Solved Exercises on Subtraction with Regrouping in Grade 3

Master subtraction with regrouping: borrowing, place value, and regrouping strategies through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Two-digit subtraction with regrouping
Exercise 1
Subtract:
\(52 - 27\)
Definition:

Regrouping: Taking a value from a higher place value to make subtraction possible in a lower place value

Subtraction method:
  1. Align numbers by place value (ones under ones, tens under tens)
  2. Start subtracting from the right (ones column)
  3. If top digit is smaller than bottom digit, borrow from the next column
  4. Subtract in each column and write the result
Numbers
\(\begin{array}{c@{}c@{}c} & 5 & 2 \\ - & 2 & 7 \\ \hline \end{array}\)
Borrow
\(\begin{array}{c@{}c@{}c} ^{4} & ^{1}2 & \\ \cancel{5} & \cancel{2} & \\ - & 2 & 7 \\ \hline & & \\ \end{array}\)
Result
\(\begin{array}{c@{}c@{}c} ^{4} & ^{1}2 & \\ \cancel{5} & \cancel{2} & \\ - & 2 & 7 \\ \hline & 2 & 5 \\ \end{array}\)
Step 1: Align by place value

\(52 - 27\) → Write 52 above 27, aligning ones and tens columns

Step 2: Try to subtract ones column

\(2 - 7\) → Since 2 is less than 7, we need to borrow from the tens column

Step 3: Borrow from tens column

Borrow 1 ten from the 5 tens, leaving 4 tens. Add 10 to the 2 ones to get 12 ones

Step 4: Subtract the ones column

\(12 - 7 = 5\), write down 5 in the ones place

Step 5: Subtract the tens column

\(4 - 2 = 2\), write down 2 in the tens place

\(52 - 27 = 25\)
Final answer:

\(52 - 27 = 25\)

Applied rules:

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

• Place value alignment: Always align digits by their place value

• Subtraction algorithm: Subtract one column at a time from right to left

2 Three-digit subtraction with regrouping
Exercise 2
Subtract:
\(421 - 186\)
Definition:

Borrowing: The process of taking value from a higher place value to enable subtraction

Numbers
\(\begin{array}{c@{}c@{}c@{}c} & 4 & 2 & 1 \\ - & 1 & 8 & 6 \\ \hline \end{array}\)
Borrow
\(\begin{array}{c@{}c@{}c@{}c} ^{3} & ^{1}1 & ^{1}1 & \\ \cancel{4} & \cancel{2} & \cancel{1} & \\ - & 1 & 8 & 6 \\ \hline & & & \\ \end{array}\)
Result
\(\begin{array}{c@{}c@{}c@{}c} ^{3} & ^{1}1 & ^{1}1 & \\ \cancel{4} & \cancel{2} & \cancel{1} & \\ - & 1 & 8 & 6 \\ \hline & 2 & 3 & 5 \\ \end{array}\)
Step 1: Align by place value

\(421 - 186\) → Write 421 above 186, aligning hundreds, tens, and ones columns

Step 2: Try to subtract ones column

\(1 - 6\) → Since 1 is less than 6, we need to borrow from the tens column

Step 3: Borrow from tens column

Try to borrow from 2 tens, but since we need to borrow and 2 is greater than 1, we can borrow 1 ten from the 2 tens to make 11 ones

Step 4: Now subtract ones column

\(11 - 6 = 5\), write down 5 in the ones place

Step 5: Try to subtract tens column

Now we have 1 ten in the tens place (since we borrowed 1), so \(1 - 8\) → Since 1 is less than 8, we need to borrow from the hundreds column

Step 6: Borrow from hundreds column

Borrow 1 hundred from the 4 hundreds, leaving 3 hundreds. Add 10 to the 1 ten to get 11 tens

Step 7: Subtract the tens column

\(11 - 8 = 3\), write down 3 in the tens place

Step 8: Subtract the hundreds column

\(3 - 1 = 2\), write down 2 in the hundreds place

\(421 - 186 = 235\)
Final answer:

\(421 - 186 = 235\)

Applied rules:

• Multiple regrouping: May need to borrow from multiple columns in one problem

• Sequential borrowing: Borrow from right to left as needed

• Place value maintenance: Keep track of changes in each column after borrowing

3 Four-digit subtraction with multiple regroupings
Exercise 3
Subtract:
\(3,002 - 1,457\)
Definition:

Zero regrouping: Special case when borrowing from a zero requires going to the next non-zero digit

Numbers
\(\begin{array}{c@{}c@{}c@{}c@{}c} & 3 & 0 & 0 & 2 \\ - & 1 & 4 & 5 & 7 \\ \hline \end{array}\)
Borrow
\(\begin{array}{c@{}c@{}c@{}c@{}c} ^{2} & ^{9} & ^{9} & ^{1}2 & \\ \cancel{3} & \cancel{0} & \cancel{0} & \cancel{2} & \\ - & 1 & 4 & 5 & 7 \\ \hline & & & & \\ \end{array}\)
Result
\(\begin{array}{c@{}c@{}c@{}c@{}c} ^{2} & ^{9} & ^{9} & ^{1}2 & \\ \cancel{3} & \cancel{0} & \cancel{0} & \cancel{2} & \\ - & 1 & 4 & 5 & 7 \\ \hline 1 & 5 & 4 & 5 \\ \end{array}\)
Step 1: Align by place value

\(3,002 - 1,457\) → Write 3,002 above 1,457, aligning thousands, hundreds, tens, and ones columns

Step 2: Try to subtract ones column

\(2 - 7\) → Since 2 is less than 7, we need to borrow from the tens column

Step 3: Cannot borrow from zero tens

The tens column has 0, so we need to go to the hundreds column to borrow

Step 4: Cannot borrow from zero hundreds

The hundreds column also has 0, so we need to go to the thousands column to borrow

Step 5: Borrow from thousands column

Borrow 1 thousand from the 3 thousands, leaving 2 thousands. This gives 10 hundreds

Step 6: Convert hundreds to tens

Take 1 hundred from the 10 hundreds, leaving 9 hundreds. This gives 10 tens

Step 7: Convert tens to ones

Take 1 ten from the 10 tens, leaving 9 tens. This gives 12 ones

Step 8: Subtract the ones column

\(12 - 7 = 5\), write down 5 in the ones place

Step 9: Subtract the tens column

\(9 - 5 = 4\), write down 4 in the tens place

Step 10: Subtract the hundreds column

\(9 - 4 = 5\), write down 5 in the hundreds place

Step 11: Subtract the thousands column

\(2 - 1 = 1\), write down 1 in the thousands place

\(3,002 - 1,457 = 1,545\)
Final answer:

\(3,002 - 1,457 = 1,545\)

Applied rules:

• Zero regrouping: When borrowing from zeros, continue to the next non-zero digit

• Sequential conversion: Convert higher place values to lower ones as needed

• Place value tracking: Keep careful track of all conversions during regrouping

Rules and methods, laws,...
\(52 - 27 = 25\)
Two-Digit Subtraction
Borrowing Rule
If top < bottom, borrow 1 from next column
When subtraction isn't possible
Place Value
Ones, Tens, Hundreds, Thousands
Each position has specific value
Regrouping
Converting higher to lower place value
Borrowing process
Key definitions:

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

Borrowing: Taking a group of ten from one column to add to the next column

Regrouping: Reorganizing numbers by converting from higher to lower place values

Complete methodology:
  1. Align numbers: Place values in the same column (ones under ones, tens under tens...)
  2. Subtract from right to left: Start with the ones column
  3. Borrow when needed: If top digit is smaller than bottom digit, borrow from next column
  4. Adjust values: Reduce borrowed column by 1, increase receiving column by 10
  5. Continue for all columns: Repeat for tens, hundreds, thousands, etc.
Tip 1: Always align numbers by place value before subtracting.
Tip 2: Cross out and rewrite digits when borrowing to keep track.
Tip 3: Check your work by adding the result to the subtracted number.
Tip 4: Practice with smaller numbers first before moving to larger ones.
Common errors: Forgetting to reduce the borrowed column, misaligning columns, subtracting smaller from larger without borrowing.
Practice strategies: Use manipulatives, draw place value charts, check with addition.
Formulas to know by heart:

• Borrowing rule: When top digit < bottom digit, borrow 1 from next column (add 10 to current column)

• Place value: Each position has 10 times the value of the position to its right

• Subtraction algorithm: Subtract column by column from right to left

Comprehensive Summary of Subtraction with Regrouping

Complete Learning Summary

Key Definitions

Subtraction: The operation of finding the difference between two numbers.

Regrouping: The process of exchanging one unit of a higher place value for ten units of a lower place value.

Borrowing: Taking value from a higher place value column to make subtraction possible in a lower place value column.

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

Core Rules & Laws

The Borrowing Rule: When the top digit in a column is smaller than the bottom digit, borrow 1 from the next higher place value column.

The Conversion Rule: When borrowing, decrease the higher place value by 1 and increase the lower place value by 10.

The Sequential Rule: If a higher place value column contains zero, continue borrowing from the next non-zero column.

The Alignment Rule: Always align numbers by place value before beginning subtraction.

Step-by-Step Method

  1. Write the numbers vertically with the larger number on top
  2. Align digits by place value (ones under ones, tens under tens, etc.)
  3. Start subtracting from the rightmost column (ones place)
  4. If the top digit is smaller than the bottom digit, borrow from the next column
  5. Reduce the higher column by 1 and add 10 to the current column
  6. Continue subtracting in each column from right to left
  7. Write the final answer below the line

Examples & Applications

Simple Example: 52 - 27 → Need to borrow in ones column

Complex Example: 421 - 186 → Need to borrow in ones and tens columns

Advanced Example: 3002 - 1457 → Need to borrow across zeros

Real-world Application: Calculating change, measuring differences, comparing quantities

Tips & Common Mistakes

Tips: Always cross out and rewrite digits when borrowing, use lined paper to keep columns aligned, check your answer by adding.

Common Mistakes: Forgetting to reduce the borrowed column, subtracting bottom from top when top is smaller, misaligning columns.

Pitfalls: Not borrowing when needed, borrowing from zeros without continuing to the next non-zero digit, arithmetic errors.

Verification: Add the result to the subtracted number to get back to the original number.

Key Takeaways

Memorize: The borrowing rule (top smaller than bottom means borrow)

Understand: Place value system and how it enables regrouping

Practice: Regular problems with increasing complexity

Verify: Always check your work by adding the result to the subtracted number

Questions & Answers

Question: I don't understand why I need to borrow when subtracting. Why can't I just subtract the smaller number from the bigger number in each column?

Answer: Great question! When we subtract, we're essentially taking away. In the problem 52 - 27, we want to take away 27 from 52. Looking at the ones place, we need to take away 7 ones from 2 ones, but we only have 2 ones to start with!

Think of it like having 2 apples but needing to give away 7 apples - it's impossible! So we need to "borrow" some apples from the tens place. We take 1 ten (which equals 10 ones) and add it to our 2 ones, giving us 12 ones total. Now we can take away 7 ones, leaving 5 ones.

The borrowing ensures we have enough in each place value to complete the subtraction properly.

Question: My child gets confused when there are zeros in the number, like in 3002 - 1457. How do I help them understand borrowing across zeros?

Answer: Borrowing across zeros is indeed challenging! Here's how to break it down:

  • Think of it like a chain reaction - when you need to borrow from a zero, you have to go to the next non-zero digit
  • In 3002, when borrowing for the ones place, you can't take from the 0 tens, so you go to the 0 hundreds, which also doesn't work, so you go to the 3 thousands
  • This converts the 3 thousands to 2 thousands, the 0 hundreds to 10 hundreds, then take 1 hundred to make 10 tens, then take 1 ten to make 12 ones
  • Use physical objects like base-10 blocks to demonstrate this concept visually

Practice with simpler problems first, like 100 - 15, before moving to more complex problems with multiple zeros.

Question: Sometimes I forget to change the number I borrowed from. How can I remember to reduce it by 1?

Answer: This is a very common mistake! Here are several strategies to help:

  • Cross out and rewrite: When you borrow, cross out the original number and write the new number immediately
  • Visual marking: Put a small mark or circle around the number you're reducing
  • Physical action: Point to the number you're borrowing from and say "one less" out loud
  • Verification: After completing the problem, double-check that all borrowed numbers were reduced by 1

Remember: when you borrow 1 from a column, that column decreases by 1. For example, if you had 5 tens and borrowed 1 ten, you now have 4 tens.