Solved Exercises on Place Value Subtraction in Grade 3

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

Solution: Exercises 1 to 3
1 Basic place value subtraction
Exercise 1
Subtract using place value:
\(57 - 23\)
Definition:

Place value subtraction: Subtract digits in the same place value positions (ones from ones, tens from tens)

Subtraction method:
  1. Identify the place value of each digit
  2. Subtract ones from ones
  3. Subtract tens from tens
  4. Combine the results
Tens
5
-
2
=
3
Ones
7
-
3
=
4
50 + 7 - 20 - 3 = 30 + 4
Step 1: Break down by place value

\(57 = 50 + 7\) (5 tens and 7 ones)

\(23 = 20 + 3\) (2 tens and 3 ones)

Step 2: Subtract tens from tens

\(50 - 20 = 30\)

Step 3: Subtract ones from ones

\(7 - 3 = 4\)

Step 4: Combine the results

\(30 + 4 = 34\)

\(57 - 23 = 34\)
Final answer:

\(57 - 23 = 34\)

Applied rules:

• Place value: Always subtract digits in the same place value

• Regrouping: Only needed when bottom digit is larger than top digit

• Combination: Add partial results together

2 Subtraction with regrouping
Exercise 2
Subtract using place value:
\(62 - 17\)
Definition:

Regrouping: When the bottom digit is larger than the top digit, borrow from the next higher place value

Tens
6
-
1
=
5
Ones
2
-
7
(Borrow needed)
62 = 50 + 12 → 5 tens and 12 ones
50 + 12 - 10 - 7 = 40 + 5
Step 1: Check if regrouping is needed

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

Step 2: Regroup/borrow from tens

Take 1 ten from 6 tens: \(62 = 50 + 12\) (5 tens and 12 ones)

Step 3: Subtract the ones

\(12 - 7 = 5\)

Step 4: Subtract the tens

\(5 - 1 = 4\)

Step 5: Combine the results

\(40 + 5 = 45\)

\(62 - 17 = 45\)
Final answer:

\(62 - 17 = 45\)

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 place value subtraction
Exercise 3
Subtract using place value:
\(245 - 128\)
Definition:

Three-digit subtraction: Same principles apply to hundreds, tens, and ones places

Hundreds
2
-
1
=
1
Tens
4
-
2
=
2
Ones
5
-
8
(Borrow needed)
200 + 40 + 5 - 100 - 20 - 8
Step 1: Break down by place value

\(245 = 200 + 40 + 5\) (2 hundreds, 4 tens, 5 ones)

\(128 = 100 + 20 + 8\) (1 hundred, 2 tens, 8 ones)

Step 2: Check for regrouping in ones

Ones: \(5 < 8\), so we need to regroup from tens

Step 3: Regroup from tens to ones

Take 1 ten from 4 tens: \(245 = 200 + 30 + 15\) (2 hundreds, 3 tens, 15 ones)

Step 4: Subtract each place value

Hundreds: \(200 - 100 = 100\)

Tens: \(30 - 20 = 10\)

Ones: \(15 - 8 = 7\)

Step 5: Combine the results

\(100 + 10 + 7 = 117\)

\(245 - 128 = 117\)
Final answer:

\(245 - 128 = 117\)

Applied rules:

• Three-digit rule: Subtract hundreds from hundreds, tens from tens, ones from ones

• Regrouping cascade: Borrow from next higher place value when needed

• Value conversion: 1 hundred = 10 tens, 1 ten = 10 ones

Rules and methods, laws,...
\(abc = a \times 100 + b \times 10 + c \times 1\)
Place Value Representation
Rule 1
\(100 = 10 \times 10\)
1 hundred = 10 tens
Rule 2
\(10 = 1 \times 10\)
1 ten = 10 ones
Rule 3
\(a - b = ?\)
When \(a < b\), regroup
Key definitions:

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

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

Complete methodology:
  1. Identify place values: Determine the value of each digit
  2. Compare digits: Check if bottom digit is larger than top digit
  3. Regroup if needed: Borrow from higher place value
  4. Subtract systematically: Start from the rightmost digit
  5. Combine results: Add the partial differences
Tip 1: Always start subtracting from the ones place.
Tip 2: Draw lines to separate place values visually.
Tip 3: Check your answer by adding the difference back to the subtracted number.
Tip 4: Practice with base-ten blocks to visualize the process.
Common errors: Forgetting to regroup, subtracting smaller number from larger regardless of position, misaligning digits.
Exam preparation: Master basic facts, practice regrouping, understand place value relationships.
Formulas to know by heart:

• Place value: \(abc = a \times 100 + b \times 10 + c \times 1\)

• Regrouping: \(1\) hundred \(= 10\) tens, \(1\) ten \(= 10\) ones

• Subtraction sequence: Start from ones, move left

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

• Verification: Difference + Subtracted = Original number

Solution: Exercises 4 to 5
4 Subtraction with multiple regroups
Exercise 4
Subtract using place value:
\(402 - 157\)
Definition:

Multiple regrouping: When multiple place values require borrowing due to zeros in higher places

Hundreds
4
-
1
=
3
Tens
0
-
5
(Borrow needed)
Ones
2
-
7
(Borrow needed)
402 = 300 + 100 + 2 → 300 + 90 + 12
300 + 90 + 12 → 200 + 190 + 12
Step 1: Identify multiple regrouping needs

Ones: \(2 < 7\), Tens: \(0 < 5\) (after borrowing from hundreds)

Step 2: First regroup from hundreds to tens

Take 1 hundred from 4 hundreds: \(402 = 300 + 100 + 2\)

Step 3: Second regroup from tens to ones

Take 1 ten from 10 tens: \(300 + 100 + 2 = 300 + 90 + 12\)

Step 4: Subtract each place value

Hundreds: \(3 - 1 = 2\)

Tens: \(9 - 5 = 4\)

Ones: \(12 - 7 = 5\)

Step 5: Combine the results

\(200 + 40 + 5 = 245\)

\(402 - 157 = 245\)
Final answer:

\(402 - 157 = 245\)

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 place value subtraction
Exercise 5
Sarah has 345 stickers. She gives 168 stickers to her friend. How many stickers does Sarah have left?
Definition:

Word problem: Translate the story into a mathematical subtraction problem using place value

Hundreds
3
-
1
=
2
Tens
4
-
6
(Borrow needed)
Ones
5
-
8
(Borrow needed)
300 + 40 + 5 - 100 - 60 - 8
Step 1: Identify the operation

"Gives away" means subtraction: \(345 - 168\)

Step 2: Set up place value columns

Hundreds: 3, Tens: 4, Ones: 5

Subtract: Hundreds: 1, Tens: 6, Ones: 8

Step 3: Regroup from tens to ones

Take 1 ten from 4 tens: \(345 = 300 + 30 + 15\)

Step 4: Regroup from hundreds to tens

Take 1 hundred from 3 hundreds: \(300 + 30 + 15 = 200 + 130 + 15\)

Step 5: Continue regrouping

\(200 + 130 + 15 = 200 + 120 + 25\)

Step 6: Subtract each place value

Hundreds: \(2 - 1 = 1\)

Tens: \(12 - 6 = 6\)

Ones: \(25 - 8 = 17\)

Step 7: Combine and finalize

\(100 + 60 + 17 = 177\)

\(345 - 168 = 177\)
Final answer:

Sarah has 177 stickers left.

Applied rules:

• Word translation: Convert story to mathematical expression

• Regrouping sequence: Handle multiple regroups systematically

• Contextual answer: Include units in final answer

Place Value Subtraction Laws, Methods, and Definitions
\(abc - def = (a-d) \times 100 + (b-e) \times 10 + (c-f) \times 1\)
Place Value Subtraction Formula
Key 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 specifically in subtraction operations

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

Subtraction: Finding the difference between two numbers

Complete methodology:
  1. Analyze the problem: Identify the numbers and their place values
  2. Align digits: Line up digits by place value (ones under ones, tens under tens, etc.)
  3. Start from the right: Begin subtraction in the ones place
  4. Check for regrouping: If bottom digit is larger than top digit, regroup from next higher place
  5. Perform subtraction: Subtract digits in each place value column
  6. Move left: Continue to tens, hundreds, and higher place values
  7. Combine results: Write the final answer combining all place values
  8. Verify: Add the difference back to the subtracted number to check accuracy
Tip 1: Always start subtracting from the ones place and work left.
Tip 2: Use base-ten blocks or drawings to visualize regrouping.
Tip 3: When you see a zero, always check if regrouping is needed from higher places.
Tip 4: Draw lines to separate place values for better organization.
Tip 5: Check your work by adding the difference back to the subtracted number.
Tip 6: Remember: 1 hundred = 10 tens, 1 ten = 10 ones.
Common errors: Forgetting to regroup, subtracting smaller number from larger regardless of position, misaligning digits, carrying over instead of borrowing.
Key concepts: Place value understanding, regrouping mechanics, sequential processing from right to left, value preservation during regrouping.
Visual aids: Base-ten blocks, place value charts, arrow notation for regrouping, expanded form representation.
Problem-solving: Read word problems carefully, identify keywords indicating subtraction, organize information systematically.
Formulas and Rules to Know:

• Place value: \(abc = a \times 100 + b \times 10 + c \times 1\)

• Regrouping: \(1\) hundred \(= 10\) tens, \(1\) ten \(= 10\) ones

• Subtraction sequence: Start from ones, move left

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

• Verification: Difference + Subtracted = Original number

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

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

Place Value Subtraction Workflow

📊
5-Step Subtraction Process
1
Align
2
Start Right
3
Check Need
4
Regroup
5
Subtract
Visual Representation
300 + 40 + 5 - 100 - 20 - 3
Practice Makes Perfect!

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.