Solved Exercises on Three-Digit Addition With Regrouping in Grade 2

Master three-digit addition with regrouping: place value, column addition, and carrying concepts through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Single regrouping
Exercise 1
Calculate:
\(257 + 168\)
Definition:

Regrouping (carrying): When the sum of a column is 10 or more, we carry the extra value to the next column

Regrouping method:
  1. Line up numbers by place value (ones under ones, tens under tens, hundreds under hundreds)
  2. Add the ones column: \(7 + 8 = 15\), write 5 in ones place, carry 1 to tens column
  3. Add the tens column: \(5 + 6 + 1 = 12\), write 2 in tens place, carry 1 to hundreds column
  4. Add the hundreds column: \(2 + 1 + 1 = 4\), write 4 in hundreds place
  5. Write the answer: \(425\)
Expression
\(257 + 168\)
Column addition with regrouping
\(\begin{array}{r} 257 \\ + 168 \\ \hline 425 \end{array}\)
Result
\(425\)
Step 1: Line up by place value

\(\begin{array}{r} H \ T \ O \\ 2 \ 5 \ 7 \\ + 1 \ 6 \ 8 \\ \hline \end{array}\)

Step 2: Add ones column

\(7 + 8 = 15\), write 5 in ones place, carry 1 to tens column

Step 3: Add tens column

\(5 + 6 + 1 = 12\), write 2 in tens place, carry 1 to hundreds column

Step 4: Add hundreds column

\(2 + 1 + 1 = 4\), write 4 in hundreds place

\(257 + 168 = 425\)
Final answer:

\(257 + 168 = 425\)

Applied rules:

• Place value: Align digits by their place values (hundreds, tens, ones)

• Regrouping rule: When column sum ≥ 10, carry the tens digit to the next column

• Carry forward: Include carried digits in the next column's addition

2 Regrouping in two columns
Exercise 2
Calculate:
\(379 + 254\)
Definition:

Multiple regrouping: When regrouping is needed in more than one column during the same addition

Expression
\(379 + 254\)
Column addition with regrouping
\(\begin{array}{r} 379 \\ + 254 \\ \hline 633 \end{array}\)
Result
\(633\)
Step 1: Line up by place value

\(\begin{array}{r} H \ T \ O \\ 3 \ 7 \ 9 \\ + 2 \ 5 \ 4 \\ \hline \end{array}\)

Step 2: Add ones column

\(9 + 4 = 13\), write 3 in ones place, carry 1 to tens column

Step 3: Add tens column

\(7 + 5 + 1 = 13\), write 3 in tens place, carry 1 to hundreds column

Step 4: Add hundreds column

\(3 + 2 + 1 = 6\), write 6 in hundreds place

\(379 + 254 = 633\)
Final answer:

\(379 + 254 = 633\)

Applied rules:

• Multiple regrouping: Regroup in each column where sum ≥ 10

• Carry accumulation: Each carried digit affects the next column's sum

• Sequential processing: Process each column from right to left

3 Regrouping with zeros
Exercise 3
Calculate:
\(495 + 308\)
Definition:

Regrouping with zeros: When adding requires regrouping and one of the numbers contains zeros

Expression
\(495 + 308\)
Column addition with regrouping
\(\begin{array}{r} 495 \\ + 308 \\ \hline 803 \end{array}\)
Result
\(803\)
Step 1: Line up by place value

\(\begin{array}{r} H \ T \ O \\ 4 \ 9 \ 5 \\ + 3 \ 0 \ 8 \\ \hline \end{array}\)

Step 2: Add ones column

\(5 + 8 = 13\), write 3 in ones place, carry 1 to tens column

Step 3: Add tens column

\(9 + 0 + 1 = 10\), write 0 in tens place, carry 1 to hundreds column

Step 4: Add hundreds column

\(4 + 3 + 1 = 8\), write 8 in hundreds place

\(495 + 308 = 803\)
Final answer:

\(495 + 308 = 803\)

Applied rules:

• Zero handling: Zeros still participate in addition and regrouping

• Regrouping with zeros: Carried digits affect zeros in the same way as other digits

• Place value preservation: Maintain correct place value even with regrouped zeros

Three-Digit Addition With Regrouping Strategies
\(\begin{array}{r} H \ T \ O \\ 2 \ 5 \ 7 \\ + 1 \ 6 \ 8 \\ \hline 4 \ 2 \ 5 \end{array}\)
Regrouping Process
Rule 1
Regroup When ≥ 10
Carry the tens digit to the next column
Rule 2
Include Carries
Add carried digits to the next column
Rule 3
Right-to-Left Processing
Process columns from ones to hundreds
Key definitions:

Three-digit number: A number with three digits representing hundreds, tens, and ones places

Place value: The value of a digit based on its position in the number

Regrouping (carrying): Moving a group of ten from one place value to the next higher place value

Addition with regrouping methodology:
  1. Align numbers: Put numbers in columns by place value
  2. Add ones column: If sum ≥ 10, write ones digit, carry tens digit to tens column
  3. Add tens column: Include any carried digit, if sum ≥ 10, write ones digit, carry tens digit to hundreds column
  4. Add hundreds column: Include any carried digit
  5. Write the answer: Combine the results from each column
Tip 1: Always regroup when the sum of a column is 10 or more.
Tip 2: Remember to add any carried digits to the next column.
Tip 3: Work from right to left (ones → tens → hundreds).
Tip 4: Circle or mark carried digits to keep track of them.
Common errors: Forgetting to carry, carrying to the wrong column, adding carries incorrectly.
Success strategies: Use lined paper, mark carries clearly, check each step.
Three-digit addition with regrouping rules:

• Regrouping condition: \(7 + 8 = 15\) → write 5, carry 1

• Carry inclusion: \(5 + 6 + 1 = 12\) → write 2, carry 1

• Sequential processing: Process each column before moving to the next

• Multiple regrouping: \(9 + 4 = 13\) → write 3, carry 1 to next column

Solution: Exercises 4 to 5
4 Multiple regrouping
Exercise 4
Calculate:
\(586 + 347\)
Definition:

Multiple regrouping: When regrouping occurs in more than one column during the same addition

Expression
\(586 + 347\)
Column addition with regrouping
\(\begin{array}{r} 586 \\ + 347 \\ \hline 933 \end{array}\)
Result
\(933\)
Step 1: Line up by place value

\(\begin{array}{r} H \ T \ O \\ 5 \ 8 \ 6 \\ + 3 \ 4 \ 7 \\ \hline \end{array}\)

Step 2: Add ones column

\(6 + 7 = 13\), write 3 in ones place, carry 1 to tens column

Step 3: Add tens column

\(8 + 4 + 1 = 13\), write 3 in tens place, carry 1 to hundreds column

Step 4: Add hundreds column

\(5 + 3 + 1 = 9\), write 9 in hundreds place

\(586 + 347 = 933\)
Final answer:

\(586 + 347 = 933\)

Applied rules:

• Multiple regrouping: Regroup in each column where sum ≥ 10

• Carry accumulation: Each carried digit affects the next column's sum

• Sequential processing: Process each column from right to left

5 Real-world application
Exercise 5
A farmer harvested 479 apples from one tree and 285 apples from another tree. How many apples did he harvest in total?
Definition:

Word problem: A real-world scenario that requires mathematical calculation to solve

Problem
Tree 1: 479, Tree 2: 285, Total: ?
Equation
479 + 285 = ?
Result
764
Step 1: Identify the numbers

Apples from tree 1 = 479, Apples from tree 2 = 285

Step 2: Set up the addition

We need to find the total, so we add: 479 + 285

Step 3: Line up by place value

\(\begin{array}{r} H \ T \ O \\ 4 \ 7 \ 9 \\ + 2 \ 8 \ 5 \\ \hline \end{array}\)

Step 4: Add each column with regrouping

Ones: \(9 + 5 = 14\) → write 4, carry 1

Tens: \(7 + 8 + 1 = 16\) → write 6, carry 1

Hundreds: \(4 + 2 + 1 = 7\) → write 7

The farmer harvested 764 apples in total.
Final answer:

The farmer harvested 764 apples in total.

Applied rules:

• Real-world application: Connect math to everyday situations

• Problem-solving: Identify what needs to be calculated

• Regrouping: Apply carrying rules when column sums ≥ 10

Three-Digit Addition With Regrouping Laws, Methods, and Definitions
\(\begin{array}{r} H \ T \ O \\ a \ b \ c \\ + d \ e \ f \\ \hline g \ h \ i \end{array}\)
General Form of Three-Digit Addition with Regrouping
Key definitions:

Three-digit number: A number with three digits representing hundreds, tens, and ones places

Place value: The value of a digit based on its position in the number (hundreds, tens, ones)

Regrouping (carrying): Moving a group of ten from one place value to the next higher place value when the sum in a column is 10 or more

Three-digit addition with regrouping methodology:
  1. Align numbers: Put numbers in columns by place value
  2. Add ones column: If sum ≥ 10, write ones digit, carry tens digit to tens column
  3. Add tens column: Include any carried digit, if sum ≥ 10, write ones digit, carry tens digit to hundreds column
  4. Add hundreds column: Include any carried digit
  5. Write the answer: Combine the results from each column
Tip 1: Always regroup when the sum of a column is 10 or more.
Tip 2: Remember to add any carried digits to the next column.
Tip 3: Work from right to left (ones → tens → hundreds).
Tip 4: Circle or mark carried digits to keep track of them.
Common errors: Forgetting to carry, carrying to the wrong column, adding carries incorrectly.
Success strategies: Use lined paper, mark carries clearly, check each step.
Three-digit addition with regrouping formulas:

• Regrouping condition: If column sum ≥ 10, carry the tens digit to the next column

• Carry inclusion: Include carried digits in the next column's addition

• Sequential processing: Process columns from right to left

• Multiple regrouping: Apply regrouping rules to each column independently

Exercise with Visualization: Regrouping Process
Exercise 6: Regrouping Visualization
Consider the regrouping process in three-digit addition:
\(f_1(x) = x\) (before regrouping)
\(f_2(x) = x \mod 10\) (digits kept)
\(f_3(x) = \lfloor x/10 \rfloor\) (carry digits)

Analysis: The chart shows how regrouping works in three-digit addition.

  • \(f_1(x) = x\) (original sum before regrouping)
  • \(f_2(x) = x \mod 10\) (digit written in current column)
  • \(f_3(x) = \lfloor x/10 \rfloor\) (digit carried to next column)

Questions & Answers

Question: When do I need to regroup in three-digit addition?

Answer: You need to regroup (or carry) whenever the sum of a column is 10 or more. Here's how to recognize it:

  • Ones column: If the ones digits add up to 10 or more (like 7 + 6 = 13)
  • Tens column: If the tens digits (plus any carry from ones) add up to 10 or more
  • Hundreds column: If the hundreds digits (plus any carry from tens) add up to 10 or more

For example, in 257 + 168:

  • Ones: 7 + 8 = 15 (≥ 10), so regroup: write 5, carry 1
  • Tens: 5 + 6 + 1 (carry) = 12 (≥ 10), so regroup: write 2, carry 1
  • Hundreds: 2 + 1 + 1 (carry) = 4 (< 10), no regrouping needed

Remember: regrouping only happens when the sum in a column is 10 or greater!

Question: I sometimes forget to add the carried number to the next column. How can I remember to do this?

Answer: This is a very common mistake! Here are some strategies to help you remember:

  1. Circle the carry: Draw a circle around the number you're carrying so it's visible
  2. Write it above: Write the carried number slightly above the next column
  3. Say it out loud: When adding the next column, say "plus the carry" to remind yourself
  4. Mark it: Put a small checkmark or dot to remind you that you have a carry

For example, in 379 + 254:

\(\begin{array}{r} 379 \\ + 254 \\ \hline \end{array}\)

When you add 9 + 4 = 13, write the 3 and put the 1 above the tens column:

\(\begin{array}{r} \phantom{0}^1 \\ 379 \\ + 254 \\ \hline \end{array}\)

Now when you add the tens column, you'll remember to include that 1: 7 + 5 + 1 = 13

With practice, this will become automatic!

Question: My child sometimes carries the wrong number. How can I help them understand what to carry?

Answer: This is a common source of confusion! Here's how to help your child understand:

Regrouping Rule: When the sum in a column is 10 or more, you "regroup" by taking groups of 10 and moving them to the next higher place value.

What to carry: Only carry the "tens" part of the sum:

  • If you get 10, carry 1 (because 10 = 1 ten)
  • If you get 11, carry 1 (because 11 = 1 ten and 1 one)
  • If you get 12, carry 1 (because 12 = 1 ten and 2 ones)
  • If you get 15, carry 1 (because 15 = 1 ten and 5 ones)
  • If you get 20, carry 2 (because 20 = 2 tens)

Think of it as: "How many groups of 10 are in my sum?"

For example, in 257 + 168:

  • Ones: 7 + 8 = 15 → There is 1 group of 10 in 15, so carry 1
  • Write 5 in the ones place, carry 1 to the tens column

Use physical objects like base-10 blocks to show this concept visually!