: Binary uses just two digits: 0 and 1 [OK]

Common Mistakes:

• Confusing binary with decimal digits
• Thinking binary uses digits 0 to 9
• Assuming binary includes digits beyond 1

2. Which of the following is the correct binary number?

easy

A. 1101

B. 1234

C. 1021

D. 2010

Solution

1. Step 1: Check each digit for binary validity

   Binary numbers only contain 0s and 1s. Check each option's digits.

2. Step 2: Identify the valid binary number

   1101 is 1101, which contains only 1s and 0s. Others have digits like 2 or 3, invalid in binary.

3. Final Answer:

   1101 -> Option A

4. Quick Check:

   Valid binary = digits 0 or 1 only [OK]

Hint: Binary digits are only 0 or 1, no other digits allowed [OK]

Common Mistakes:

• Choosing numbers with digits other than 0 or 1
• Confusing binary with decimal or octal numbers
• Ignoring invalid digits in options

3. What is the decimal value of the binary number 1011?

medium

A. 7

B. 11

C. 13

D. 9

Solution

1. Step 1: Assign powers of 2 to each bit

   From right to left, bits represent 2^0=1, 2^1=2, 2^2=4, 2^3=8.

2. Step 2: Calculate decimal value by adding bits with 1

   Bits: 1(8) + 0(4) + 1(2) + 1(1) = 8 + 0 + 2 + 1 = 11.

3. Final Answer:

   11 -> Option B

4. Quick Check:

   Binary 1011 = Decimal 11 [OK]

Hint: Add powers of 2 where bit is 1, starting from right [OK]

Common Mistakes:

• Misaligning bit positions and powers of 2
• Adding all bits instead of weighted values
• Confusing binary digits with decimal digits

4. Identify the error in this binary number: 11012

medium

A. Binary numbers cannot start with 1

B. Too many digits for a binary number

C. Contains digit '2' which is invalid in binary

D. Missing leading zeros

Solution

1. Step 1: Check each digit for binary validity

   Binary digits must be only 0 or 1. The digit '2' is not allowed.

2. Step 2: Confirm the invalid digit

   The presence of '2' makes the number invalid as a binary number.

3. Final Answer:

   Contains digit '2' which is invalid in binary -> Option C

4. Quick Check:

   Binary digits = 0 or 1 only [OK]

Hint: Binary digits never include 2 or higher [OK]

Common Mistakes:

• Thinking binary numbers have digit limits on length
• Believing binary cannot start with 1
• Assuming missing zeros is an error

5. Convert the decimal number 18 to binary.

hard

A. 10010

B. 11001

C. 10110

D. 10001

Solution

1. Step 1: Divide decimal number by 2 repeatedly

   18 ÷ 2 = 9 remainder 0; 9 ÷ 2 = 4 remainder 1; 4 ÷ 2 = 2 remainder 0; 2 ÷ 2 = 1 remainder 0; 1 ÷ 2 = 0 remainder 1.

2. Step 2: Write remainders in reverse order

   Reading remainders from last to first: 1 0 0 1 0.

3. Final Answer:

   10010 -> Option A

4. Quick Check:

   Decimal 18 = Binary 10010 [OK]

Hint: Divide by 2, collect remainders backward [OK]

Common Mistakes:

• Reading remainders in wrong order
• Mixing up division and subtraction steps
• Confusing binary digits with decimal digits