Place Value Error in a Subtraction
A student computes 5,042 − 2,768 and writes the answer 3,386. The teacher asks the student to explain each step. The student says: "2 minus 8, I borrowed so it's 12 minus 8 = 4. Then 3 minus 6, I borrowed so 13 minus 6 = 7. Then 9 minus 7 = 2. Then 5 minus 2 = 3." The teacher notices the student's written answer does not match the spoken steps, and that the borrowing was applied inconsistently. A second student claims the correct answer is 2,274. The class wants to know which answer is right and exactly where the first student's reasoning broke down.
1. Find the correct difference and identify the specific place-value errors in the first student's reasoning.
Read the worked answer
Compute carefully using regrouping:
- Ones: 2 − 8 requires regrouping. Borrow from the tens; but the tens digit is 4, so 5,042 becomes 4,132 effectively for the ones: 12 − 8 = 4.
- Tens: now 3 − 6 requires regrouping from the hundreds. The hundreds digit is 0, so borrow from the thousands: 5,042 → 4,932 in the hundreds/tens region. Tens become 13 − 6 = 7.
- Hundreds: 9 − 7 = 2.
- Thousands: 4 − 2 = 2.
Correct difference: 2,274.
The first student's errors: (1) the spoken steps produced 3,274, not the written 3,386 - an inconsistency; (2) the student borrowed in the tens but then used 9 in the hundreds without ever regrouping through the zero, which is the key trap; (3) the final thousands digit should be 4, not 5, because a thousand was borrowed. The second student is correct.
2. Explain why the zero in the hundreds place makes this subtraction harder, and describe a reliable check the class can use.
Read the worked answer
The zero in the hundreds place cannot lend directly, so regrouping must cascade: a thousand is broken into 10 hundreds, then one hundred is broken into 10 tens. Skipping that chain is exactly what produced the wrong digits. Students often "borrow across the zero" as if the zero were a 9 or 10 without adjusting the thousands digit.
A reliable check is addition (inverse operation): 2,274 + 2,768. Add ones 4 + 8 = 12, write 2 carry 1; tens 7 + 6 + 1 = 14, write 4 carry 1; hundreds 2 + 7 + 1 = 10, write 0 carry 1; thousands 2 + 2 + 1 = 5. Result 5,042, which matches the minuend. Estimation also flags errors: 5,000 − 2,800 = 2,200, so an answer near 3,300 is clearly too large.