Number bases interactive worksheet - Free Printable

Let's solve each of the four problems step by step. These are arithmetic operations in different number bases: base 5, base 2 (binary), base 8 (octal), and base 9.

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1. $ 234_5 + 42_5 $

We are adding two numbers in base 5.

#### Step 1: Understand base 5
In base 5, digits go from 0 to 4. When you reach 5, you carry over.

Let’s write the numbers vertically:

```
2 3 4
+ 4 2
-------
```

Start from the rightmost digit.

#### Rightmost column (units place):
- $ 4 + 2 = 6 $
- But 6 in base 5 is $ 1 \times 5 + 1 $ → so write down 1, carry 1

#### Middle column (fives place):
- $ 3 + 4 = 7 $, plus carry 1 → $ 8 $
- $ 8 $ in base 5: $ 1 \times 5 + 3 $ → write 3, carry 1

#### Leftmost column (twenty-fives place):
- $ 2 + 0 = 2 $, plus carry 1 → $ 3 $

So result is: 3 3 1

✔ Answer: $ \boxed{331_5} $

---

2. $ 10101_2 + 111_2 $

Binary addition (base 2)

Write them aligned:

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

Align to the right:

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

Now add column by column (right to left):

#### Column 1 (rightmost):
- $ 1 + 1 = 10_2 $ → write 0, carry 1

#### Column 2:
- $ 0 + 1 = 1 $, plus carry 1 → $ 10 $ → write 0, carry 1

#### Column 3:
- $ 1 + 1 = 10 $, plus carry 1 → $ 11 $ → write 1, carry 1

#### Column 4:
- $ 0 + 0 = 0 $, plus carry 1 → $ 1 $

#### Column 5:
- $ 1 + 0 = 1 $

Result: 1 1 0 1 0

✔ Answer: $ \boxed{11010_2} $

---

3. $ 1764_8 - 72_8 $

Octal subtraction (base 8)

Write vertically:

```
1 7 6 4
- 7 2
---------
```

Align:

```
1 7 6 4
- 0 7 2
---------
```

Subtract column by column from right to left.

#### Column 1 (units):
- $ 4 - 2 = 2 $

#### Column 2 (eights):
- $ 6 - 7 $ → can't do, borrow from next digit

Borrow from the 7 (which is in the 64s place). The 7 becomes 6, and the 6 becomes $ 6 + 8 = 14 $ in base 8.

Now subtract: $ 14 - 7 = 7 $

#### Column 3 (64s):
- Now we have $ 6 - 0 = 6 $

#### Column 4 (512s):
- $ 1 - 0 = 1 $

So result: 1 6 7 2

✔ Answer: $ \boxed{1672_8} $

---

4. $ 831_9 - 18_9 $

Base 9 subtraction

Write aligned:

```
8 3 1
- 1 8
-------
```

As:

```
8 3 1
- 0 1 8
-------
```

Now subtract:

#### Column 1 (units):
- $ 1 - 8 $ → can’t do, borrow from 3

But 3 is in the 9s place. So borrow 1 from 3 → 3 becomes 2, and 1 becomes $ 1 + 9 = 10 $ in base 9.

Now $ 10 - 8 = 2 $

#### Column 2 (nines):
- $ 2 - 1 = 1 $

#### Column 3 (81s):
- $ 8 - 0 = 8 $

So result: 8 1 2

✔ Answer: $ \boxed{812_9} $

---

✔ Final Answers:

1. $ 234_5 + 42_5 = \boxed{331_5} $
2. $ 10101_2 + 111_2 = \boxed{11010_2} $
3. $ 1764_8 - 72_8 = \boxed{1672_8} $
4. $ 831_9 - 18_9 = \boxed{812_9} $

Let me know if you'd like verification by converting to decimal!