Solved Exercises on Three-Digit Subtraction Without Regrouping in Grade 2

Master three-digit subtraction without regrouping: place value, column subtraction, and borrowing concepts through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic three-digit subtraction
Exercise 1
Calculate:
\(456 - 234\)
Definition:

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

Column subtraction method:
  1. Line up numbers by place value (ones under ones, tens under tens, hundreds under hundreds)
  2. Subtract the ones column: \(6 - 4 = 2\)
  3. Subtract the tens column: \(5 - 3 = 2\)
  4. Subtract the hundreds column: \(4 - 2 = 2\)
  5. Write the answer: \(222\)
Expression
\(456 - 234\)
Column subtraction
\(\begin{array}{r} 456 \\ - 234 \\ \hline 222 \end{array}\)
Result
\(222\)
Step 1: Line up by place value

\(\begin{array}{r} H \ T \ O \\ 4 \ 5 \ 6 \\ - 2 \ 3 \ 4 \\ \hline \end{array}\)

Step 2: Subtract ones column

\(6 - 4 = 2\) (write 2 in the ones place)

Step 3: Subtract tens column

\(5 - 3 = 2\) (write 2 in the tens place)

Step 4: Subtract hundreds column

\(4 - 2 = 2\) (write 2 in the hundreds place)

\(456 - 234 = 222\)
Final answer:

\(456 - 234 = 222\)

Applied rules:

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

• Column subtraction: Subtract digits in the same place value columns separately

• No regrouping: Since each top digit is ≥ bottom digit, no borrowing is needed

2 Subtraction with zeros
Exercise 2
Calculate:
\(508 - 203\)
Definition:

Zero in place value: Zero holds a place in the number but has no value when subtracted

Expression
\(508 - 203\)
Column subtraction
\(\begin{array}{r} 508 \\ - 203 \\ \hline 305 \end{array}\)
Result
\(305\)
Step 1: Line up by place value

\(\begin{array}{r} H \ T \ O \\ 5 \ 0 \ 8 \\ - 2 \ 0 \ 3 \\ \hline \end{array}\)

Step 2: Subtract ones column

\(8 - 3 = 5\) (write 5 in the ones place)

Step 3: Subtract tens column

\(0 - 0 = 0\) (write 0 in the tens place)

Step 4: Subtract hundreds column

\(5 - 2 = 3\) (write 3 in the hundreds place)

\(508 - 203 = 305\)
Final answer:

\(508 - 203 = 305\)

Applied rules:

• Zero property: Subtracting zero doesn't change the value (\(0 - 0 = 0\))

• Place value: Even if there's a zero, still align by place value

• No regrouping: Since each top digit is ≥ bottom digit, no borrowing is needed

3 Subtraction with different digit counts
Exercise 3
Calculate:
\(642 - 31\)
Definition:

Borrowing (regrouping): When the top digit is smaller than the bottom digit in a column, we borrow from the next higher place value

Expression
\(642 - 31\)
Column subtraction
\(\begin{array}{r} 642 \\ - \phantom{0}31 \\ \hline 611 \end{array}\)
Result
\(611\)
Step 1: Line up by place value

Write the numbers so ones are under ones, tens under tens, etc.:\(\begin{array}{r} H \ T \ O \\ 6 \ 4 \ 2 \\ - \phantom{6} \ 3 \ 1 \\ \hline \end{array}\)

Step 2: Subtract ones column

\(2 - 1 = 1\) (write 1 in the ones place)

Step 3: Subtract tens column

\(4 - 3 = 1\) (write 1 in the tens place)

Step 4: Subtract hundreds column

\(6 - 0 = 6\) (write 6 in the hundreds place)

\(642 - 31 = 611\)
Final answer:

\(642 - 31 = 611\)

Applied rules:

• Place value alignment: Align digits by their place values even when numbers have different digit counts

• Treating missing digits: Think of 31 as 031 when subtracting from 642

• No regrouping: Since each top digit is ≥ bottom digit, no borrowing is needed

Three-Digit Subtraction Without Regrouping Strategies
\(\begin{array}{r} H \ T \ O \\ 4 \ 5 \ 6 \\ - 2 \ 3 \ 4 \\ \hline 2 \ 2 \ 2 \end{array}\)
Column Subtraction Method
Rule 1
Align by Place Value
Ones under ones, tens under tens, hundreds under hundreds
Rule 2
Subtract Each Column Separately
Ones first, then tens, then hundreds
Rule 3
No Regrouping
Each top digit must be ≥ bottom digit
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 (borrowing): Moving a group of ten from one place value to the next lower place value

Subtraction methodology:
  1. Align numbers: Put numbers in columns by place value
  2. Subtract ones column: Subtract the bottom digit from the top digit
  3. Subtract tens column: Subtract the bottom digit from the top digit
  4. Subtract hundreds column: Subtract the bottom digit from the top digit
  5. Write the answer: Combine the results from each column
Tip 1: Always align numbers by place value. Use lined paper to keep columns straight.
Tip 2: When subtracting, go from right to left (ones, tens, hundreds).
Tip 3: If any top digit is smaller than the bottom digit, regrouping is needed (not covered in this section).
Tip 4: Check your work by adding the answer to the number you subtracted.
Common errors: Misaligning columns, subtracting smaller from larger, forgetting place value.
Success strategies: Use lined paper, check each column individually, verify with addition.
Three-digit subtraction rules:

• Place value alignment: \(456 - 234 = \begin{array}{r} 456 \\ - 234 \\ \hline 222 \end{array}\)

• Column-wise subtraction: Subtract ones, tens, then hundreds separately

• No regrouping condition: Each top digit must be ≥ bottom digit

• Zero handling: \(508 - 203 = \begin{array}{r} 508 \\ - 203 \\ \hline 305 \end{array}\)

Solution: Exercises 4 to 5
4 Subtraction with multiple zeros
Exercise 4
Calculate:
\(700 - 205\)
Definition:

Multiple zeros: When subtracting numbers with zeros in different place values

Expression
\(700 - 205\)
Column subtraction
\(\begin{array}{r} 700 \\ - 205 \\ \hline 495 \end{array}\)
Result
\(495\)
Step 1: Line up by place value

\(\begin{array}{r} H \ T \ O \\ 7 \ 0 \ 0 \\ - 2 \ 0 \ 5 \\ \hline \end{array}\)

Step 2: Subtract ones column

\(0 - 5\): Since 0 < 5, regrouping is needed. This violates our no-regrouping rule.

Step 3: Since regrouping is needed

This problem requires regrouping and cannot be solved without it.

Step 4: Correct problem without regrouping

Let's solve \(705 - 200\) instead: \(5 - 0 = 5\), \(0 - 0 = 0\), \(7 - 2 = 5\)

\(705 - 200 = 505\)
Final answer:

\(705 - 200 = 505\)

Applied rules:

• No regrouping limitation: Only subtract when top digit ≥ bottom digit

• Problem selection: Choose problems that don't require regrouping

5 Real-world application
Exercise 5
A school has 876 students. 342 students went on a field trip. How many students stayed at school?
Definition:

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

Problem
Total: 876, Trip: 342, Stayed: ?
Equation
876 - 342 = ?
Result
534
Step 1: Identify the numbers

Total students = 876, Students on trip = 342

Step 2: Set up the subtraction

We need to find those who stayed, so we subtract: 876 - 342

Step 3: Line up by place value

\(\begin{array}{r} H \ T \ O \\ 8 \ 7 \ 6 \\ - 3 \ 4 \ 2 \\ \hline \end{array}\)

Step 4: Subtract each column

Ones: \(6 - 2 = 4\), Tens: \(7 - 4 = 3\), Hundreds: \(8 - 3 = 5\)

534 students stayed at school.
Final answer:

534 students stayed at school.

Applied rules:

• Real-world application: Connect math to everyday situations

• Problem-solving: Identify what needs to be calculated

• No regrouping: Each top digit is ≥ bottom digit

Three-Digit Subtraction Without 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 Subtraction
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 (borrowing): Moving a group of ten from one place value to the next lower place value

Three-digit subtraction methodology:
  1. Align numbers: Put numbers in columns by place value
  2. Subtract ones column: Subtract the bottom digit from the top digit
  3. Subtract tens column: Subtract the bottom digit from the top digit
  4. Subtract hundreds column: Subtract the bottom digit from the top digit
  5. Write the answer: Combine the results from each column
Tip 1: Always align numbers by place value. Use lined paper to keep columns straight.
Tip 2: When subtracting, go from right to left (ones, tens, hundreds).
Tip 3: If any top digit is smaller than the bottom digit, regrouping is needed (not covered in this section).
Tip 4: Check your work by adding the answer to the number you subtracted.
Common errors: Misaligning columns, subtracting smaller from larger, forgetting place value.
Success strategies: Use lined paper, check each column individually, verify with addition.
Three-digit subtraction formulas:

• Place value alignment: \(456 - 234 = \begin{array}{r} 456 \\ - 234 \\ \hline 222 \end{array}\)

• Column-wise subtraction: Subtract ones, tens, then hundreds separately

• No regrouping condition: Each top digit must be ≥ bottom digit

• Zero handling: \(508 - 203 = \begin{array}{r} 508 \\ - 203 \\ \hline 305 \end{array}\)

Exercise with Visualization: Place Value Components
Exercise 6: Place Value Visualization
Consider the place value components of three-digit numbers:
\(f_1(x) = 100x\) (hundreds)
\(f_2(x) = 10x\) (tens)
\(f_3(x) = x\) (ones)

Analysis: The chart shows how three-digit numbers decompose into place value components.

  • \(f_1(x) = 100x\) (Hundreds place: 100 × digit value)
  • \(f_2(x) = 10x\) (Tens place: 10 × digit value)
  • \(f_3(x) = x\) (Ones place: 1 × digit value)

Questions & Answers

Question: Why do we have to line up the numbers when subtracting three-digit numbers?

Answer: Lining up numbers by place value is crucial because each digit has a different value depending on its position:

  • The rightmost digit is the ones place (value × 1)
  • The middle digit is the tens place (value × 10)
  • The leftmost digit is the hundreds place (value × 100)

For example, in 456:

  • 6 is worth 6 ones = 6
  • 5 is worth 5 tens = 50
  • 4 is worth 4 hundreds = 400

If we didn't line up the numbers properly, we might accidentally subtract tens from hundreds or ones from tens, which would give us the wrong answer!

Think of it like organizing coins: you wouldn't subtract dimes from pennies when counting!

Question: What happens if I subtract 642 - 31 but forget to line up the digits properly? I got 611, but I'm not sure if I did it right.

Answer: Your answer of 611 is actually correct for 642 - 31! Let me show you the proper way to line up these numbers:

Right way (proper alignment):

\(\begin{array}{r} 642 \\ - \phantom{0}31 \\ \hline 611 \end{array}\)

Notice that the 1 (ones) lines up with the 2 (ones), and the 3 (tens) lines up with the 4 (tens). The 6 (hundreds) stands alone.

If you had lined them up differently, like:

\(\begin{array}{r} 642 \\ - 31 \\ \hline \end{array}\)

You might accidentally subtract 3 (from 31) from 6 (from 642), which would be subtracting tens from hundreds!

Your answer is correct, but make sure you're lining up the digits properly to build good habits for more complex problems.

Question: My child sometimes subtracts diagonally instead of vertically. How can I help them understand the correct method?

Answer: This is a common mistake! Here are some strategies to help your child:

  1. Use visual aids: Draw vertical lines between columns to clearly separate ones, tens, and hundreds
  2. Color coding: Use different colored pencils for each column (red for ones, blue for tens, green for hundreds)
  3. Physical objects: Use base-10 blocks to show that you only subtract like items from like items (ones from ones, tens from tens)
  4. Slow down: Have your child point to each column as they subtract it
  5. Practice with grid paper: This helps keep numbers aligned properly

For example, with 456 - 234:

\(\begin{array}{|c|c|c|} \hline H & T & O \\ \hline 4 & 5 & 6 \\ \hline - & - & - \\ \hline 2 & 3 & 4 \\ \hline 2 & 2 & 2 \\ \hline \end{array}\)

Emphasize that we only subtract vertically within each column, never diagonally across columns!