Binary System Exercises - Addition | Download Free PDF ... - Free Printable

Let's solve each of the binary addition and subtraction problems step by step. I'll go through them one by one, explaining the process for both binary addition and binary subtraction, including borrowing and carrying as needed.

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🔹 Binary Addition Rules:

- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 10 (write down 0, carry 1)
- 1 + 1 + 1 = 11 (write down 1, carry 1)

🔹 Binary Subtraction Rules:

- 0 - 0 = 0
- 1 - 0 = 1
- 1 - 1 = 0
- 0 - 1 = 1 (with borrow from the next higher bit, making it 10 - 1 = 1)

We’ll now solve each problem:

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1) 10110 - 01101

Align:
```
10110
- 01101
-------
```

Start from right to left:

- 0 - 1 → borrow needed → 10 - 1 = 1, borrow from left
- Next: 0 (after borrow) - 0 → but we borrowed, so it becomes 0 → need to borrow again → 10 - 0 = 10? Wait — better to do column by column.

Let’s write with positions:

```
1 0 1 1 0
- 0 1 1 0 1
------------
```

Step-by-step:

- Rightmost (bit 0): 0 - 1 → borrow → 10 - 1 = 1, mark borrow
- Bit 1: 1 → becomes 0 after borrow; 0 - 0 = 0? But we have 1 → wait:

Actually:

- Bit 0: 0 - 1 → can't do → borrow from bit 1 → bit 1 is 1 → becomes 0, bit 0 becomes 10 → 10 - 1 = 1
- Bit 1: 0 (after borrow) - 0 = 0
- Bit 2: 1 - 1 = 0
- Bit 3: 0 - 1 → can't do → borrow from bit 4 → bit 4 is 1 → becomes 0, bit 3 becomes 10 → 10 - 1 = 1
- Bit 4: 0 (after borrow) - 0 = 0

Wait! Let’s write it clearly:

```
1 0 1 1 0
- 0 1 1 0 1
------------
```

Now compute:

- Bit 0: 0 - 1 → borrow → 10 - 1 = 1 → borrow from bit 1
- Bit 1: was 1 → becomes 0 → 0 - 0 = 0
- Bit 2: 1 - 1 = 0
- Bit 3: 0 - 1 → can’t → borrow from bit 4 → bit 4 is 1 → becomes 0, bit 3 becomes 10 → 10 - 1 = 1
- Bit 4: 0 - 0 = 0

So result: 01001

But leading zero can be dropped → 1001

✔ Answer: 1001

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2) 10110 + 11000

```
10110
+ 11000
-------
```

Add:

- 0 + 0 = 0
- 1 + 0 = 1
- 1 + 0 = 1
- 0 + 1 = 1
- 1 + 1 = 10 → write 0, carry 1

Now carry over: 1 (carry) → total: 101110

Let’s write:

```
10110
+ 11000
-------
101110
```

✔ Answer: 101110

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3) 11011 - 11000

```
11011
- 11000
-------
```

Subtract:

- 1 - 0 = 1
- 1 - 0 = 1
- 0 - 0 = 0
- 1 - 1 = 0
- 1 - 1 = 0

Result: 00011 → 11

✔ Answer: 11

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4) 11011 + 01011

```
11011
+ 01011
-------
```

Add:

- 1 + 1 = 10 → write 0, carry 1
- 1 + 1 + 1 = 11 → write 1, carry 1
- 0 + 0 + 1 = 1
- 1 + 1 = 10 → write 0, carry 1
- 1 + 0 + 1 = 10 → write 0, carry 1

Final carry: 1

So: 100110

Let’s write:

```
11011
+ 01011
-------
100110
```

✔ Answer: 100110

---

5) 11111 - 01011

```
11111
- 01011
-------
```

Subtract:

- 1 - 1 = 0
- 1 - 1 = 0
- 1 - 0 = 1
- 1 - 1 = 0
- 1 - 0 = 1

→ 10100

✔ Answer: 10100

---

6) 11111 + 01100

```
11111
+ 01100
-------
```

Add:

- 1 + 0 = 1
- 1 + 0 = 1
- 1 + 1 = 10 → write 0, carry 1
- 1 + 1 + 1 = 11 → write 1, carry 1
- 1 + 0 + 1 = 10 → write 0, carry 1

Final carry: 1 → 101011

✔ Answer: 101011

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7) 01100 + 01001

```
01100
+ 01001
-------
```

Add:

- 0 + 1 = 1
- 0 + 0 = 0
- 1 + 0 = 1
- 1 + 1 = 10 → write 0, carry 1
- 0 + 0 + 1 = 1

→ 10101

✔ Answer: 10101

---

8) 11100 - 01001

```
11100
- 01001
-------
```

Subtract:

- 0 - 1 → borrow → 10 - 1 = 1 → borrow from left
- 0 (after borrow) - 0 → but we borrowed → so 0 → need to borrow again → 10 - 0 = 10? Let's do carefully:

Bit positions (from right):

- Bit 0: 0 - 1 → borrow → 10 - 1 = 1 → borrow from bit 1
- Bit 1: 0 → becomes 0 after borrow → but needs to subtract 0 → 0 - 0 = 0 → but we already used it? Wait:

Better:

```
1 1 1 0 0
- 0 1 0 0 1
------------
```

- Bit 0: 0 - 1 → borrow → 10 - 1 = 1 → borrow from bit 1
- Bit 1: was 0 → becomes 10 (after borrow), then subtract 0 → 10 - 0 = 10? No — after borrow, it's 10 (but we use only one digit). Actually:

After borrow, bit 1 becomes 0 (since it was 0), but we must borrow from bit 2.

So:

- Bit 0: 10 - 1 = 1 → borrow from bit 1
- Bit 1: 0 → can't lend → borrow from bit 2 → bit 2 is 1 → becomes 0, bit 1 becomes 10 → then we give 1 to bit 0 → so bit 1 becomes 9? No — in binary.

Standard way:

- Bit 0: 0 - 1 → borrow → 10 - 1 = 1 → bit 1 reduces by 1
- Bit 1: 0 → becomes -1 → can't → so borrow from bit 2 → bit 2 is 1 → becomes 0, bit 1 becomes 10 → now subtract 0 → 10 - 0 = 10 → but we only write one digit → actually:

Wait: after borrowing:

- Bit 1: 10 (after borrow) → but we need to subtract 0 → so 10 - 0 = 10 → but we are at bit 1 → we write 0 and carry?

No — let's do it properly.

Better: write step by step:

```
1 1 1 0 0
- 0 1 0 0 1
------------
```

From right:

- Bit 0: 0 - 1 → borrow → 10 - 1 = 1 → borrow from bit 1
- Bit 1: 0 → becomes 0 after borrow → but needs to lend → so borrow from bit 2 → bit 2 is 1 → becomes 0, bit 1 becomes 10 → now subtract 0 → 10 - 0 = 10 → but we only need to write one digit → so bit 1 = 0, carry 1 to next?
Wait — no.

Actually, after borrowing:

- Bit 1: 10 (after borrow from bit 2) → then subtract 0 → 10 - 0 = 10 → so write 0, carry 1 to bit 2?

No — that's not how it works.

Let me do it manually:

We can convert to decimal to verify:

- 11100₂ = 28
- 01001₂ = 9
- 28 - 9 = 19 → 19₁₀ = 10011₂

So answer should be 10011

Let’s check via binary subtraction:

```
1 1 1 0 0
- 0 1 0 0 1
------------
```

We want to get 10011

Do it:

- Bit 0: 0 - 1 → borrow → 10 - 1 = 1 → borrow from bit 1
- Bit 1: 0 → becomes 0 → can't lend → borrow from bit 2 → bit 2 is 1 → becomes 0, bit 1 becomes 10 → now subtract 0 → 10 - 0 = 10 → but we only write 0, carry 1? No.

Wait — after borrowing, bit 1 becomes 10 → then we subtract 0 → so 10 - 0 = 10 → but we are writing digits from right.

Actually:

After borrow chain:

- Bit 0: 10 - 1 = 1 → borrow from bit 1
- Bit 1: 0 → but we need to borrow → so borrow from bit 2 → bit 2 is 1 → becomes 0, bit 1 becomes 10 → now we give 1 to bit 0 → so bit 1 becomes 1 (after giving) → then subtract 0 → 1 - 0 = 1
- Bit 2: 0 (after borrow) - 0 = 0
- Bit 3: 1 - 1 = 0
- Bit 4: 1 - 0 = 1

So result: 1 0 0 1 1 → 10011

✔ Answer: 10011

---

9) 11100 - 01000

```
11100
- 01000
-------
```

- 0 - 0 = 0
- 0 - 0 = 0
- 1 - 0 = 1
- 1 - 1 = 0
- 1 - 0 = 1

→ 10100

✔ Answer: 10100

---

10) 10111 - 01000

```
10111
- 01000
-------
```

- 1 - 0 = 1
- 1 - 0 = 1
- 1 - 0 = 1
- 0 - 1 → borrow → 10 - 1 = 1 → borrow from left
- 1 (after borrow) - 0 = 1

So:

- Bit 0: 1
- Bit 1: 1
- Bit 2: 1
- Bit 3: 10 - 1 = 1 → borrow → bit 4 becomes 0
- Bit 4: 0 - 0 = 0

Wait: original bits:

- Bit 4: 1 → after borrow → 0
- Bit 3: 0 → becomes 10 → subtract 1 → 1
- So: 01111

But leading zero → 1111

Wait: let's write:

```
1 0 1 1 1
- 0 1 0 0 0
------------
```

- Bit 0: 1 - 0 = 1
- Bit 1: 1 - 0 = 1
- Bit 2: 1 - 0 = 1
- Bit 3: 0 - 1 → borrow → 10 - 1 = 1 → borrow from bit 4
- Bit 4: 1 → becomes 0 → 0 - 0 = 0

So result: 01111 → 1111

✔ Answer: 1111

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11) 10111 + 10000

```
10111
+ 10000
-------
```

- 1 + 0 = 1
- 1 + 0 = 1
- 1 + 0 = 1
- 0 + 0 = 0
- 1 + 1 = 10 → write 0, carry 1

→ 100111

✔ Answer: 100111

---

12) 10010 - 10000

```
10010
- 10000
-------
```

- 0 - 0 = 0
- 1 - 0 = 1
- 0 - 0 = 0
- 0 - 0 = 0
- 1 - 1 = 0

→ 00010 → 10

✔ Answer: 10

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13) 10010 + 10011

```
10010
+ 10011
-------
```

- 0 + 1 = 1
- 1 + 1 = 10 → write 0, carry 1
- 0 + 0 + 1 = 1
- 0 + 0 = 0
- 1 + 1 = 10 → write 0, carry 1

Final carry: 1 → 100101

✔ Answer: 100101

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14) 10011 + 10101

```
10011
+ 10101
-------
```

- 1 + 1 = 10 → write 0, carry 1
- 1 + 0 + 1 = 10 → write 0, carry 1
- 0 + 1 + 1 = 10 → write 0, carry 1
- 0 + 0 + 1 = 1
- 1 + 1 = 10 → write 0, carry 1

Final carry: 1 → 100100

Wait: let's write:

```
10011
+ 10101
-------
100100
```

Yes.

✔ Answer: 100100

---

15) 10101 + 11010

```
10101
+ 11010
-------
```

- 1 + 0 = 1
- 0 + 1 = 1
- 1 + 0 = 1
- 0 + 1 = 1
- 1 + 1 = 10 → write 0, carry 1

→ 101111

✔ Answer: 101111

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16) 11010 + 11001

```
11010
+ 11001
-------
```

- 0 + 1 = 1
- 1 + 0 = 1
- 0 + 0 = 0
- 1 + 1 = 10 → write 0, carry 1
- 1 + 1 + 1 = 11 → write 1, carry 1

Final carry: 1 → 110011

✔ Answer: 110011

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## ✔ Final Answers:

| Problem | Answer |
|--------|-------------|
| 1) | 1001 |
| 2) | 101110 |
| 3) | 11 |
| 4) | 100110 |
| 5) | 10100 |
| 6) | 101011 |
| 7) | 10101 |
| 8) | 10011 |
| 9) | 10100 |
| 10) | 1111 |
| 11) | 100111 |
| 12) | 10 |
| 13) | 100101 |
| 14) | 100100 |
| 15) | 101111 |
| 16) | 110011 |

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Let me know if you'd like these verified with decimal conversions or a printable version!