Practice · Recover hidden digits using regrouping · Sensim Math

Variant 1 easy answer: A = 2, B = 7, C = 7 (since 450 - 273 = 177)

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

\begin{array}{r} 4\;5\;0 \\ -\;A\;B\;3 \\ \hline 1\;7\;C \end{array}

Using regrouping (borrowing) in the three-digit subtraction $450 - \overline{AB3} = \overline{17C}$ , work out the digit in each place to determine $A$ , $B$ , and $C$ .

Show solution

1 · Understandwhat's really being asked

In the column subtraction 450 minus the three-digit number A B 3 equals 1 7 C, recover the missing digits A, B, and C by working each place value using borrowing.

Givens

The minuend is 450.
The subtrahend is the three-digit number whose digits are A, B, and 3.
The difference is the three-digit number whose digits are 1, 7, 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).

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Units place

#11 Work Backwards 2.NBT.B.7

In the units column we need 0 minus 3, which is impossible without borrowing, so we borrow 1 ten: 10 minus 3 equals 7. So C = 7, and the tens digit of 450 drops from 5 to 4.

10 - 3 = 7 \Rightarrow C = 7

Borrowing one ten is the same regrouping kids learn for subtracting within 1000.

2Tens place

#11 Work Backwards 3.NBT.A.2

After the units work the tens digit of 450 is 4. We need 4 minus B to give 7, which is impossible, so we borrow again: 14 minus B equals 7, giving B = 7. The hundreds digit of 450 drops from 4 to 3.

14 - B = 7 \Rightarrow B = 7

Reading the tens column backwards turns the subtraction into a simple missing-number fact.

3Hundreds place

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

After the tens work the hundreds digit of 450 is 3. We need 3 minus A to equal 1, so A = 2.

3 - A = 1 \Rightarrow A = 2

Finding the unknown in a place-value fact is a basic determine-the-unknown step.

4Verify

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

Substitute back: the subtrahend is 273 and 450 minus 273 equals 177, which matches 1 7 C with C = 7.

450 - 273 = 177

Re-subtracting confirms every place lines up.

Answer: A = 2, B = 7, C = 7 (since 450 - 273 = 177)

4 · Reviewdoes it hold up?

The recovered subtrahend 273 is a valid three-digit number, and 450 - 273 = 177 reproduces the given difference exactly, with every borrow accounted for.

Another way: Instead of working backwards place-by-place, you could compute the subtrahend directly as 450 - 177 = 273, then read off A = 2, B = 7, and the shown units force C from 450 - 273 = 177.

Standardsmin grade 3

2.NBT.B.7 Add and subtract within 1000 using models or strategies — Regrouping (borrowing) across place values in the 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.

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

Variant 2 easy answer: A = 1, B = 7, C = 4 (since 542 - 178 = 364)

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

\begin{array}{r} 5\;4\;2 \\ -\;A\;B\;8 \\ \hline 3\;6\;C \end{array}

Using regrouping (borrowing) in the three-digit subtraction $542 - \overline{AB8} = \overline{36C}$ , work out the digit in each place to determine $A$ , $B$ , and $C$ .

Show solution

1 · Understandwhat's really being asked

In the column subtraction 542 minus the three-digit number A B 8 equals 3 6 C, recover the missing digits A, B, and C by working each place value using borrowing.

Givens

The minuend is 542.
The subtrahend is the three-digit number whose digits are A, B, and 8.
The difference is the three-digit number whose digits are 3, 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).

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Units place

#11 Work Backwards 2.NBT.B.7

In the units column we need 2 minus 8, which is impossible without borrowing, so we borrow 1 ten: 12 minus 8 equals 4. So C = 4, and the tens digit of 542 drops from 4 to 3.

12 - 8 = 4 \Rightarrow C = 4

Borrowing one ten is the same regrouping kids learn for subtracting within 1000.

2Tens place

#11 Work Backwards 3.NBT.A.2

After the units work the tens digit of 542 is 3. We need 3 minus B to give 6, which is impossible, so we borrow again: 13 minus B equals 6, giving B = 7. The hundreds digit of 542 drops from 5 to 4.

13 - B = 6 \Rightarrow B = 7

Reading the tens column backwards turns the subtraction into a simple missing-number fact.

3Hundreds place

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

After the tens work the hundreds digit of 542 is 4. We need 4 minus A to equal 3, so A = 1.

4 - A = 3 \Rightarrow A = 1

Finding the unknown in a place-value fact is a basic determine-the-unknown step.

4Verify

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

Substitute back: the subtrahend is 178 and 542 minus 178 equals 364, which matches 3 6 C with C = 4.

542 - 178 = 364

Re-subtracting confirms every place lines up.

Answer: A = 1, B = 7, C = 4 (since 542 - 178 = 364)

4 · Reviewdoes it hold up?

The recovered subtrahend 178 is a valid three-digit number, and 542 - 178 = 364 reproduces the given difference exactly, with every borrow accounted for.

Another way: Instead of working backwards place-by-place, you could compute the subtrahend directly as 542 - 364 = 178, then read off A = 1, B = 7, and the shown units force C from 542 - 178 = 364.

Standardsmin grade 3

2.NBT.B.7 Add and subtract within 1000 using models or strategies — Regrouping (borrowing) across place values in the 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.

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

Variant 3 easy answer: A = 3, B = 5, C = 7 (since 621 - 354 = 267)

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

\begin{array}{r} 6\;2\;1 \\ -\;A\;B\;4 \\ \hline 2\;6\;C \end{array}

Using regrouping (borrowing) in the three-digit subtraction $621 - \overline{AB4} = \overline{26C}$ , work out the digit in each place to determine $A$ , $B$ , and $C$ .

Show solution

1 · Understandwhat's really being asked

In the column subtraction 621 minus the three-digit number A B 4 equals 2 6 C, recover the missing digits A, B, and C by working each place value using borrowing.

Givens

The minuend is 621.
The subtrahend is the three-digit number whose digits are A, B, and 4.
The difference is the three-digit number whose digits are 2, 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).

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Units place

#11 Work Backwards 2.NBT.B.7

In the units column we need 1 minus 4, which is impossible without borrowing, so we borrow 1 ten: 11 minus 4 equals 7. So C = 7, and the tens digit of 621 drops from 2 to 1.

11 - 4 = 7 \Rightarrow C = 7

Borrowing one ten is the same regrouping kids learn for subtracting within 1000.

2Tens place

#11 Work Backwards 3.NBT.A.2

After the units work the tens digit of 621 is 1. We need 1 minus B to give 6, which is impossible, so we borrow again: 11 minus B equals 6, giving B = 5. The hundreds digit of 621 drops from 6 to 5.

11 - B = 6 \Rightarrow B = 5

Reading the tens column backwards turns the subtraction into a simple missing-number fact.

3Hundreds place

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

After the tens work the hundreds digit of 621 is 5. We need 5 minus A to equal 2, so A = 3.

5 - A = 2 \Rightarrow A = 3

Finding the unknown in a place-value fact is a basic determine-the-unknown step.

4Verify

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

Substitute back: the subtrahend is 354 and 621 minus 354 equals 267, which matches 2 6 C with C = 7.

621 - 354 = 267

Re-subtracting confirms every place lines up.

Answer: A = 3, B = 5, C = 7 (since 621 - 354 = 267)

4 · Reviewdoes it hold up?

The recovered subtrahend 354 is a valid three-digit number, and 621 - 354 = 267 reproduces the given difference exactly, with every borrow accounted for.

Another way: Instead of working backwards place-by-place, you could compute the subtrahend directly as 621 - 267 = 354, then read off A = 3, B = 5, and the shown units force C from 621 - 354 = 267.

Standardsmin grade 3

2.NBT.B.7 Add and subtract within 1000 using models or strategies — Regrouping (borrowing) across place values in the 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.

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

Variant 4 medium answer: A = 2, B = 8, C = 8 (since 623 - 285 = 338)

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

\begin{array}{r} 6\;2\;3 \\ -\;A\;B\;5 \\ \hline 3\;3\;C \end{array}

Using regrouping (borrowing) in the three-digit subtraction $623 - \overline{AB5} = \overline{33C}$ , work out the digit in each place to determine $A$ , $B$ , and $C$ .

Show solution

1 · Understandwhat's really being asked

In the column subtraction 623 minus the three-digit number A B 5 equals 3 3 C, recover the missing digits A, B, and C by working each place value using borrowing.

Givens

The minuend is 623.
The subtrahend is the three-digit number whose digits are A, B, and 5.
The difference is the three-digit number whose digits are 3, 3, 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).

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Units place

#11 Work Backwards 2.NBT.B.7

In the units column we need 3 minus 5, which is impossible without borrowing, so we borrow 1 ten: 13 minus 5 equals 8. So C = 8, and the tens digit of 623 drops from 2 to 1.

13 - 5 = 8 \Rightarrow C = 8

Borrowing one ten is the same regrouping kids learn for subtracting within 1000.

2Tens place

#11 Work Backwards 3.NBT.A.2

After the units work the tens digit of 623 is 1. We need 1 minus B to give 3, which is impossible, so we borrow again: 11 minus B equals 3, giving B = 8. The hundreds digit of 623 drops from 6 to 5.

11 - B = 3 \Rightarrow B = 8

Reading the tens column backwards turns the subtraction into a simple missing-number fact.

3Hundreds place

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

After the tens work the hundreds digit of 623 is 5. We need 5 minus A to equal 3, so A = 2.

5 - A = 3 \Rightarrow A = 2

Finding the unknown in a place-value fact is a basic determine-the-unknown step.

4Verify

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

Substitute back: the subtrahend is 285 and 623 minus 285 equals 338, which matches 3 3 C with C = 8.

623 - 285 = 338

Re-subtracting confirms every place lines up.

Answer: A = 2, B = 8, C = 8 (since 623 - 285 = 338)

4 · Reviewdoes it hold up?

The recovered subtrahend 285 is a valid three-digit number, and 623 - 285 = 338 reproduces the given difference exactly, with every borrow accounted for.

Another way: Instead of working backwards place-by-place, you could compute the subtrahend directly as 623 - 338 = 285, then read off A = 2, B = 8, and the shown units force C from 623 - 285 = 338.

Standardsmin grade 3

2.NBT.B.7 Add and subtract within 1000 using models or strategies — Regrouping (borrowing) across place values in the 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.

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

Variant 5 medium answer: A = 2, B = 4, C = 4 (since 700 - 246 = 454)

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

\begin{array}{r} 7\;0\;0 \\ -\;A\;B\;6 \\ \hline 4\;5\;C \end{array}

Using regrouping (borrowing) in the three-digit subtraction $700 - \overline{AB6} = \overline{45C}$ , work out the digit in each place to determine $A$ , $B$ , and $C$ .

Show solution

1 · Understandwhat's really being asked

In the column subtraction 700 minus the three-digit number A B 6 equals 4 5 C, recover the missing digits A, B, and C by working each place value using borrowing.

Givens

The minuend is 700.
The subtrahend is the three-digit number whose digits are A, B, and 6.
The difference is the three-digit number whose digits are 4, 5, 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).

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Units place

#11 Work Backwards 2.NBT.B.7

In the units column we need 0 minus 6, which is impossible without borrowing, so we borrow 1 ten: 10 minus 6 equals 4. So C = 4, and the tens digit of 700 drops from 0 to -1.

10 - 6 = 4 \Rightarrow C = 4

Borrowing one ten is the same regrouping kids learn for subtracting within 1000.

2Tens place

#11 Work Backwards 3.NBT.A.2

After the units work the tens digit of 700 is -1. We need -1 minus B to give 5, which is impossible, so we borrow again: 9 minus B equals 5, giving B = 4. The hundreds digit of 700 drops from 7 to 6.

9 - B = 5 \Rightarrow B = 4

Reading the tens column backwards turns the subtraction into a simple missing-number fact.

3Hundreds place

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

After the tens work the hundreds digit of 700 is 6. We need 6 minus A to equal 4, so A = 2.

6 - A = 4 \Rightarrow A = 2

Finding the unknown in a place-value fact is a basic determine-the-unknown step.

4Verify

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

Substitute back: the subtrahend is 246 and 700 minus 246 equals 454, which matches 4 5 C with C = 4.

700 - 246 = 454

Re-subtracting confirms every place lines up.

Answer: A = 2, B = 4, C = 4 (since 700 - 246 = 454)

4 · Reviewdoes it hold up?

The recovered subtrahend 246 is a valid three-digit number, and 700 - 246 = 454 reproduces the given difference exactly, with every borrow accounted for.

Another way: Instead of working backwards place-by-place, you could compute the subtrahend directly as 700 - 454 = 246, then read off A = 2, B = 4, and the shown units force C from 700 - 246 = 454.

Standardsmin grade 3

2.NBT.B.7 Add and subtract within 1000 using models or strategies — Regrouping (borrowing) across place values in the 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.

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

Variant 6 medium answer: A = 1, B = 6, C = 7 (since 742 - 165 = 577)

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

\begin{array}{r} 7\;4\;2 \\ -\;A\;B\;5 \\ \hline 5\;7\;C \end{array}

Using regrouping (borrowing) in the three-digit subtraction $742 - \overline{AB5} = \overline{57C}$ , work out the digit in each place to determine $A$ , $B$ , and $C$ .

Show solution

1 · Understandwhat's really being asked

In the column subtraction 742 minus the three-digit number A B 5 equals 5 7 C, recover the missing digits A, B, and C by working each place value using borrowing.

Givens

The minuend is 742.
The subtrahend is the three-digit number whose digits are A, B, and 5.
The difference is the three-digit number whose digits are 5, 7, 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).

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Units place

#11 Work Backwards 2.NBT.B.7

In the units column we need 2 minus 5, which is impossible without borrowing, so we borrow 1 ten: 12 minus 5 equals 7. So C = 7, and the tens digit of 742 drops from 4 to 3.

12 - 5 = 7 \Rightarrow C = 7

Borrowing one ten is the same regrouping kids learn for subtracting within 1000.

2Tens place

#11 Work Backwards 3.NBT.A.2

After the units work the tens digit of 742 is 3. We need 3 minus B to give 7, which is impossible, so we borrow again: 13 minus B equals 7, giving B = 6. The hundreds digit of 742 drops from 7 to 6.

13 - B = 7 \Rightarrow B = 6

Reading the tens column backwards turns the subtraction into a simple missing-number fact.

3Hundreds place

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

After the tens work the hundreds digit of 742 is 6. We need 6 minus A to equal 5, so A = 1.

6 - A = 5 \Rightarrow A = 1

Finding the unknown in a place-value fact is a basic determine-the-unknown step.

4Verify

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

Substitute back: the subtrahend is 165 and 742 minus 165 equals 577, which matches 5 7 C with C = 7.

742 - 165 = 577

Re-subtracting confirms every place lines up.

Answer: A = 1, B = 6, C = 7 (since 742 - 165 = 577)

4 · Reviewdoes it hold up?

The recovered subtrahend 165 is a valid three-digit number, and 742 - 165 = 577 reproduces the given difference exactly, with every borrow accounted for.

Another way: Instead of working backwards place-by-place, you could compute the subtrahend directly as 742 - 577 = 165, then read off A = 1, B = 6, and the shown units force C from 742 - 165 = 577.

Standardsmin grade 3

2.NBT.B.7 Add and subtract within 1000 using models or strategies — Regrouping (borrowing) across place values in the 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.

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

Variant 7 medium answer: A = 3, B = 6, C = 4 (since 831 - 367 = 464)

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

1 · Understandwhat's really being asked

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).

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Units place

#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 is the same regrouping kids learn for subtracting within 1000.

2Tens place

#11 Work Backwards 3.NBT.A.2

After the units work 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.

3Hundreds place

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

After the tens work 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 a place-value fact is a basic determine-the-unknown step.

4Verify

#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-subtracting confirms every place lines up.

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

4 · Reviewdoes it hold up?

The recovered subtrahend 367 is a valid three-digit number, and 831 - 367 = 464 reproduces the given difference exactly, with every borrow accounted for.

Another way: 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 shown units force C from 831 - 367 = 464.

Standardsmin grade 3

2.NBT.B.7 Add and subtract within 1000 using models or strategies — Regrouping (borrowing) across place values in the 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.

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

Variant 8 medium answer: A = 4, B = 8, C = 6 (since 835 - 489 = 346)

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

\begin{array}{r} 8\;3\;5 \\ -\;A\;B\;9 \\ \hline 3\;4\;C \end{array}

Using regrouping (borrowing) in the three-digit subtraction $835 - \overline{AB9} = \overline{34C}$ , work out the digit in each place to determine $A$ , $B$ , and $C$ .

Show solution

1 · Understandwhat's really being asked

In the column subtraction 835 minus the three-digit number A B 9 equals 3 4 C, recover the missing digits A, B, and C by working each place value using borrowing.

Givens

The minuend is 835.
The subtrahend is the three-digit number whose digits are A, B, and 9.
The difference is the three-digit number whose digits are 3, 4, 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).

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Units place

#11 Work Backwards 2.NBT.B.7

In the units column we need 5 minus 9, which is impossible without borrowing, so we borrow 1 ten: 15 minus 9 equals 6. So C = 6, and the tens digit of 835 drops from 3 to 2.

15 - 9 = 6 \Rightarrow C = 6

Borrowing one ten is the same regrouping kids learn for subtracting within 1000.

2Tens place

#11 Work Backwards 3.NBT.A.2

After the units work the tens digit of 835 is 2. We need 2 minus B to give 4, which is impossible, so we borrow again: 12 minus B equals 4, giving B = 8. The hundreds digit of 835 drops from 8 to 7.

12 - B = 4 \Rightarrow B = 8

Reading the tens column backwards turns the subtraction into a simple missing-number fact.

3Hundreds place

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

After the tens work the hundreds digit of 835 is 7. We need 7 minus A to equal 3, so A = 4.

7 - A = 3 \Rightarrow A = 4

Finding the unknown in a place-value fact is a basic determine-the-unknown step.

4Verify

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

Substitute back: the subtrahend is 489 and 835 minus 489 equals 346, which matches 3 4 C with C = 6.

835 - 489 = 346

Re-subtracting confirms every place lines up.

Answer: A = 4, B = 8, C = 6 (since 835 - 489 = 346)

4 · Reviewdoes it hold up?

The recovered subtrahend 489 is a valid three-digit number, and 835 - 489 = 346 reproduces the given difference exactly, with every borrow accounted for.

Another way: Instead of working backwards place-by-place, you could compute the subtrahend directly as 835 - 346 = 489, then read off A = 4, B = 8, and the shown units force C from 835 - 489 = 346.

Standardsmin grade 3

2.NBT.B.7 Add and subtract within 1000 using models or strategies — Regrouping (borrowing) across place values in the 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.

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

Variant 9 hard answer: A = 5, B = 3, C = 5 (since 903 - 538 = 365)

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

\begin{array}{r} 9\;0\;3 \\ -\;A\;B\;8 \\ \hline 3\;6\;C \end{array}

Using regrouping (borrowing) in the three-digit subtraction $903 - \overline{AB8} = \overline{36C}$ , work out the digit in each place to determine $A$ , $B$ , and $C$ .

Show solution

1 · Understandwhat's really being asked

In the column subtraction 903 minus the three-digit number A B 8 equals 3 6 C, recover the missing digits A, B, and C by working each place value using borrowing.

Givens

The minuend is 903.
The subtrahend is the three-digit number whose digits are A, B, and 8.
The difference is the three-digit number whose digits are 3, 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).

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Units place

#11 Work Backwards 2.NBT.B.7

In the units column we need 3 minus 8, which is impossible without borrowing, so we borrow 1 ten: 13 minus 8 equals 5. So C = 5, and the tens digit of 903 drops from 0 to -1.

13 - 8 = 5 \Rightarrow C = 5

Borrowing one ten is the same regrouping kids learn for subtracting within 1000.

2Tens place

#11 Work Backwards 3.NBT.A.2

After the units work the tens digit of 903 is -1. We need -1 minus B to give 6, which is impossible, so we borrow again: 9 minus B equals 6, giving B = 3. The hundreds digit of 903 drops from 9 to 8.

9 - B = 6 \Rightarrow B = 3

Reading the tens column backwards turns the subtraction into a simple missing-number fact.

3Hundreds place

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

After the tens work the hundreds digit of 903 is 8. We need 8 minus A to equal 3, so A = 5.

8 - A = 3 \Rightarrow A = 5

Finding the unknown in a place-value fact is a basic determine-the-unknown step.

4Verify

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

Substitute back: the subtrahend is 538 and 903 minus 538 equals 365, which matches 3 6 C with C = 5.

903 - 538 = 365

Re-subtracting confirms every place lines up.

Answer: A = 5, B = 3, C = 5 (since 903 - 538 = 365)

4 · Reviewdoes it hold up?

The recovered subtrahend 538 is a valid three-digit number, and 903 - 538 = 365 reproduces the given difference exactly, with every borrow accounted for.

Another way: Instead of working backwards place-by-place, you could compute the subtrahend directly as 903 - 365 = 538, then read off A = 5, B = 3, and the shown units force C from 903 - 538 = 365.

Standardsmin grade 3

2.NBT.B.7 Add and subtract within 1000 using models or strategies — Regrouping (borrowing) across place values in the 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.

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

Variant 10 hard answer: A = 6, B = 5, C = 7 (since 914 - 657 = 257)

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

\begin{array}{r} 9\;1\;4 \\ -\;A\;B\;7 \\ \hline 2\;5\;C \end{array}

Using regrouping (borrowing) in the three-digit subtraction $914 - \overline{AB7} = \overline{25C}$ , work out the digit in each place to determine $A$ , $B$ , and $C$ .

Show solution

1 · Understandwhat's really being asked

In the column subtraction 914 minus the three-digit number A B 7 equals 2 5 C, recover the missing digits A, B, and C by working each place value using borrowing.

Givens

The minuend is 914.
The subtrahend is the three-digit number whose digits are A, B, and 7.
The difference is the three-digit number whose digits are 2, 5, 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).

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Units place

#11 Work Backwards 2.NBT.B.7

In the units column we need 4 minus 7, which is impossible without borrowing, so we borrow 1 ten: 14 minus 7 equals 7. So C = 7, and the tens digit of 914 drops from 1 to 0.

14 - 7 = 7 \Rightarrow C = 7

Borrowing one ten is the same regrouping kids learn for subtracting within 1000.

2Tens place

#11 Work Backwards 3.NBT.A.2

After the units work the tens digit of 914 is 0. We need 0 minus B to give 5, which is impossible, so we borrow again: 10 minus B equals 5, giving B = 5. The hundreds digit of 914 drops from 9 to 8.

10 - B = 5 \Rightarrow B = 5

Reading the tens column backwards turns the subtraction into a simple missing-number fact.

3Hundreds place

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

After the tens work the hundreds digit of 914 is 8. We need 8 minus A to equal 2, so A = 6.

8 - A = 2 \Rightarrow A = 6

Finding the unknown in a place-value fact is a basic determine-the-unknown step.

4Verify

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

Substitute back: the subtrahend is 657 and 914 minus 657 equals 257, which matches 2 5 C with C = 7.

914 - 657 = 257

Re-subtracting confirms every place lines up.

Answer: A = 6, B = 5, C = 7 (since 914 - 657 = 257)

4 · Reviewdoes it hold up?

The recovered subtrahend 657 is a valid three-digit number, and 914 - 657 = 257 reproduces the given difference exactly, with every borrow accounted for.

Another way: Instead of working backwards place-by-place, you could compute the subtrahend directly as 914 - 257 = 657, then read off A = 6, B = 5, and the shown units force C from 914 - 657 = 257.

Standardsmin grade 3

2.NBT.B.7 Add and subtract within 1000 using models or strategies — Regrouping (borrowing) across place values in the 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.

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

Variant 11 hard answer: A = 5, B = 9, C = 7 (since 986 - 599 = 387)

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

\begin{array}{r} 9\;8\;6 \\ -\;A\;B\;9 \\ \hline 3\;8\;C \end{array}

Using regrouping (borrowing) in the three-digit subtraction $986 - \overline{AB9} = \overline{38C}$ , work out the digit in each place to determine $A$ , $B$ , and $C$ .

Show solution

1 · Understandwhat's really being asked

In the column subtraction 986 minus the three-digit number A B 9 equals 3 8 C, recover the missing digits A, B, and C by working each place value using borrowing.

Givens

The minuend is 986.
The subtrahend is the three-digit number whose digits are A, B, and 9.
The difference is the three-digit number whose digits are 3, 8, 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).

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Units place

#11 Work Backwards 2.NBT.B.7

In the units column we need 6 minus 9, which is impossible without borrowing, so we borrow 1 ten: 16 minus 9 equals 7. So C = 7, and the tens digit of 986 drops from 8 to 7.

16 - 9 = 7 \Rightarrow C = 7

Borrowing one ten is the same regrouping kids learn for subtracting within 1000.

2Tens place

#11 Work Backwards 3.NBT.A.2

After the units work the tens digit of 986 is 7. We need 7 minus B to give 8, which is impossible, so we borrow again: 17 minus B equals 8, giving B = 9. The hundreds digit of 986 drops from 9 to 8.

17 - B = 8 \Rightarrow B = 9

Reading the tens column backwards turns the subtraction into a simple missing-number fact.

3Hundreds place

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

After the tens work the hundreds digit of 986 is 8. We need 8 minus A to equal 3, so A = 5.

8 - A = 3 \Rightarrow A = 5

Finding the unknown in a place-value fact is a basic determine-the-unknown step.

4Verify

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

Substitute back: the subtrahend is 599 and 986 minus 599 equals 387, which matches 3 8 C with C = 7.

986 - 599 = 387

Re-subtracting confirms every place lines up.

Answer: A = 5, B = 9, C = 7 (since 986 - 599 = 387)

4 · Reviewdoes it hold up?

The recovered subtrahend 599 is a valid three-digit number, and 986 - 599 = 387 reproduces the given difference exactly, with every borrow accounted for.

Another way: Instead of working backwards place-by-place, you could compute the subtrahend directly as 986 - 387 = 599, then read off A = 5, B = 9, and the shown units force C from 986 - 599 = 387.

Standardsmin grade 3

2.NBT.B.7 Add and subtract within 1000 using models or strategies — Regrouping (borrowing) across place values in the 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.

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

Recover hidden digits using regrouping · 11 practice problems

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