Two-Digit Addition and Subtraction with Regrouping

Master both two-digit addition with carrying and subtraction with borrowing through these 5 detailed exercises with step-by-step solutions and visual explanations.

Addition Exercises 1 to 2
1 Addition with Carrying
Exercise 1: Addition with Carrying
Calculate:
\(47 + 38\)
Addition Process Addition
47
+
38
→
85
Carrying Rule: When the sum of ones digits is 10 or more, carry the tens digit to the tens place.
Definition:

Carrying: Also known as "regrouping" in addition, it's taking groups of ten from the ones place to add to the tens place when the sum of ones digits is 10 or more

Addition with carrying method:
  1. Align numbers by place value (ones under ones, tens under tens)
  2. Add ones digits: if sum ≥ 10, carry the tens digit to the tens column
  3. Add tens digits: include any carried tens
  4. Write the final result
Setup
\(47 + 38\)
Add ones
\(7 + 8 = 15\)
Carry
\(1\) to tens, \(5\) in ones
Step 1: Set up the problem vertically

\(\begin{array}{c@{}c@{}c} & 4 & 7 \\ + & 3 & 8 \\ \hline \end{array}\)

Step 2: Add the ones digits

\(7 + 8 = 15\), which is 1 ten and 5 ones

Step 3: Carry the tens digit

Write 5 in the ones place and carry 1 to the tens column

Step 4: Add the tens digits

\(4 + 3 + 1 = 8\) (including the carried 1)

Step 5: Write the final answer

\(8\) tens and \(5\) ones equals \(85\)

\(47 + 38 = 85\)
Final answer:

\(47 + 38 = 85\)

Applied rules:

• Carrying rule: When ones sum ≥ 10, carry the tens digit to tens column

• Place value: Maintain proper alignment throughout

• Carried digit: Include carried digit in tens addition

2 Subtraction with Borrowing
Exercise 2: Subtraction with Borrowing
Calculate:
\(63 - 27\)
Subtraction Process Subtraction
63
-
27
→
36
Borrowing Rule: When the top digit in any place value is smaller than the bottom digit, borrow from the next higher place value.
Setup
\(63 - 27\)
Borrow
\(63 = 5 tens + 13 ones\)
Subtract
\(13-7=6, 5-2=3\)
Step 1: Set up the problem vertically

\(\begin{array}{c@{}c@{}c} & 6 & 3 \\ - & 2 & 7 \\ \hline \end{array}\)

Step 2: Compare ones digits

\(3 < 7\), so we need to borrow from tens place

Step 3: Borrow from tens place

Take 1 ten from the tens place (6 becomes 5) and add 10 to the ones place (3 becomes 13)

Step 4: Subtract the ones digits

\(13 - 7 = 6\)

Step 5: Subtract the tens digits

\(5 - 2 = 3\)

Step 6: Write the final answer

\(3\) tens and \(6\) ones equals \(36\)

\(63 - 27 = 36\)
Final answer:

\(63 - 27 = 36\)

Applied rules:

• Borrowing rule: When top digit is smaller, borrow 1 from next higher place value

• Place value: 1 ten = 10 ones

• Adjustment: Reduce the higher place value by 1 after borrowing

Addition vs Subtraction Comparison Guide
\(AB + CD = (A \times 10 + B) + (C \times 10 + D)\)
\(AB - CD = (A \times 10 + B) - (C \times 10 + D)\)
Addition and Subtraction Formulas
Addition with Carrying

When to carry: When ones sum ≥ 10

What to do: Write ones digit, carry tens digit to tens column

Example: \(58 + 37 = 95\)

Subtraction with Borrowing

When to borrow: When top digit < bottom digit

What to do: Borrow 1 from higher place, add 10 to lower place

Example: \(72 - 46 = 26\)

Addition Carrying
If B + D ≥ 10
Carry 1 to tens
Subtraction Borrowing
If B < D
Borrow 1 from tens
Place Value
Tens + Ones
Align by place value
1
Align
2
Check
3
Regroup
4
Calculate
5
Answer
Key definitions:

Regrouping: The general term for both carrying (in addition) and borrowing (in subtraction)

Carrying: Moving a group of ten from ones to tens in addition

Borrowing: Taking a group of ten from tens to ones in subtraction

Place value: Position of a digit determines its value (tens, ones)

Minuend: The number being subtracted from (in subtraction)

Subtrahend: The number being subtracted (in subtraction)

Sum: The result of addition

Difference: The result of subtraction

Complete methodology:
  1. Align numbers: Put ones under ones, tens under tens
  2. Check for regrouping: Determine if carrying/borrowing is needed
  3. Regroup if necessary: Carry or borrow as appropriate
  4. Perform operation: Add or subtract each place value
  5. Combine results: Write final answer with proper place value
Tip 1: Always check if regrouping is needed before starting the operation.
Tip 2: Remember that carrying adds to the tens, borrowing takes from the tens.
Tip 3: Draw lines to separate tens and ones to stay organized.
Tip 4: Check your answer by using the inverse operation.
Tip 5: Practice basic addition and subtraction facts to build fluency.
Common errors: Forgetting to carry/borrow, carrying/borrowing when not needed, misaligning digits, forgetting to include carried digits.
Key notes: Practice both operations together to understand the relationship, always check your work, remember place value rules.
Formulas to know by heart:

• Addition: \(AB + CD = (A \times 10 + B) + (C \times 10 + D)\)

• Subtraction: \(AB - CD = (A \times 10 + B) - (C \times 10 + D)\)

• Carrying rule: When ones sum ≥ 10, carry 1 to tens place

• Borrowing rule: When top digit < bottom digit, borrow 1 ten = 10 ones

• Verification: Sum - Addend = Other Addend, Minuend - Difference = Subtrahend

Mixed Operations Exercises 3 to 5
3 Mixed Addition and Subtraction
Exercise 3: Mixed Operations
Solve both operations:
\(56 + 28\) and \(73 - 39\)
Addition: \(56 + 28\) Addition
Add ones
\(6 + 8 = 14\)
Carry
\(1\) to tens
Add tens
\(5 + 2 + 1 = 8\)
\(56 + 28 = 84\)
Subtraction: \(73 - 39\) Subtraction
Borrow
\(73 = 6 tens + 13 ones\)
Subtract ones
\(13 - 9 = 4\)
Subtract tens
\(6 - 3 = 3\)
\(73 - 39 = 34\)
Definition:

Mixed operations: Problems that involve both addition and subtraction, requiring careful attention to which operation to perform

Addition Steps:

\(\begin{array}{c@{}c@{}c} & 5 & 6 \\ + & 2 & 8 \\ \hline & 8 & 4 \end{array}\)

Subtraction Steps:

\(\begin{array}{c@{}c@{}c} & 7 & 3 \\ - & 3 & 9 \\ \hline & 3 & 4 \end{array}\)

Final answers:

\(56 + 28 = 84\)

\(73 - 39 = 34\)

Applied rules:

• Operation identification: Recognize whether to add or subtract

• Carrying rule: Apply when ones sum ≥ 10

• Borrowing rule: Apply when top digit < bottom digit

4 Word Problem Application
Exercise 4: Real-Life Problem
Emma had 85 dollars. She earned 37 more dollars. Then she spent 49 dollars. How much money does she have now?
Problem Breakdown

Step 1: Add money earned: \(85 + 37\)

Step 2: Subtract money spent: \((85 + 37) - 49\)

Step 1: Calculate \(85 + 37\)

\(\begin{array}{c@{}c@{}c} & 8 & 5 \\ + & 3 & 7 \\ \hline & 1 & 2 & 2 \end{array}\)

\(5 + 7 = 12\), carry 1. \(8 + 3 + 1 = 12\)

Step 2: Calculate \(122 - 49\)

\(\begin{array}{c@{}c@{}c} & 1 & 2 & 2 \\ - & & 4 & 9 \\ \hline & & 7 & 3 \end{array}\)

\(2 < 9\), so borrow: \(122 = 11\) tens + \(12\) ones

\(12 - 9 = 3\), \(11 - 4 = 7\)

\(85 + 37 - 49 = 73\)
Final answer:

Emma has 73 dollars now.

Applied rules:

• Problem solving: Identify operations needed from context

• Carrying: Apply when ones sum ≥ 10

• Borrowing: Apply when top digit < bottom digit

5 Complex Mixed Operations
Exercise 5: Multi-Step Problem
A store had 94 books. They received 28 more books, then sold 37 books. How many books are left?
Multi-Step Solution

Step 1: Add received books: \(94 + 28\)

Step 2: Subtract sold books: \((94 + 28) - 37\)

Step 1: Calculate \(94 + 28\)

\(\begin{array}{c@{}c@{}c} & 9 & 4 \\ + & 2 & 8 \\ \hline & 1 & 2 & 2 \end{array}\)

\(4 + 8 = 12\), carry 1. \(9 + 2 + 1 = 12\)

Step 2: Calculate \(122 - 37\)

\(\begin{array}{c@{}c@{}c} & 1 & 2 & 2 \\ - & & 3 & 7 \\ \hline & & 8 & 5 \end{array}\)

\(2 < 7\), so borrow: \(122 = 11\) tens + \(12\) ones

\(12 - 7 = 5\), \(11 - 3 = 8\)

\(94 + 28 - 37 = 85\)
Final answer:

The store has 85 books left.

Applied rules:

• Multi-step problems: Break into smaller parts and solve sequentially

• Carrying: Apply when ones sum ≥ 10

• Borrowing: Apply when top digit < bottom digit

Complete Guide to Two-Digit Operations with Regrouping
Complete Methodology

Master both addition and subtraction with regrouping through systematic practice.

\(AB + CD = (A \times 10 + B) + (C \times 10 + D)\)
\(AB - CD = (A \times 10 + B) - (C \times 10 + D)\)
Two-Digit Operation Formulas
Addition with Carrying
  1. Align numbers by place value
  2. Add ones digits
  3. If sum ≥ 10, carry tens digit to tens column
  4. Add tens digits (include carried digit)
  5. Write final result
Subtraction with Borrowing
  1. Align numbers by place value
  2. Compare ones digits
  3. If top < bottom, borrow from tens column
  4. Subtract ones digits
  5. Subtract tens digits (include borrowed adjustment)
  6. Write final result
Key definitions:

Two-digit numbers: Numbers from 10 to 99 with tens and ones place values

Carrying: Taking groups of ten from ones to tens in addition when sum ≥ 10

Borrowing: Taking one group of ten from tens to ones in subtraction when top digit < bottom digit

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

Regrouping: The general term for both carrying and borrowing

Sum: The result of addition

Difference: The result of subtraction

Complete methodology:
  1. Read the problem: Understand what operation is needed
  2. Set up the equation: Write numbers vertically with proper alignment
  3. Align by place value: Ones under ones, tens under tens
  4. Check for regrouping: Determine if carrying or borrowing is needed
  5. Regroup if necessary: Carry or borrow as appropriate
  6. Perform operation: Add or subtract each place value
  7. Combine results: Write final answer with proper place value
  8. Check your answer: Use inverse operation to verify
Tip 1: Practice basic addition and subtraction facts (0-10) to build fluency.
Tip 2: Always write problems vertically to keep digits aligned.
Tip 3: Check if regrouping is needed before starting the operation.
Tip 4: Verify your answer by using the inverse operation.
Tip 5: Practice both operations together to understand the relationship.
Common errors: Forgetting to carry/borrow, carrying/borrowing when not needed, misaligning digits, forgetting to include carried digits in addition or borrowed adjustments in subtraction.
Memory aids: "Carry the 1 when sum is 10 or more", "Borrow when top is smaller", "Always check your work".
Rules to know by heart:

• Place value: Always align digits by place value

• Carrying rule: When ones sum ≥ 10, carry 1 to tens place

• Borrowing rule: When top digit < bottom digit, borrow 1 ten = 10 ones

• Operation sequence: Add/subtract ones first, then tens

• Verification: Sum - Addend = Other Addend, Minuend - Difference = Subtrahend

Example Patterns
Pattern 1: Addition with carrying

\(58 + 37\): Since \(8 + 7 = 15 ≥ 10\), carry 1 to tens place

Pattern 2: Subtraction with borrowing

\(62 - 38\): Since \(2 < 8\), borrow 1 from tens place

Pattern 3: No regrouping needed

\(75 + 23\): Since \(5 + 3 = 8 < 10\), no carrying needed

Questions & Answers

Question: How do I know when to carry in addition and when to borrow in subtraction?

Answer: Great question! The rules are different for addition and subtraction:

  • For addition (carrying): Carry when the sum of ones digits is 10 or more. For example, in 58 + 37, since 8 + 7 = 15, you carry the 1.
  • For subtraction (borrowing): Borrow when the top digit in any place value is smaller than the bottom digit. For example, in 63 - 27, since 3 < 7, you borrow from the tens place.
  • Memory tricks: "Carry the 1 when sum is 10 or more" and "Borrow when top is smaller".

Practice with both operations to develop the ability to quickly identify which regrouping is needed!

Question: How can I help my students understand the relationship between addition and subtraction with regrouping?

Answer: Understanding the relationship between addition and subtraction is crucial! Here are effective strategies:

  • Fact families: Show how addition and subtraction facts are related (e.g., 47 + 38 = 85, so 85 - 38 = 47)
  • Inverse operations: Emphasize that subtraction undoes addition and vice versa
  • Verification: Teach students to check addition by subtracting and check subtraction by adding
  • Visual models: Use number lines and bar models to show the connection
  • Real-world connections: Use problems that demonstrate both operations (e.g., starting amount, change, ending amount)

The key is helping students see that regrouping in both operations maintains the balance of the equation.

Question: My child confuses carrying and borrowing. How can I help them remember which to use?

Answer: This is a very common confusion! Here are memory aids and strategies to help:

  • Operation identification: First, clearly identify if the problem is addition (+) or subtraction (-)
  • Carrying (addition): "Adding more" - when sum ≥ 10, carry to the next column
  • Borrowing (subtraction): "Taking away" - when top digit < bottom digit, borrow from next column
  • Visual cues: Use color coding - red for subtraction/borrowing, blue for addition/carrying
  • Practice sequence: Master one operation completely before introducing the other
  • Check work: Always verify answers using the inverse operation

Practice with problems labeled as "addition" or "subtraction" first, then gradually mix them together as your child gains confidence.