Problem
Recover hidden digits using regrouping
In the column subtraction 831 minus the three-digit number A B 7 equals 4 6 C, recover the missing digits A, B, and C by working each place value using borrowing.
Base-ten numbersOperations
Your answer
Watch out After borrowing in the units place, the tens digit of 831 becomes 2, not 3 -- forgetting this borrow-adjusted digit gives the wrong value for B.
How to solve
Strategy Work Backwards — The result of the subtraction is given and the hidden digits sit inside the problem, so we reason backwards one place at a time from the units, tracking each borrow, and confirm each recovered digit against the column.
1STEP 1
Units place
Units: 1 minus 7 needs a borrow, so borrow 1 ten: 11 minus 7 is 4, giving C = 4. The tens digit of 831 now drops from 3 to 2.
11 - 7 = 4 \Rightarrow C = 4
Borrowing one ten to make 11 is the same regrouping kids learn for subtracting within 1000.
2.NBT.B.7Work BackwardsIn the units column, because the top digit 1 is smaller than 7, we borrow one ten from the tens place and add it to the ones as ten more units, making 11 before we take 7 away.
Why?
We are allowed to move value from the tens place to the ones place because rewriting one ten as ten ones does not change how much 831 is worth.
Why?
Ten ones bundle into exactly one ten, so trading one ten for ten ones swaps equal amounts.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
The number 831 is just its hundreds, tens, and ones added together, so shifting some of that value from the tens pile into the ones pile leaves the whole number unchanged.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
We are allowed to handle the ones by themselves because the full subtraction is the same as subtracting hundreds from hundreds, tens from tens, and ones from ones.
Why?
Both numbers split into place-value parts with no gaps or overlap, so the whole difference is the sum of the differences found in each place.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
We have to add ten more ones first because you cannot take 7 ones away from a single one, so the top needs at least as many ones as we remove.
Why?
One borrowed ten is worth ten ones, so it hands the ones column more than enough units to take 7 away.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
2STEP 2
Tens place
The tens digit is now 2, so 2 minus B can't reach 6: borrow again, 12 minus B equals 6, giving B = 6. The hundreds digit drops from 8 to 7.
12 - B = 6 \Rightarrow B = 6
Reading the tens column backwards turns the subtraction into a simple missing-number fact.
3.NBT.A.2Work Backwards3STEP 3
Hundreds place
After the second borrow the hundreds digit of 831 is 7. We need 7 minus A to equal 4, so A = 3.
7 - A = 4 \Rightarrow A = 3
Finding the unknown in 7 minus A equals 4 is a basic determine-the-unknown fact.
3.OA.A.4Guess And Check4STEP 4
Verify
Substitute back: the subtrahend is 367 and 831 minus 367 equals 464, which matches 4 6 C with C = 4.
831 - 367 = 464
Re-adding or re-subtracting confirms every place lines up.
3.NBT.A.2Guess And CheckAnswer
A = 3, B = 6, C = 4 (since 831 - 367 = 464)
831 - 367 = 464
The recovered subtrahend 367 is a valid three-digit number, and 831 - 367 = 464 reproduces the given difference exactly, with both borrows accounted for. Magnitudes are sensible: 831 - 367 should be a bit under 500, and 464 fits.
Takeaway
This only needs Grade 3 borrowing sense: chase each place from the ones up and the hidden digits pop out!
- Units place
- Tens place
- Hundreds place
- Verify
Where next?
Level upsuggested
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Step back and solidifysuggested