Depth · one concept from first principles · step 2 of 4
In the subtraction below, find the digit that belongs in each of , , and .
Using regrouping (borrowing) in the three-digit subtraction , work out the digit in each place to determine , , and .
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.
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.
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.
Ten ones bundle into exactly one ten, so trading one ten for ten ones swaps equal amounts.
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.
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.
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.
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.
One borrowed ten is worth ten ones, so it hands the ones column more than enough units to take 7 away.
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.
Reading the tens column backwards turns the subtraction into a simple missing-number fact.
3.NBT.A.2Work BackwardsAfter the second borrow the hundreds digit of 831 is 7. We need 7 minus A to equal 4, so A = 3.
Finding the unknown in 7 minus A equals 4 is a basic determine-the-unknown fact.
3.OA.A.4Guess And CheckSubstitute back: the subtrahend is 367 and 831 minus 367 equals 464, which matches 4 6 C with C = 4.
Re-adding or re-subtracting confirms every place lines up.
3.NBT.A.2Guess And CheckThis only needs Grade 3 borrowing sense: chase each place from the ones up and the hidden digits pop out!
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The same shape shows up later on the AMC, as Small-Domain Constrained Search. Why the shapes repeat →