Binary/Hexadecimal Worksheet for practicing number system conversions.

Let's solve each problem step by step, filling in the missing values for Decimal, Hexadecimal, and Binary. We'll go through each row and convert between the number systems as needed.

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🔁 Conversion Rules:

- Binary to Decimal: Use powers of 2.
- Binary to Hex: Group bits into sets of 4 (from right), convert each group to hex digit.
- Decimal to Hex: Divide by 16 repeatedly.
- Hex to Decimal: Multiply each digit by 16^n.
- Hex to Binary: Convert each hex digit to 4-bit binary.

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✔ Problem 1: `10111010₂` → Decimal = 186, Hex = BA

We are given:
- Binary: `10111010₂`
- Decimal: `186`
- Hex: `BA`

✔ All correct.

Now let’s verify and fill in Binary column — but it’s already filled? Wait, no — the binary is given, so we just confirm.

But the binary is already provided: `10111010₂`, so nothing to do here.

Wait — actually, looking at the worksheet:

> Each row has one value given in one system, and others are blank or partially filled.

But in this case, all three are filled? Let’s check:

1. Binary: `10111010₂`
Decimal: `186`
Hex: `BA`

Let’s verify:

- `10111010₂` = 1×128 + 0×64 + 1×32 + 1×16 + 1×8 + 0×4 + 1×2 + 0×1 = 128+32+16+8+2 = 186 ✔
- Hex: Split into groups: `1011 1010` → B A → `BA` ✔

So all correct. No need to fill anything.

---

✔ Problem 2: `11110011₂` → Decimal = 243, Hex = F3

Given:
- Binary: `11110011₂`
- Decimal: `243`
- Hex: `F3`

Verify:

- Binary: 128+64+32+16+0+0+2+1 = 243 ✔
- Hex: `1111 0011` → F 3 → `F3` ✔

All correct.

---

✔ Problem 3: `10101010₂` → Decimal = 170, Hex = AA

- Binary: `10101010₂` → 128+32+8+2 = 170 ✔
- Hex: `1010 1010` → A A → `AA` ✔

Good.

---

✔ Problem 4: `11111110₂` → Decimal = 254, Hex = FE

- Binary: 128+64+32+16+8+4+2+0 = 254 ✔
- Hex: `1111 1110` → F E → `FE` ✔

Good.

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✔ Problem 5: `11011011₂` → Decimal = 219, Hex = DB

- Binary: 128+64+0+16+8+0+2+1 = 128+64=192, +16=208, +8=216, +2=218, +1=219 ✔
- Hex: `1101 1011` → D B → `DB` ✔

Good.

---

❓ Problem 6: `2E9₁₆` → Decimal = 745, Binary = `1011101001`

Given:
- Hex: `2E9₁₆`
- Decimal: `745`
- Binary: `1011101001`

Let’s verify:

#### Hex → Decimal:
- `2E9₁₆` = 2×256 + 14×16 + 9 = 512 + 224 + 9 = 745 ✔

#### Hex → Binary:
- `2` → `0010`
- `E` → `1110`
- `9` → `1001`
→ So `0010 1110 1001` → remove leading zeros → `1011101001` ✔

So all correct.

---

❓ Problem 7: `ACE₁₆` → Decimal = 2766, Binary = `101011001110`

Given:
- Hex: `ACE₁₆`
- Decimal: `2766`
- Binary: `101011001110`

Check:

#### Hex → Decimal:
- A = 10, C = 12, E = 14
- `ACE₁₆` = 10×256 + 12×16 + 14 = 2560 + 192 + 14 = 2766 ✔

#### Hex → Binary:
- A = 1010
- C = 1100
- E = 1110
→ `1010 1100 1110` → `101011001110` ✔

Perfect.

---

❓ Problem 8: `FF₁₆` → Decimal = 255, Binary = `11111111`

Given:
- Hex: `FF₁₆`
- Decimal: `255`
- Binary: `11111111`

Check:
- FF₁₆ = 15×16 + 15 = 240 + 15 = 255 ✔
- Binary: F = 1111 → `1111 1111` → `11111111` ✔

Good.

---

❓ Problem 9: `2E₁₆ - 16₁₆` → Result = 18, Binary = `11000`

Given:
- Hex subtraction: `2E₁₆ - 16₁₆`
- Result decimal: `18`
- Binary: `11000`

Let’s compute:

- `2E₁₆` = 2×16 + 14 = 32 + 14 = 46
- `16₁₆` = 1×16 + 6 = 22
- 46 - 22 = 24 → But it says result is 18?

Wait! There's a mismatch.

It says:

```
46 - 22 =
18
```

But 46 - 22 = 24, not 18.

Wait — maybe it's a typo?

Let’s look again:

> 9. 2E₁₆ - 16₁₆ → 46 - 22 = 18 → 43 + 170

This seems wrong.

Wait — perhaps the result is meant to be in hex, and the line "46 - 22 = 18" is misleading.

Wait — let's re-express:

- `2E₁₆` = 46₁₀
- `16₁₆` = 22₁₀
- 46 - 22 = 24₁₀

But it says 18 — that can't be.

Unless the subtraction is not in decimal.

Wait — maybe the answer is 18₁₆? But 18₁₆ = 24₁₀.

Yes!

Ah! So `2E₁₆ - 16₁₆ = ?`

Compute in hex:

- `2E₁₆` = 46₁₀
- `16₁₆` = 22₁₀
- Difference = 24₁₀ = 18₁₆

Because 1×16 + 8 = 24.

So yes: `2E₁₆ - 16₁₆ = 18₁₆`

And `18₁₆` = 24₁₀

But the worksheet says:

> 46 - 22 = 18

That’s confusing — because 46 - 22 = 24, not 18.

But likely, it means:

- 46₁₀ - 22₁₀ = 24₁₀ = 18₁₆

So the result is `18₁₆` (hex), which is 24 in decimal.

Then it says:

> 43 + 170

Wait — what? That doesn’t make sense.

Wait — let’s read carefully:

> 9. 2E₁₆ - 16₁₆ → ________ → ________ → 11000

And below it says:

> 46 - 22 =
> 18
> 43 + 170

Hmm — this looks like a mistake in formatting.

Possibly, the line:

> 46 - 22 =
> 18

is meant to say: `46 - 22 = 24`, but they wrote 18 by mistake?

No — wait: if the answer is 18₁₆, then it’s correct.

So:

- `2E₁₆ - 16₁₆ = 18₁₆` (hex)
- In decimal: 24
- In binary: `11000₂` (since 16 + 8 = 24)

Yes! `11000₂` = 16 + 8 = 24 ✔

So the binary is given: `11000`

So we need to fill in:

- Decimal: 24
- Hexadecimal: `18`

But the worksheet shows:

> 9. 2E₁₆ - 16₁₆ → ________ → ________ → 11000

So:

- Decimal: 24
- Hex: 18
- Binary: `11000` ✔

But why does it say “46 - 22 = 18”? That must be a typo — probably meant to write “= 24” or indicate that the result is `18₁₆`.

So correct answer:

- Decimal: 24
- Hex: 18
- Binary: `11000` ✔

---

❓ Problem 10: `2B₁₆ + AA₁₆` → Decimal = 213, Binary = `11010101`

Given:
- Hex addition: `2B₁₆ + AA₁₆`
- Decimal: `213`
- Binary: `11010101`

Compute:

- `2B₁₆` = 2×16 + 11 = 32 + 11 = 43
- `AA₁₆` = 10×16 + 10 = 160 + 10 = 170
- Sum = 43 + 170 = 213 ✔

Now, binary of 213:

- 128 + 64 + 16 + 4 + 1 = 128+64=192, +16=208, +4=212, +1=213
- So: `11010101₂` ✔

Also, verify from hex:

- `2B₁₆` → `0010 1011`
- `AA₁₆` → `1010 1010`
- Add:
```
0010 1011
+ 1010 1010
------------
1101 0101
```
→ `11010101₂` ✔

So all good.

---

❓ Problem 11: `74₁₀` → Hex = `4A`, Binary = `1001010`

Given:
- Decimal: `74`
- Hex: `4A`
- Binary: `1001010`

Check:

- 74 ÷ 16 = 4 remainder 10 → 4A₁₆ ✔
- Binary: 64 + 8 + 2 = 74 → `1001010₂` ✔

Wait: `1001010` is 7 bits: positions from left: 64, 32, 16, 8, 4, 2, 1

- 64 + 8 + 2 = 74 ✔

So yes.

But note: `1001010₂` is correct.

So all correct.

---

❓ Problem 12: `100₁₀` → Hex = `64`, Binary = `1100100`

Given:
- Decimal: `100`
- Hex: `64`
- Binary: `1100100`

Check:

- 100 ÷ 16 = 6 rem 4 → `64₁₆` ✔
- Binary: 64 + 32 = 96, need 4 more → 64 + 32 + 4 = 100 → `1100100₂` ✔

Yes.

---

## ✔ Final Answer Table

Let’s fill in all missing values:

| # | Binary | Decimal | Hexadecimal | Notes |
|---|---------------|---------|-------------|-------|
| 1 | 10111010 | 186 | BA | Given |
| 2 | 11110011 | 243 | F3 | Given |
| 3 | 10101010 | 170 | AA | Given |
| 4 | 11111110 | 254 | FE | Given |
| 5 | 11011011 | 219 | DB | Given |
| 6 | 1011101001 | 745 | 2E9 | Hex: 2E9 |
| 7 | 101011001110 | 2766 | ACE | Hex: ACE |
| 8 | 11111111 | 255 | FF | Given |
| 9 | 11000 | 24 | 18 | 2E - 16 = 18₁₆ |
|10 | 11010101 | 213 | 2B + AA = 131? Wait — 2B + AA = 43 + 170 = 213 → hex: 213 ÷ 16 = 13×16=208, rem 5 → D5? Wait! |

⚠️ Problem 10: Mistake Detected

Wait — we made a mistake here.

In Problem 10, we have:

- `2B₁₆ + AA₁₆` → sum = 43 + 170 = 213
- But hexadecimal of 213 is not given — the hex is missing.

But the worksheet says:

> 10. 2B₁₆ + AA₁₆ → ________ → ________ → 11010101

So we need to find:
- Decimal: 213 ✔
- Hex: ??? → 213 ÷ 16 = 13×16 = 208, remainder 5 → `D5₁₆`

But the worksheet says the binary is `11010101` → which is 213 ✔

So hex should be `D5`, not something else.

But in the table, it's blank.

So:

- Hex: `D5`

Similarly, Problem 9:

- We said hex is `18`, decimal is 24

But the worksheet has:

> 9. 2E₁₆ - 16₁₆ → ________ → ________ → 11000

So:

- Decimal: 24
- Hex: 18
- Binary: `11000` ✔

Now fix the missing hex in Problem 10.

---

✔ Final Corrections:

| # | Binary | Decimal | Hexadecimal | Notes |
|---|------------------|---------|-------------|-------|
| 1 | 10111010 | 186 | BA | OK |
| 2 | 11110011 | 243 | F3 | OK |
| 3 | 10101010 | 170 | AA | OK |
| 4 | 11111110 | 254 | FE | OK |
| 5 | 11011011 | 219 | DB | OK |
| 6 | 1011101001 | 745 | 2E9 | OK |
| 7 | 101011001110 | 2766 | ACE | OK |
| 8 | 11111111 | 255 | FF | OK |
| 9 | 11000 | 24 | 18 | 2E - 16 = 18₁₆ |
|10 | 11010101 | 213 | D5 | 2B + AA = D5₁₆ |
|11 | 1001010 | 74 | 4A | OK |
|12 | 1100100 | 100 | 64 | OK |

---

✔ Final Answers to Fill In:

Now, let’s list only the missing values:

1. Binary: 10111010 → already filled
2. Binary: 11110011 → already filled
3. Binary: 10101010 → already filled
4. Binary: 11111110 → already filled
5. Binary: 11011011 → already filled
6. Hex: 2E9
7. Hex: ACE
8. Hex: FF
9. Decimal: 24, Hex: 18
10. Hex: D5
11. Decimal: 74
12. Decimal: 100

Wait — but some are already filled.

Let’s re-check the original worksheet:

Each row has one value given, others blank.

But many rows have multiple filled — probably the worksheet is partially completed, and we need to fill in the blanks.

Let’s go row by row and identify what’s missing.

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🔍 Row-by-Row Analysis

#### 1. `10111010₂` → Decimal: 186, Hex: BA, Binary: ___
- Binary is given → so blank is Binary → but it’s already written → no need
- Actually, all three are filled → skip

#### 2. `11110011₂` → Decimal: 243, Hex: F3, Binary: ___
- Binary is missing? Wait — no, the binary is given at start → so Binary is known
- But the blank is under Binary column → but it’s empty → so we fill it with `11110011`

Wait — the format is:

```
Decimal Hexadecimal Binary
1. 10111010₂ 186 BA ____
```

So Binary column is blank → but the binary is given as input → so we copy it.

So:

- Binary: 10111010

Similarly:

#### 1. Binary: 10111010
#### 2. Binary: 11110011
#### 3. Binary: 10101010
#### 4. Binary: 11111110
#### 5. Binary: 11011011

These are already in the input — so just copy.

#### 6. `2E9₁₆` → Decimal: 745, Hex: ___, Binary: 1011101001
- Hex is missing → 2E9
- Binary is given → already there

So fill Hex: 2E9

#### 7. `ACE₁₆` → Decimal: 2766, Hex: ___, Binary: 101011001110
- Hex: ACE
- Already given

#### 8. `FF₁₆` → Decimal: 255, Hex: ___, Binary: 11111111
- Hex: FF
- Binary: 11111111

#### 9. `2E₁₆ - 16₁₆` → Decimal: ___, Hex: ___, Binary: 11000
- Compute: 2E₁₆ = 46, 16₁₆ = 22 → 46 - 22 = 24
- 24₁₀ = 18₁₆
- Binary: 11000₂ → given
- So:
- Decimal: 24
- Hex: 18

#### 10. `2B₁₆ + AA₁₆` → Decimal: 213, Hex: ___, Binary: 11010101
- 2B = 43, AA = 170 → 213
- 213 ÷ 16 = 13×16=208, rem 5 → D5₁₆
- So Hex: D5

#### 11. `74₁₀` → Decimal: ___, Hex: 4A, Binary: 1001010
- Decimal: 74
- Hex: 4A → already given
- Binary: 1001010 → already given

#### 12. `100₁₀` → Decimal: ___, Hex: 64, Binary: 1100100
- Decimal: 100
- Hex: 64 → given
- Binary: 1100100 → given

---

✔ Final Answers to Fill In:

| # | Binary | Decimal | Hexadecimal |
|---|----------------|---------|-------------|
| 1 | 10111010 | 186 | BA |
| 2 | 11110011 | 243 | F3 |
| 3 | 10101010 | 170 | AA |
| 4 | 11111110 | 254 | FE |
| 5 | 11011011 | 219 | DB |
| 6 | 1011101001 | 745 | 2E9 |
| 7 | 101011001110 | 2766 | ACE |
| 8 | 11111111 | 255 | FF |
| 9 | 11000 | 24 | 18 |
|10 | 11010101 | 213 | D5 |
|11 | 1001010 | 74 | 4A |
|12 | 1100100 | 100 | 64 |

---

✔ Summary of Missing Values:

- 6. Hex: `2E9`
- 7. Hex: `ACE`
- 8. Hex: `FF`
- 9. Decimal: `24`, Hex: `18`
- 10. Hex: `D5`
- 11. Decimal: `74`
- 12. Decimal: `100`

Note: Rows 1–5 already complete.

---

🟩 Final Answer:

Fill in the blanks as follows:

1. Binary: 10111010
2. Binary: 11110011
3. Binary: 10101010
4. Binary: 11111110
5. Binary: 11011011
6. Hex: 2E9
7. Hex: ACE
8. Hex: FF
9. Decimal: 24, Hex: 18
10. Hex: D5
11. Decimal: 74
12. Decimal: 100

✔ All problems solved and verified.