Students identify and correct mistakes in two-digit by two-digit multiplication problems.

Let’s solve each problem step by step.

---

Problem a:

The student tried to multiply 54 × 23 like this:

```
54
× 23
-----
162 ← This is 54 × 3 → correct!
+1,080 ← This should be 54 × 20 → but they wrote 1,080? Let’s check.
-----
1,232
```

Wait — let’s calculate 54 × 20 correctly.

54 × 20 = 54 × 2 × 10 = 108 × 10 = 1,080 → That part is actually correct numerically.

But here’s the mistake: In multiplication, when you multiply by the tens digit (the “2” in 23), you’re really multiplying by 20. So you must shift the result one place to the left — meaning you add a zero at the end OR write it starting under the tens column.

In the student’s work, they wrote “+1,080” aligned as if it were ones — but 1,080 should be written shifted left by one digit (so the 0 is under the 2 in 162). But even worse — look at how they added:

They did:
```
162
+1080
------
1232
```

That addition is wrong!

Let’s add 162 + 1,080 properly:

162
+1080
= ?

Start from right:

2 + 0 = 2
6 + 8 = 14 → write 4, carry 1
1 + 0 + 1 = 2
1 + nothing = 1

So total = 1,242

But wait — let’s double-check what 54 × 23 really is.

We can do it another way:

54 × 23 = 54 × (20 + 3) = (54 × 20) + (54 × 3) = 1,080 + 162 = 1,242

So the student got 1,232 — which is 10 less than it should be.

Why? Because when adding 162 + 1,080, they probably misaligned or miscalculated.

Actually, looking again — maybe they didn’t misalign — perhaps they just added wrong.

162 + 1080:

If you line them up:

  162
+1080
------
 1242 ← correct sum

But student wrote 1,232 — so they likely did:

162
+1080
→ thought 6+8=13? Or forgot to carry?

Anyway — the real error is in the addition step, not the multiplication steps.

BUT — there’s another possible mistake: sometimes students forget to shift the second row.

In standard multiplication format:

```
54
× 23
-----
162 ← 54 × 3
108 ← 54 × 2 (but this is actually 54 × 20, so it should be 1080 — but we write it as 108 with a space or zero implied)
-----
1242
```

Ah! Here’s the key: In traditional vertical multiplication, when you multiply by the tens digit, you don’t write “1,080” — you write “108” and shift it left (meaning it’s understood to be 1,080). Then you add:

```
54
× 23
-----
162
108 ← this is actually 1080, but written without the trailing zero, shifted left
-----
1242
```

But in the student’s version, they wrote “+1,080” — which is fine numerically — but then they added incorrectly to get 1,232 instead of 1,242.

So the mistake is in the final addition: 162 + 1,080 ≠ 1,232 — it equals 1,242.

Alternatively, if they meant to write “108” (without the zero) and forgot to shift, that would also cause an error — but since they wrote “1,080”, I think the main error is arithmetic in addition.

Let me verify once more:

54 × 23:

Break it down:

50 × 23 = 1,150
4 × 23 = 92
Total = 1,150 + 92 = 1,242 ✔

Or:

23 × 50 = 1,150
23 × 4 = 92
Same thing.

So correct answer is 1,242

Student got 1,232 — off by 10.

Most likely, when adding 162 + 1,080, they did:

162
+1080
→ added 2+0=2, 6+8=14 → wrote 4, carried 1, then 1+0=1, plus carry 1 = 2, then 1 → 1,242? Wait no — unless they messed up the hundreds.

Wait — maybe they lined it up wrong?

If they wrote:

  162
+1080
and accidentally aligned the 1 under the 6? No — that doesn’t make sense.

Perhaps they calculated 54 × 20 as 1,070? But they wrote 1,080.

I think the simplest explanation: they added 162 + 1,080 and made a simple addition error — got 1,232 instead of 1,242.

So for part a:

Mistake: The student added 162 and 1,080 incorrectly. They got 1,232, but the correct sum is 1,242.

Correct solution:

```
54
× 23
-----
162 (54 × 3)
+1080 (54 × 20)
-----
1242
```

Final Answer for a: 1,242

---

Problem b:

Student did:

```
99
× 12
-----
188 ← 99 × 2? Should be 198!
+ 990 ← 99 × 10 = 990 → correct
-----
1,178
```

First, check 99 × 2:

99 × 2 = 198 — but student wrote 188. That’s the first mistake!

Then they added 188 + 990 = 1,178 — which is correct *if* 188 were right — but it’s not.

So two errors:

1. Wrong product for 99 × 2 → should be 198, not 188.
2. Therefore, final sum is wrong.

Let’s compute correctly:

99 × 12 = 99 × (10 + 2) = (99 × 10) + (99 × 2) = 990 + 198 = ?

Add 990 + 198:

990
+198
= ?

0 + 8 = 8
9 + 9 = 18 → write 8, carry 1
9 + 1 + 1 = 11 → write 11

So 1,188

Check another way:

100 × 12 = 1,200
Minus 1 × 12 = 12
So 1,200 - 12 = 1,188 ✔

So correct answer is 1,188

Student had 1,178 — because they used 188 instead of 198.

Difference: 198 - 188 = 10 → so their answer is 10 too low.

For part b:

Mistake: The student calculated 99 × 2 as 188, but it should be 198. This caused the final answer to be wrong.

Correct solution:

```
99
× 12
-----
198 (99 × 2)
+ 990 (99 × 10)
-----
1188
```

Final Answer for b: 1,188

---

Final Answer:
a. 1,242
b. 1,188