Recover hidden digits using regrouping

3.NBT.A.23.OA.A.42.NBT.B.7 · take · grade 3

Archetype: Recover Hidden Digits from Carries · step in a 5-type progression

In the subtraction below, find the digit that belongs in each of $A$ , $B$ , and $C$ .

\begin{array}{r} 8\;3\;1 \\ -\;A\;B\;7 \\ \hline 4\;6\;C \end{array}

Using regrouping (borrowing) in the three-digit subtraction $831 - \overline{AB7} = \overline{46C}$ , work out the digit in each place to determine $A$ , $B$ , and $C$ .

Show solution

Understand

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.

Givens

The minuend is 831.
The subtrahend is the three-digit number whose digits are A, B, and 7.
The difference is the three-digit number whose digits are 4, 6, and C.

Unknowns

The hundreds digit A of the subtrahend.
The tens digit B of the subtrahend.
The units digit C of the difference.

Constraints

Each of A, B, and C is a single digit (0-9).
The subtraction must hold place-by-place with regrouping (borrowing).

Plan

#11 Work Backwards · also uses: #6 Guess and Check

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.

Execute

#11 Work Backwards 2.NBT.B.7

In the units column we need 1 minus 7, which is impossible without borrowing, so we borrow 1 ten: 11 minus 7 equals 4. So C = 4, and the tens digit of 831 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.

#11 Work Backwards 3.NBT.A.2

After the borrow the tens digit of 831 is 2. We need 2 minus B to give 6, which is impossible, so we borrow again: 12 minus B equals 6, giving B = 6. The hundreds digit of 831 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.

#6 Guess and Check 3.OA.A.4

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.

#6 Guess and Check 3.NBT.A.2

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.

Answer: A = 3, B = 6, C = 4 (since 831 - 367 = 464)

Review

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.

Instead of working backwards place-by-place, you could compute the subtrahend directly as 831 - 464 = 367, then read off A = 3, B = 6, and the units 7 forces C from 831 - 367 = 464.

Standards · min grade 3

2.NBT.B.7 Add and subtract within 1000 using models or strategies — Regrouping (borrowing) across place values in the three-digit subtraction.
3.NBT.A.2 Fluently add and subtract within 1000 — Carrying out and verifying the full three-digit subtraction.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding each unknown digit as the missing number in a place-value fact.

💡 This only needs Grade 3 borrowing sense: chase each place from the ones up and the hidden digits pop out!