Let's solve each of these base conversion problems step by step, converting them to
base ten (decimal).
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🔢 General Rule for Base Conversion:
To convert a number from any base $ b $ to base 10:
$$
\text{Number}_{10} = d_n \times b^n + d_{n-1} \times b^{n-1} + \dots + d_0 \times b^0
$$
Where $ d $ are the digits and $ b $ is the base.
---
✔ Solutions:
---
1. $ 10_{\text{two}} $
Binary (base 2):
$$
1 \times 2^1 + 0 \times 2^0 = 2 + 0 = \boxed{2}
$$
---
2. $ 21_{\text{three}} $
Base 3:
$$
2 \times 3^1 + 1 \times 3^0 = 6 + 1 = \boxed{7}
$$
---
3. $ 101_{\text{two}} $
Binary:
$$
1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 4 + 0 + 1 = \boxed{5}
$$
---
4. $ 240_{\text{five}} $
Base 5:
$$
2 \times 5^2 + 4 \times 5^1 + 0 \times 5^0 = 2 \times 25 + 4 \times 5 + 0 = 50 + 20 = \boxed{70}
$$
---
5. $ 10110_{\text{two}} $
Binary:
$$
1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 16 + 0 + 4 + 2 + 0 = \boxed{22}
$$
---
6. $ 33_{\text{six}} $
Base 6:
$$
3 \times 6^1 + 3 \times 6^0 = 18 + 3 = \boxed{21}
$$
---
7. $ 43_{\text{five}} $
Base 5:
$$
4 \times 5^1 + 3 \times 5^0 = 20 + 3 = \boxed{23}
$$
---
8. $ 314_{\text{five}} $
Base 5:
$$
3 \times 5^2 + 1 \times 5^1 + 4 \times 5^0 = 3 \times 25 + 5 + 4 = 75 + 5 + 4 = \boxed{84}
$$
---
9. $ 203_{\text{four}} $
Base 4:
$$
2 \times 4^2 + 0 \times 4^1 + 3 \times 4^0 = 2 \times 16 + 0 + 3 = 32 + 3 = \boxed{35}
$$
---
10. $ 2012_{\text{three}} $
Base 3:
$$
2 \times 3^3 + 0 \times 3^2 + 1 \times 3^1 + 2 \times 3^0 = 2 \times 27 + 0 + 3 + 2 = 54 + 3 + 2 = \boxed{59}
$$
---
11. $ 155_{\text{six}} $
Base 6:
$$
1 \times 6^2 + 5 \times 6^1 + 5 \times 6^0 = 36 + 30 + 5 = \boxed{71}
$$
---
12. $ 403_{\text{five}} $
Base 5:
$$
4 \times 5^2 + 0 \times 5^1 + 3 \times 5^0 = 4 \times 25 + 0 + 3 = 100 + 3 = \boxed{103}
$$
---
✔ Final Answers:
| Problem | Answer |
|--------|--------|
| 1. $10_{\text{two}}$ |
2 |
| 2. $21_{\text{three}}$ |
7 |
| 3. $101_{\text{two}}$ |
5 |
| 4. $240_{\text{five}}$ |
70 |
| 5. $10110_{\text{two}}$ |
22 |
| 6. $33_{\text{six}}$ |
21 |
| 7. $43_{\text{five}}$ |
23 |
| 8. $314_{\text{five}}$ |
84 |
| 9. $203_{\text{four}}$ |
35 |
| 10. $2012_{\text{three}}$ |
59 |
| 11. $155_{\text{six}}$ |
71 |
| 12. $403_{\text{five}}$ |
103 |
Let me know if you'd like an explanation in a visual format or with place value charts! 🎄
Parent Tip: Review the logic above to help your child master the concept of base 5 worksheet.
