Question

rebecca tries to find 436 - 271. rebecca breaks apart the numbers by place value, regroups, and subtracts. her work is shown. in which step does rebecca make the first error? step 1 step 2 step 3 step 4 rebeccas work step 1: 436 = 400 + 130 + 6 step 2: - 271 = 200 + 70 + 1 step 3: 200 + 60 + 5 step 4: the difference is 265.

Explanation:

Step 1: Analyze Step 1

Rebecca breaks 436 into place values. 436 has 4 hundreds, 3 tens, and 6 ones. But she wrote \(436 = 400 + 130 + 6\). Wait, actually, when regrouping for subtraction, maybe she is trying to adjust the tens place. Wait, no, let's check Step 3. Wait, Step 3 is the subtraction: \(436 - 271\) should be done by place values. Wait, Step 3: She has \(200 + 60 + 5\). Let's check the subtraction:

Original numbers: 436 (400 + 30 + 6) and 271 (200 + 70 + 1). To subtract, we need to regroup if necessary. 30 - 70 is not possible, so we borrow from the hundreds place: 400 becomes 300, and 30 becomes 130. So 436 is \(300 + 130 + 6\)? Wait, no, 4 hundreds is 400, if we borrow 1 hundred (100) to the tens place, 400 becomes 300, and 30 (tens) becomes 30 + 100 = 130? Wait, no, tens place: 3 tens is 30, to subtract 7 tens (70), we need to borrow 1 hundred (100) as 10 tens, so 30 + 100 = 130 tens? No, wait, 1 hundred is 10 tens, so 436 = 300 + 130 + 6 (because we borrow 1 hundred (100) from the hundreds place to the tens place, making the tens place 30 + 100 = 130? Wait, no, 436: hundreds digit 4, tens digit 3, ones digit 6. To subtract 271 (hundreds 2, tens 7, ones 1), we need to do:

Ones place: 6 - 1 = 5

Tens place: 3 - 7. Can't do, so borrow 1 from hundreds place: 4 becomes 3, and 3 becomes 3 + 10 = 13. Then 13 - 7 = 6

Hundreds place: 3 - 2 = 1

So the result should be 100 + 60 + 5 = 165? Wait, no, that's not right. Wait, 436 - 271: 436 - 200 = 236; 236 - 70 = 166; 166 - 1 = 165. Wait, but Rebecca's Step 4 says 265. Wait, but the question is which step is the first error.

Wait, Step 3: She has \(200 + 60 + 5\). Let's check the subtraction:

If 436 is \(400 + 130 + 6\) (Step 1) and 271 is \(200 + 70 + 1\) (Step 2), then subtracting:

Hundreds: 400 - 200 = 200? Wait, no, if we didn't borrow, but we need to borrow. Wait, Step 1: Is 436 correctly broken down? 436 is 400 + 30 + 6. If we need to subtract 70 (from 271), we need to borrow, so 436 becomes 300 + 130 + 6 (because 400 - 100 = 300, and 30 + 100 = 130). So Step 1: \(436 = 400 + 130 + 6\) – is that correct? Wait, 400 + 130 + 6 = 536? No! Wait, 400 + 130 is 530, plus 6 is 536. That's wrong! Oh! So Step 1 is incorrect. Wait, 436 is 400 + 30 + 6. If she is trying to regroup, she should have 300 + 130 + 6 (because 400 - 100 = 300, and 30 + 100 = 130), but 300 + 130 + 6 = 436? Wait, 300 + 130 = 430, plus 6 is 436. Oh, right! 300 + 130 + 6 = 436. So Step 1 is correct? Wait, 400 + 130 + 6: 400 + 130 is 530? No, 400 + 130 is 530? Wait, 400 + 100 is 500, plus 30 is 530. Oh! Wait, that's the mistake. 436 is 400 + 30 + 6. If she writes 400 + 130 + 6, that's 400 + 130 = 530 + 6 = 536, which is not 436. So Step 1 is wrong. Wait, but maybe she is regrouping for subtraction. Let's check Step 3. Step 3: She has 200 + 60 + 5. Let's calculate that: 200 + 60 = 260 + 5 = 265. But the correct result of 436 - 271 is 165. So Step 3 is wrong. Wait, but when does the first error occur?

Wait, Step 1: 436 = 400 + 130 + 6. Let's compute that: 400 + 130 = 530 + 6 = 536. That's not 436. So Step 1 is incorrect. But maybe she meant 300 + 130 + 6? Let's check: 300 + 130 = 430 + 6 = 436. Ah, so maybe a typo? Wait, the problem says "Rebecca breaks apart the numbers by place value, regroups, and subtracts". So when regrouping, to subtract 271 (which has 70 in tens place), she needs to borrow from the hundreds place. So 436: hundreds digit 4, tens digit 3, ones digit 6. To subtract 70 (tens place), we need to borrow 1 from the hundreds place, so hundreds digit becomes 3, and tens digit becomes 3 + 10 = 13 (tens). So 436 = 300 (hundreds) + 130 (tens) + 6 (ones). So Step 1: 436 = 400 + 130 + 6. Wait, 400 is the original hundreds digit, but after borrowing, it should be 300. So Step 1 is incorrect because she didn't adjust the hundreds place. Wait, but maybe the error is in Step 3. Let's check the subtraction:

If Step 1 is 436 = 400 + 130 + 6 (even though that's 536, which is wrong), and Step 2 is 271 = 200 + 70 + 1, then Step 3 should be (400 - 200) + (130 - 70) + (6 - 1) = 200 + 60 + 5 = 265. But the correct result of 436 - 271 is 165, so where is the mistake?

Ah! The mistake is in Step 1. 436 is 400 + 30 + 6, not 400 + 130 + 6. The 130 comes from borrowing 100 from the hundreds place to the tens place, but that would make the hundreds place 300, not 400. So Step 1 is incorrect because she kept the hundreds place as 400 while changing the tens place to 130, which is a contradiction. Therefore, the first error is in Step 1.

Wait, but let's verify:

Correct breakdown for subtraction:

436 - 271:

- Ones place: 6 - 1 = 5

- Tens place: 3 - 7. Can't do, so borrow 1 from hundreds place: 4 becomes 3, and 3 becomes 3 + 10 = 13. So tens place: 13 - 7 = 6

- Hundreds place: 3 - 2 = 1

So result: 100 + 60 + 5 = 165

Rebecca's Step 3: 200 + 60 + 5 = 265. So her hundreds place subtraction is 400 - 200 = 200, but she should have 300 - 200 = 100 (because she borrowed 100 from the hundreds place to the tens place). So the error in Step 3 comes from Step 1, where she didn't adjust the hundreds place. But the question is "in which step does Rebecca make the first error?" So Step 1 is the first error because she incorrectly broke down 436 as 400 + 130 + 6 (which is actually 536, not 436) or, if she was regrouping, she should have 300 + 130 + 6, not 400 + 130 + 6. Therefore, the first error is in Step 1.

Wait, but let's check the numbers again. 400 + 130 + 6 = 536, which is not 436. So Step 1 is wrong. So the first error is Step 1.

Step 2: Confirm the error

Step 1: 436 = 400 + 130 + 6. Calculating that: 400 + 130 = 530 + 6 = 536 ≠ 436. So this is incorrect. Therefore, the first error is in Step 1.

Answer:

Step 1