Solved Exercises on Place Value Subtraction with Regrouping in Grade 3

Master place value subtraction with regrouping: borrowing, tens and ones, mathematical foundations for 3rd grade students through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic regrouping
Exercise 1
Subtract using regrouping:
\(52 - 17\)
Definition:

Regrouping: Converting 1 unit from a higher place value to 10 units of the next lower place value

Subtraction method:
  1. Identify the place value of each digit
  2. Check if regrouping is needed (bottom digit > top digit)
  3. Regroup/borrow from next higher place value
  4. Subtract the ones
  5. Subtract the tens
Tens
5
-
1
=
4
Ones
2
-
7
(Regrouping needed)
52 = 40 + 12 → 4 tens and 12 ones
50 + 2 → 40 + 12
Step 1: Check if regrouping is needed

Ones: \(2 < 7\), so we need to regroup from tens

Step 2: Regroup from tens to ones

Take 1 ten from 5 tens: \(52 = 40 + 12\) (4 tens and 12 ones)

Step 3: Subtract the ones

\(12 - 7 = 5\)

Step 4: Subtract the tens

\(4 - 1 = 3\)

Step 5: Combine the results

\(30 + 5 = 35\)

\(52 - 17 = 35\)
Final answer:

\(52 - 17 = 35\)

Applied rules:

• Regrouping rule: Borrow 1 from higher place value when needed

• Value preservation: 1 ten = 10 ones

• Subtraction sequence: Start from the rightmost digit

2 Two-digit regrouping
Exercise 2
Subtract using regrouping:
\(71 - 26\)
Definition:

Two-digit regrouping: When the bottom digit in the ones place is larger than the top digit

Tens
7
-
2
=
5
Ones
1
-
6
(Regrouping needed)
71 = 60 + 11 → 6 tens and 11 ones
70 + 1 → 60 + 11
Step 1: Check if regrouping is needed

Ones: \(1 < 6\), so we need to regroup from tens

Step 2: Regroup from tens to ones

Take 1 ten from 7 tens: \(71 = 60 + 11\) (6 tens and 11 ones)

Step 3: Subtract the ones

\(11 - 6 = 5\)

Step 4: Subtract the tens

\(6 - 2 = 4\)

Step 5: Combine the results

\(40 + 5 = 45\)

\(71 - 26 = 45\)
Final answer:

\(71 - 26 = 45\)

Applied rules:

• Regrouping rule: Borrow 1 from higher place value when needed

• Value preservation: 1 ten = 10 ones

• Subtraction sequence: Start from the rightmost digit

3 Three-digit regrouping
Exercise 3
Subtract using regrouping:
\(304 - 127\)
Definition:

Three-digit regrouping: When multiple place values require borrowing, especially when zeros are involved

Hundreds
3
-
1
=
2
Tens
0
-
2
(Regrouping needed)
Ones
4
-
7
(Regrouping needed)
300 + 0 + 4 → 200 + 10 + 4 → 200 + 9 + 14
Step 1: Identify multiple regrouping needs

Ones: \(4 < 7\), Tens: \(0 < 2\) (after borrowing from hundreds)

Step 2: First regroup from hundreds to tens

Take 1 hundred from 3 hundreds: \(304 = 200 + 100 + 4\)

Step 3: Second regroup from tens to ones

Take 1 ten from 10 tens: \(200 + 100 + 4 = 200 + 90 + 14\)

Step 4: Subtract each place value

Hundreds: \(2 - 1 = 1\)

Tens: \(9 - 2 = 7\)

Ones: \(14 - 7 = 7\)

Step 5: Combine the results

\(100 + 70 + 7 = 177\)

\(304 - 127 = 177\)
Final answer:

\(304 - 127 = 177\)

Applied rules:

• Multiple regrouping: Regroup step by step from highest needed place

• Zero handling: When encountering zero, regroup from next higher place

• Sequential regrouping: Each regroup affects subsequent calculations

Complete Summary: Place Value Subtraction with Regrouping
Key Concepts and Definitions

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

Regrouping: Converting 1 unit from a higher place value to 10 units of the next lower place value

Borrowing: Another term for regrouping in subtraction operations

Base-Ten System: Our number system where each place value is 10 times the previous one

Core Rules and Principles

Regrouping Rule: When the bottom digit is larger than the top digit in a place value, you must regroup from the next higher place value

Value Conversion: 1 hundred = 10 tens, 1 ten = 10 ones

Subtraction Sequence: Always start from the rightmost digit (ones place) and move left

Zero Handling: When you encounter a zero and need to regroup, continue to the next non-zero place value

Step-by-Step Method
  1. Align Numbers: Write the numbers vertically with place values aligned
  2. Start from Right: Begin subtraction in the ones place
  3. Check Need: Compare digits in each place value (bottom vs top)
  4. Regroup if Needed: If bottom digit is larger, borrow from next higher place
  5. Subtract Digit: Perform subtraction in that place value
  6. Move Left: Continue to next place value (tens, then hundreds)
  7. Combine Results: Write the final answer combining all place values
  8. Verify: Add the difference back to the subtracted number to check accuracy
Examples - From Simple to Complex

Simple Example: \(42 - 15\)

Ones: \(2 < 5\), so regroup: \(42 = 30 + 12\), then \(12 - 5 = 7\), \(3 - 1 = 2\), answer: \(27\)

Intermediate Example: \(203 - 147\)

Ones: \(3 < 7\), regroup from tens (but tens is 0), so regroup from hundreds: \(203 = 100 + 100 + 3 = 100 + 90 + 13\), then \(13 - 7 = 6\), \(9 - 4 = 5\), \(1 - 1 = 0\), answer: \(56\)

Complex Example: \(500 - 289\)

Ones: \(0 < 9\), regroup from tens (0), regroup from hundreds: \(500 = 400 + 100 = 400 + 90 + 10\), then \(10 - 9 = 1\), \(9 - 8 = 1\), \(4 - 2 = 2\), answer: \(211\)

Tips, Tricks, and Common Pitfalls

Tips:

  • Always start subtracting from the ones place
  • Draw lines to separate place values visually
  • Check your answer by adding the difference back to the subtracted number
  • Practice with base-ten blocks to visualize the process
  • When you see a zero, always check if regrouping is needed from higher places

Common Mistakes:

  • Forgetting to regroup when needed
  • Subtracting smaller number from larger regardless of position
  • Misaligning digits when writing numbers vertically
  • Forgetting to reduce the higher place value after borrowing
  • Not continuing regrouping when there are zeros in higher places
Key Notes for Memorization

Memory Aids:

  • "More on the floor, go next door" - when bottom number is bigger, borrow from next place
  • "1 hundred = 10 tens = 100 ones" - know these conversions
  • Always start from the right (ones place)
  • After borrowing, the higher place value decreases by 1
  • Check your work: Difference + Subtracted = Original number
Solution: Exercises 4 to 5
4 Regrouping with zeros
Exercise 4
Subtract using regrouping:
\(400 - 158\)
Definition:

Regrouping with zeros: When there are zeros in higher place values, continue regrouping to the next non-zero place value

Hundreds
4
-
1
=
3
Tens
0
-
5
(Regrouping needed)
Ones
0
-
8
(Regrouping needed)
400 = 300 + 100 → 300 + 90 + 10
300 + 90 + 10 → 200 + 190 + 10
Step 1: Identify multiple regrouping needs

Ones: \(0 < 8\), Tens: \(0 < 5\) (after borrowing from hundreds)

Step 2: First regroup from hundreds to tens

Take 1 hundred from 4 hundreds: \(400 = 300 + 100\)

Step 3: Second regroup from tens to ones

Take 1 ten from 10 tens: \(300 + 100 = 300 + 90 + 10\)

Step 4: Continue regrouping

Since we still need to regroup in tens: \(300 + 90 + 10 = 200 + 190 + 10\)

Step 5: Subtract each place value

Hundreds: \(3 - 1 = 2\)

Tens: \(9 - 5 = 4\)

Ones: \(10 - 8 = 2\)

Step 6: Combine the results

\(200 + 40 + 2 = 242\)

\(400 - 158 = 242\)
Final answer:

\(400 - 158 = 242\)

Applied rules:

• Multiple regrouping: Regroup step by step from highest needed place

• Zero handling: When encountering zero, regroup from next higher place

• Sequential regrouping: Each regroup affects subsequent calculations

5 Word problem with regrouping
Exercise 5
Tom had 250 marbles. He gave 87 marbles to his friend. How many marbles does Tom have left?
Definition:

Word problem: Translate the story into a mathematical subtraction problem requiring regrouping

Hundreds
2
-
0
=
2
Tens
5
-
8
(Regrouping needed)
Ones
0
-
7
(Regrouping needed)
200 + 50 + 0 - 0 - 80 - 7
Step 1: Identify the operation

"Gave away" means subtraction: \(250 - 87\)

Step 2: Set up place value columns

Hundreds: 2, Tens: 5, Ones: 0

Subtract: Tens: 8, Ones: 7

Step 3: Regroup from tens to ones

Take 1 ten from 5 tens: \(250 = 200 + 40 + 10\)

Step 4: Regroup from hundreds to tens

Take 1 hundred from 2 hundreds: \(200 + 40 + 10 = 100 + 140 + 10\)

Step 5: Continue regrouping

\(100 + 140 + 10 = 100 + 130 + 20\)

Step 6: Subtract each place value

Hundreds: \(1 - 0 = 1\)

Tens: \(13 - 8 = 5\)

Ones: \(20 - 7 = 13\)

Step 7: Combine and finalize

\(100 + 50 + 13 = 163\)

\(250 - 87 = 163\)
Final answer:

Tom has 163 marbles left.

Applied rules:

• Word translation: Convert story to mathematical expression

• Regrouping sequence: Handle multiple regroups systematically

• Contextual answer: Include units in final answer

Questions & Answers

Question: I don't understand why I have to regroup when the bottom number is bigger than the top number in subtraction. Why can't I just subtract the smaller from the larger?

Answer: Great question! In subtraction, we always subtract the bottom number FROM the top number. When the bottom digit is larger than the top digit in a place value, we can't take away more than we have. That's why we need to regroup.

Think of it like this: If you have 2 apples and someone asks for 7 apples, you can't give them 7 because you only have 2. You'd need to get more apples first.

In math: When you see 2 ones and need to subtract 7 ones, you need to borrow 1 ten (which equals 10 ones) from the tens place. Now you have 12 ones, and you can subtract 7 ones to get 5 ones.

Example: In 52 - 17, we can't do 2 - 7 in the ones place, so we borrow 1 ten from the 5 tens, making it 4 tens and 12 ones. Then we can do 12 - 7 = 5 in the ones place.

Question: My child gets confused when there's a zero in the middle of a number like 402. How can I help them understand regrouping across zeros?

Answer: Numbers with zeros can be tricky! When there's a zero in a place value, your child needs to keep going to the next non-zero place value to borrow from. Think of it like a chain reaction.

For 402 - 157, here's the process:

  • Ones place: 2 - 7 (can't do this, need to borrow)
  • Tens place: There's a 0, so can't borrow from here
  • Hundreds place: Borrow 1 hundred (4 becomes 3), giving 10 tens
  • Now tens place: 10 tens, borrow 1 ten (10 becomes 9), giving 10 ones
  • Now ones place: 12 - 7 = 5

Use visual aids like base-ten blocks or draw the process out. Explain that the zero is like a "bridge" - you can't stop there, you have to go further to get what you need.

Question: I learned the standard algorithm, but this place value approach seems more complex. Why do we teach it this way now?

Answer: The place value approach builds conceptual understanding rather than just procedural fluency. Students who understand why regrouping works are better prepared for advanced math.

Benefits of the place value approach:

  • Students understand the base-ten system and why algorithms work
  • They can apply the concept to different number bases
  • They develop number sense and mental math skills
  • They can explain their thinking and catch errors more easily
  • It prepares them for algebra and higher-level mathematics

The standard algorithm is still taught, but building understanding first makes the procedure more meaningful. Students who understand place value can better adapt when faced with unusual problems or when learning decimals later.