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Show Answer Key & Explanations
Step-by-step solution for: STEM
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Show Answer Key & Explanations
Step-by-step solution for: STEM
Let's solve each of the binary addition problems step by step. Binary addition follows the same principles as decimal addition, but with only two digits: 0 and 1. The rules are:
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 10 (write down 0, carry over 1)
- 1 + 1 + 1 = 11 (write down 1, carry over 1)
We'll go through each problem and also provide an optional check by converting to base 10.
---
```
011100
+ 010001
---------
```
Start from the rightmost bit:
- 0 + 1 = 1
- 0 + 0 = 0
- 1 + 0 = 1
- 1 + 0 = 1
- 1 + 1 = 10 → write 0, carry 1
- 0 + 0 + carry 1 = 1
So:
```
011100
+ 010001
---------
101101
```
✔ Answer: 101101
Optional Check (Base 10):
- 011100₂ = 0×32 + 1×16 + 1×8 + 1×4 + 0×2 + 0×1 = 16 + 8 + 4 = 28
- 010001₂ = 0×32 + 1×16 + 0×8 + 0×4 + 0×2 + 1×1 = 16 + 1 = 17
- 28 + 17 = 45
- 101101₂ = 32 + 8 + 4 + 1 = 45 ✔
---
```
001110
+ 110111
---------
```
Add column by column (right to left):
- 0 + 1 = 1
- 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
- 0 + 1 + carry 1 = 10 → write 0, carry 1
- 0 + 1 + carry 1 = 10 → write 0, carry 1
Now carry 1 goes to the next position:
```
001110
+ 110111
----------
1000101
```
✔ Answer: 1000101
Optional Check (Base 10):
- 001110₂ = 0×32 + 0×16 + 1×8 + 1×4 + 1×2 + 0×1 = 8 + 4 + 2 = 14
- 110111₂ = 1×32 + 1×16 + 0×8 + 1×4 + 1×2 + 1×1 = 32 + 16 + 4 + 2 + 1 = 55
- 14 + 55 = 69
- 1000101₂ = 64 + 4 + 1 = 69 ✔
---
```
111110
+ 111100
---------
```
Add:
- 0 + 0 = 0
- 1 + 0 = 1
- 1 + 1 = 10 → write 0, carry 1
- 1 + 1 + carry 1 = 11 → write 1, carry 1
- 1 + 1 + carry 1 = 11 → write 1, carry 1
- 1 + 1 + carry 1 = 11 → write 1, carry 1
Carry 1 at the front:
```
111110
+ 111100
---------
1111010
```
✔ Answer: 1111010
Optional Check (Base 10):
- 111110₂ = 32 + 16 + 8 + 4 + 2 = 62
- 111100₂ = 32 + 16 + 8 + 4 = 60
- 62 + 60 = 122
- 1111010₂ = 64 + 32 + 16 + 8 + 2 = 122 ✔
---
```
0111011
+ 1101111
----------
```
Align and add:
```
0111011
+ 1101111
----------
```
Column by column (right to left):
- 1 + 1 = 10 → write 0, carry 1
- 1 + 1 + carry 1 = 11 → write 1, carry 1
- 0 + 1 + carry 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
- 0 + 1 + carry 1 = 10 → write 0, carry 1
Final carry: 1
```
0111011
+ 1101111
----------
10101010
```
✔ Answer: 10101010
Optional Check (Base 10):
- 0111011₂ = 0×64 + 1×32 + 1×16 + 1×8 + 0×4 + 1×2 + 1×1 = 32 + 16 + 8 + 2 + 1 = 59
- 1101111₂ = 1×64 + 1×32 + 0×16 + 1×8 + 1×4 + 1×2 + 1×1 = 64 + 32 + 8 + 4 + 2 + 1 = 111
- 59 + 111 = 170
- 10101010₂ = 128 + 32 + 8 + 2 = 170 ✔
---
```
1111111
+ 1110000
----------
```
Add:
- 1 + 0 = 1
- 1 + 0 = 1
- 1 + 0 = 1
- 1 + 0 = 1
- 1 + 1 = 10 → write 0, carry 1
- 1 + 1 + carry 1 = 11 → write 1, carry 1
- 1 + 1 + carry 1 = 11 → write 1, carry 1
Final carry: 1
```
1111111
+ 1110000
----------
11101111
```
✔ Answer: 11101111
Optional Check (Base 10):
- 1111111₂ = 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127
- 1110000₂ = 64 + 32 + 16 = 112
- 127 + 112 = 239
- 11101111₂ = 128 + 64 + 32 + 8 + 4 + 2 + 1? Wait:
- 11101111 = 128 + 64 + 32 + 0 + 8 + 4 + 2 + 1 = 128 + 64 = 192, +32=224, +8=232, +4=236, +2=238, +1=239 ✔
Yes, 239 ✔
---
1. 101101
2. 1000101
3. 1111010
4. 10101010
5. 11101111
All verified with base-10 checks.
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 10 (write down 0, carry over 1)
- 1 + 1 + 1 = 11 (write down 1, carry over 1)
We'll go through each problem and also provide an optional check by converting to base 10.
---
1. 011100 + 010001
```
011100
+ 010001
---------
```
Start from the rightmost bit:
- 0 + 1 = 1
- 0 + 0 = 0
- 1 + 0 = 1
- 1 + 0 = 1
- 1 + 1 = 10 → write 0, carry 1
- 0 + 0 + carry 1 = 1
So:
```
011100
+ 010001
---------
101101
```
✔ Answer: 101101
Optional Check (Base 10):
- 011100₂ = 0×32 + 1×16 + 1×8 + 1×4 + 0×2 + 0×1 = 16 + 8 + 4 = 28
- 010001₂ = 0×32 + 1×16 + 0×8 + 0×4 + 0×2 + 1×1 = 16 + 1 = 17
- 28 + 17 = 45
- 101101₂ = 32 + 8 + 4 + 1 = 45 ✔
---
2. 001110 + 110111
```
001110
+ 110111
---------
```
Add column by column (right to left):
- 0 + 1 = 1
- 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
- 0 + 1 + carry 1 = 10 → write 0, carry 1
- 0 + 1 + carry 1 = 10 → write 0, carry 1
Now carry 1 goes to the next position:
```
001110
+ 110111
----------
1000101
```
✔ Answer: 1000101
Optional Check (Base 10):
- 001110₂ = 0×32 + 0×16 + 1×8 + 1×4 + 1×2 + 0×1 = 8 + 4 + 2 = 14
- 110111₂ = 1×32 + 1×16 + 0×8 + 1×4 + 1×2 + 1×1 = 32 + 16 + 4 + 2 + 1 = 55
- 14 + 55 = 69
- 1000101₂ = 64 + 4 + 1 = 69 ✔
---
3. 111110 + 111100
```
111110
+ 111100
---------
```
Add:
- 0 + 0 = 0
- 1 + 0 = 1
- 1 + 1 = 10 → write 0, carry 1
- 1 + 1 + carry 1 = 11 → write 1, carry 1
- 1 + 1 + carry 1 = 11 → write 1, carry 1
- 1 + 1 + carry 1 = 11 → write 1, carry 1
Carry 1 at the front:
```
111110
+ 111100
---------
1111010
```
✔ Answer: 1111010
Optional Check (Base 10):
- 111110₂ = 32 + 16 + 8 + 4 + 2 = 62
- 111100₂ = 32 + 16 + 8 + 4 = 60
- 62 + 60 = 122
- 1111010₂ = 64 + 32 + 16 + 8 + 2 = 122 ✔
---
4. 0111011 + 1101111
```
0111011
+ 1101111
----------
```
Align and add:
```
0111011
+ 1101111
----------
```
Column by column (right to left):
- 1 + 1 = 10 → write 0, carry 1
- 1 + 1 + carry 1 = 11 → write 1, carry 1
- 0 + 1 + carry 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
- 0 + 1 + carry 1 = 10 → write 0, carry 1
Final carry: 1
```
0111011
+ 1101111
----------
10101010
```
✔ Answer: 10101010
Optional Check (Base 10):
- 0111011₂ = 0×64 + 1×32 + 1×16 + 1×8 + 0×4 + 1×2 + 1×1 = 32 + 16 + 8 + 2 + 1 = 59
- 1101111₂ = 1×64 + 1×32 + 0×16 + 1×8 + 1×4 + 1×2 + 1×1 = 64 + 32 + 8 + 4 + 2 + 1 = 111
- 59 + 111 = 170
- 10101010₂ = 128 + 32 + 8 + 2 = 170 ✔
---
5. 1111111 + 1110000
```
1111111
+ 1110000
----------
```
Add:
- 1 + 0 = 1
- 1 + 0 = 1
- 1 + 0 = 1
- 1 + 0 = 1
- 1 + 1 = 10 → write 0, carry 1
- 1 + 1 + carry 1 = 11 → write 1, carry 1
- 1 + 1 + carry 1 = 11 → write 1, carry 1
Final carry: 1
```
1111111
+ 1110000
----------
11101111
```
✔ Answer: 11101111
Optional Check (Base 10):
- 1111111₂ = 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127
- 1110000₂ = 64 + 32 + 16 = 112
- 127 + 112 = 239
- 11101111₂ = 128 + 64 + 32 + 8 + 4 + 2 + 1? Wait:
- 11101111 = 128 + 64 + 32 + 0 + 8 + 4 + 2 + 1 = 128 + 64 = 192, +32=224, +8=232, +4=236, +2=238, +1=239 ✔
Yes, 239 ✔
---
✔ Final Answers:
1. 101101
2. 1000101
3. 1111010
4. 10101010
5. 11101111
All verified with base-10 checks.
Parent Tip: Review the logic above to help your child master the concept of binary addition worksheet.
