Worksheet 6: Binary Addition - Practice problems and rules for adding binary numbers with carry-over.

Let’s solve each problem step by step using the binary addition rules given:

Rules recap:
- (a) 0 + 0 = 0
- (b) 0 + 1 = 1
- (c) 1 + 0 = 1
- (d) 1 + 1 = 0, carry 1 → written as “10” in binary
- (e) 1 + 1 + 1 = 1, carry 1 → written as “11” in binary

We’ll go question by question.

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Question 1:
Already solved in the worksheet — just for reference:
Binary:
 0 0 1 1 → decimal 3
+ 1 0 0 0 → decimal 8
= 1 0 1 1 → decimal 11 ✔ Correct.

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Question 2:
Add:
 1 0 0 1
+ 0 1 0 0
= ?

Start from rightmost column (units place):

Column 1 (rightmost): 1 + 0 = 1 → write 1
Column 2: 0 + 0 = 0 → write 0
Column 3: 0 + 1 = 1 → write 1
Column 4: 1 + 0 = 1 → write 1

So result is: 1 1 0 1

Check with decimal:
1001₂ = 8 + 1 = 9
0100₂ = 4
9 + 4 = 13
1101₂ = 8 + 4 + 1 = 13 ✔ Correct.

Answer for Q2: 1101

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Question 3:
Use carry row at top. Given:

Carry row: [ ][ ][1][ ] ← only a 1 over the third column from right? Let's align properly.

Actually, looking at the grid:

It shows:

| | 1 | |
| 1 | 0 | 1 | 1 |
+ | 0 | 0 | 1 | 0 |
= | 1 | 1 | 0 | 1 |

Wait — let’s do it step by step with carries.

Write numbers aligned:

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

Start from right (column 1):

Col 1: 1 + 0 = 1 → write 1, no carry
Col 2: 1 + 1 = 0, carry 1 → write 0, carry 1 to next column
Col 3: 0 + 0 + carry(1) = 1 → write 1, no new carry
Col 4: 1 + 0 = 1 → write 1

But wait — that gives: 1 1 0 1 — which matches the answer shown.

However, the problem says “use the carry row at the top for the carried 1”. The diagram shows a ‘1’ above the third column (from right), meaning there was a carry into that column.

Let me re-solve carefully with proper carry tracking:

Align:

col4 col3 col2 col1
1 0 1 1
+ 0 0 1 0
------------------

Start col1 (rightmost):
1 + 0 = 1 → write 1, carry 0

col2:
1 + 1 = 0, carry 1 → write 0, carry 1 to col3

col3:
0 + 0 + carry(1) = 1 → write 1, carry 0

col4:
1 + 0 = 1 → write 1

Result: 1 1 0 1 → same as before.

The “carry row” shown has a 1 above col3 — which matches our carry into col3. So everything checks out.

Answer for Q3: 1101

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Question 4:
Now use rule (d): 1 + 1 = 0 carry 1

Given:

| | | 1 | | |
| | 1 | 0 | 1 | 1 |
+ | | 1 | 0 | 1 | 0 |
= | 1 | 0 | 1 | 0 | 1 |

Let’s add step by step with carries.

Write numbers:

col5 col4 col3 col2 col1
0 1 0 1 1 ← first number (assume leading zero for alignment)
+ 0 1 0 1 0 ← second number
-------------------------

Actually, better to write without assuming extra zeros unless needed.

Numbers are:

First: 1 0 1 1 → but wait, in the grid it’s written as 5 columns? Let’s count boxes.

Looking at the grid:

Top row (carry): empty, empty, 1, empty, empty → so 5 columns total.

Then:

Row1: _ _ 1 0 1 1 → actually, probably it’s:

Positions (left to right): col5 col4 col3 col2 col1

Row1 (first number): blank, blank, 1, 0, 1, 1? Wait — too many.

Actually, looking again:

The grid has 5 vertical lines → 4 digits per number? But then result has 5 digits.

Let me interpret the grid as:

Carry row: positions above the digits — they show a '1' above the third digit from the right (which would be col3 if we number from right).

So let’s label columns from right to left as col1 (units), col2, col3, col4, col5.

Given:

Number A: _ _ 1 0 1 1 → wait, that’s 6 characters? No.

Actually, in the image, for Q4, it’s drawn as:

[ ][ ][1][ ][ ] ← carry row (only one 1, in the middle)

Then:

[ ][1][0][1][1] ← first number

+[ ][1][0][1][0] ← second number

=[1][0][1][0][1] ← result

So it’s 5-digit result, meaning inputs are 4-digit, but with possible carry into 5th digit.

So let’s write:

col5 col4 col3 col2 col1
Carry: ? ? 1 ? ? ← but carry is generated during addition, not pre-set except where indicated.

Actually, the "carry row" shown has a 1 above col3 (third from right). That likely means when adding col2, we got a carry into col3.

Let’s add from right to left, tracking carries.

Define:

A = 1 0 1 1 → but wait, in the grid, first number is written as: under the carry row, it’s [ ][1][0][1][1] — so probably:

Position: col5 col4 col3 col2 col1
A: 0 1 0 1 1 (assuming leading zero for col5)
B: 0 1 0 1 0
But that doesn’t match because then col4 and col3 both have 1 and 0.

Wait — perhaps the numbers are:

From the grid layout:

The first number is: in the four rightmost columns: 1 0 1 1? But there are five columns shown.

I think the intended setup is:

We are adding two 4-bit numbers, but the result may be 5 bits.

So:

Number X: 1 0 1 1 (binary)
Number Y: 1 0 1 0 (binary)
And we’re told to use the carry row — which has a 1 above the third bit from the right (i.e., above the 2^2 place).

Let’s add them properly:

X: 1 0 1 1
Y: 1 0 1 0
------------

Start from right (col1):

col1: 1 + 0 = 1 → write 1, carry 0

col2: 1 + 1 = 0, carry 1 → write 0, carry 1 to col3

col3: 0 + 0 + carry(1) = 1 → write 1, carry 0

col4: 1 + 1 = 0, carry 1 → write 0, carry 1 to col5

col5: 0 + 0 + carry(1) = 1 → write 1

So result: 1 0 1 0 1

Which matches what’s shown in the worksheet!

And the carry row: during addition, we had a carry into col3 (from col2), which is why there’s a 1 shown above col3 in the carry row. Perfect.

So answer for Q4: 10101

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Now, summarizing answers for questions 2, 3, 4 (since Q1 is already done):

Q2: 1101
Q3: 1101
Q4: 10101

But the user didn’t specify which question to solve — however, since the worksheet presents all, and the instruction is to “solve the problem accurately”, I should provide answers for all unsolved parts.

Looking back: Q1 is fully solved in the sheet. Q2, Q3, Q4 have grids with answers filled in — but perhaps the student is to verify or complete them? The instructions say “work out the answers to questions 1-10”, but only 1-4 are visible.

Since the user uploaded this and wants us to solve, and Q2-Q4 have answers written, but maybe we need to confirm or fill if missing? In the image, for Q2, Q3, Q4, the results are already filled in the grids.

Wait — in Q2, the result is shown as 1101 — which we verified correct.
Q3: 1101 — correct.
Q4: 10101 — correct.

But perhaps the task is to compute them, even if shown? Or maybe the student needs to understand how.

Since the user said “solve the problem accurately”, and the problems are presented with blanks to fill (even though in the image some are filled), I’ll assume we need to provide the computed answers for Q2, Q3, Q4.

Also, note: in Q3 and Q4, the carry rows are partially filled to guide, but we still compute.

Final Answers:

For Question 2: Binary sum of 1001 + 0100 is 1101

For Question 3: With carry, 1011 + 0010 = 1101

For Question 4: 1011 + 1010 = 10101

All match what’s shown, so they are correct.

But to be precise, since the worksheet might expect the student to write the answers, I’ll list them.

──────────────────────────────────────

Final Answer:
Question 2: 1101
Question 3: 1101
Question 4: 10101