Solved Exercises on Multiplying and Dividing by 10s in Grade 3

Master multiplying and dividing by 10s: place value shifts, inverse operations, and real-world applications through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Multiplying by 10
Exercise 1
What is \(8 \times 10\)? Show how the digits move.
Definition:

Multiplying by 10: Each digit moves one place value position to the left

Multiply by 10 method:
  1. Take the original number
  2. Add a zero to the end
  3. Or move each digit one place to the left
Original
\(8\)
After ×10
\(80\)
Multiplication
8 → 80
Digit moves left
Add Zero
8 → 80
Same result
Step 1: Start with the original number

8 in the ones place

Step 2: Multiply by 10

Each digit moves one place to the left

Step 3: Fill empty places with zeros

8 moves to tens place, ones place gets 0

\(8 \times 10 = 80\)
Final answer:

\(8 \times 10 = 80\)

Applied rules:

• Place value shift: Each digit moves one place to the left

• Zero rule: Add a zero to the end of the number

• Digit movement: 8 ones become 8 tens

2 Dividing by 10
Exercise 2
What is \(70 \div 10\)? Show how the digits move.
Definition:

Dividing by 10: Each digit moves one place value position to the right

Division
70 → 7
Digit moves right
Remove Zero
70 → 7
Same result
Original
\(70\)
After ÷10
\(7\)
Step 1: Start with the original number

70 has 7 in the tens place and 0 in the ones place

Step 2: Divide by 10

Each digit moves one place to the right

Step 3: Remove trailing zeros

7 moves from tens place to ones place, 0 is dropped

\(70 \div 10 = 7\)
Final answer:

\(70 \div 10 = 7\)

Applied rules:

• Place value shift: Each digit moves one place to the right

• Zero removal: Remove one zero from the end of the number

• Digit movement: 7 tens become 7 ones

3 Inverse operations
Exercise 3
Show that multiplying by 10 and dividing by 10 are inverse operations using 5 as an example.
Definition:

Inverse operations: Operations that undo each other

Inverse Operations Example
5
×10
5
0
÷10
5

Starting number = Ending number

Start
\(5\)
×10
\(50\)
÷10
\(5\)
Step 1: Start with the original number

Begin with 5

Step 2: Multiply by 10

\(5 \times 10 = 50\)

Step 3: Divide by 10

\(50 \div 10 = 5\)

Step 4: Verify the result

We end up with the same number we started with!

\(5 \times 10 = 50\), \(50 \div 10 = 5\)
Final answer:

Multiplying by 10 and dividing by 10 are inverse operations

Applied rules:

• Inverse operations: One operation undoes the other

• Place value: Multiplication moves digits left, division moves right

• Verification: Starting number equals ending number

Multiplication and Division by 10 Relationships
Multiplying by 10
\(n \times 10 = n0\)
Digits move left
Dividing by 10
\(n0 \div 10 = n\)
Digits move right
Inverse Operations
\(a \times 10 \div 10 = a\)
\(b \div 10 \times 10 = b\)
Thousands
1000
Hundreds
100
Tens
10
Ones
1
↓ ×10 ↓
Ten Thousands
10000
Thousands
1000
Hundreds
100
Tens
10
Key definitions:

Inverse operations: Operations that undo each other (multiplication and division)

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

Digit shift: Movement of digits when multiplying or dividing by powers of 10

Operation relationships:
  1. Multiplication by 10: Moves digits left, increases value by factor of 10
  2. Division by 10: Moves digits right, decreases value by factor of 10
  3. Inverse relationship: One operation undoes the other
Tip 1: Think of multiplication as making numbers bigger and division as making them smaller.
Tip 2: Remember: what you do with one operation, you undo with the other.
Tip 3: Practice with simple numbers first, then move to larger ones.
Tip 4: Use place value charts to visualize digit movement.
Key insight: Multiplying by 10 is the opposite of dividing by 10.
Common error: Mixing up direction of digit movement in multiplication vs division.
Examples and Patterns
\(n \times 10 = n0\)
Multiply by 10
\(n0 \div 10 = n\)
Divide by 10
Multiplication Examples
\(3 \times 10 = 30\)
\(6 \times 10 = 60\)
\(9 \times 10 = 90\)
Division Examples
\(30 \div 10 = 3\)
\(60 \div 10 = 6\)
\(90 \div 10 = 9\)
Larger Numbers
\(12 \times 10 = 120\)
\(45 \times 10 = 450\)
\(100 \times 10 = 1000\)
Larger Numbers
\(120 \div 10 = 12\)
\(450 \div 10 = 45\)
\(1000 \div 10 = 100\)
Check Your Understanding
\(7 \times 10 = 70\), \(70 \div 10 = 7\)
\(40 \div 10 = 4\), \(4 \times 10 = 40\)
Multiplying and Dividing by 10 Tips
\(\text{Multiply by 10} \leftrightarrow \text{Divide by 10}\)
Inverse Operations
Key concepts:

Inverse operations: Multiplication and division are opposites that undo each other

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

Digit movement: When multiplying by 10, digits shift left; when dividing, they shift right

Steps for success:
  1. Identify the operation: Determine if multiplying or dividing by 10
  2. Apply the rule: Add/remove zero or shift digits
  3. Check your work: Use the inverse operation to verify
  4. State the answer: Write the final result
Tip 1: Remember: multiplying by 10 makes numbers 10 times larger.
Tip 2: Remember: dividing by 10 makes numbers 10 times smaller.
Tip 3: Use the inverse operation to check your answers.
Tip 4: Practice with both operations together to see the relationship.
Common mistake: Confusing the direction of digit movement between multiplication and division.
Success strategy: Practice both operations together to reinforce the inverse relationship.
Important facts to remember:

• Multiplying by 10 adds a zero to the end of a number

• Dividing by 10 removes a zero from the end of a number

• These operations are inverses of each other

• Multiplication makes numbers larger, division makes them smaller

• Both operations change place value by one position

Questions & Answers

Question: Why are multiplying and dividing by 10 considered inverse operations? What does that mean?

Answer: Inverse operations are operations that undo each other. When you multiply by 10 and then divide by 10, you get back to where you started!

For example:

  • Start with 5
  • Multiply by 10: \(5 \times 10 = 50\)
  • Divide by 10: \(50 \div 10 = 5\)
  • You end up with 5 again!

It's like going forward and then backward - you return to your starting point.

Question: My child sometimes confuses multiplying and dividing by 10. How can I help them remember which is which?

Answer: Here are helpful strategies to remember:

  • Multiplying by 10: Makes numbers BIGGER (think: "more" zeros)
  • Dividing by 10: Makes numbers SMALLER (think: "fewer" zeros)
  • Direction: Multiplication moves digits LEFT, division moves digits RIGHT
  • Memory trick: "Times 10" adds a zero, "Divide 10" takes away a zero

Practice with simple numbers first and emphasize the direction of digit movement!

Question: What happens if I try to divide a number like 7 by 10? It doesn't have a zero to remove!

Answer: Great question! When you divide 7 by 10, you get 0.7, which is less than 1. This introduces decimal numbers, which you'll learn more about in higher grades.

For now, we focus on whole numbers that are multiples of 10 when dividing by 10. For example:

  • \(70 \div 10 = 7\)
  • \(50 \div 10 = 5\)
  • \(30 \div 10 = 3\)

When we divide by 10, we're looking for how many groups of 10 are in the number.

Detailed Summary: Multiplying and Dividing by 10s

Definitions of Key Concepts

  • Multiplying by 10: A mathematical operation that increases a number by a factor of 10
  • Dividing by 10: A mathematical operation that decreases a number by a factor of 10
  • Inverse Operations: Operations that undo each other (multiplication and division)
  • Place Value: The value of a digit based on its position in a number (ones, tens, hundreds, etc.)
  • Digit Shift: The movement of digits to the left (multiplication) or right (division) when working with powers of 10

Core Rules and Principles

  • Multiplication Rule: When multiplying by 10, add one zero to the end of the number
  • Division Rule: When dividing by 10, remove one zero from the end of the number
  • Digit Movement: Multiplication moves digits left, division moves digits right
  • Inverse Relationship: Multiplying by 10 and dividing by 10 are opposite operations

Step-by-Step Methods and Procedures

  1. For Multiplication by 10:
    • Identify the number to be multiplied
    • Add a zero to the end of the number
    • Alternatively, move each digit one place value to the left
  2. For Division by 10:
    • Identify the number to be divided
    • Remove one zero from the end of the number
    • Alternatively, move each digit one place value to the right
  3. For Verification:
    • Use the inverse operation to check your answer
    • If you multiplied by 10, divide by 10 to return to the original number

Concrete Examples (Simple to Advanced)

  • Simple Multiplication: \(2 \times 10 = 20\), \(5 \times 10 = 50\), \(9 \times 10 = 90\)
  • Simple Division: \(20 \div 10 = 2\), \(50 \div 10 = 5\), \(90 \div 10 = 9\)
  • Moderate Multiplication: \(12 \times 10 = 120\), \(25 \times 10 = 250\), \(47 \times 10 = 470\)
  • Moderate Division: \(120 \div 10 = 12\), \(250 \div 10 = 25\), \(470 \div 10 = 47\)
  • Advanced Inverse Operations: \(7 \times 10 = 70\), \(70 \div 10 = 7\); \(34 \times 10 = 340\), \(340 \div 10 = 34\)

Tips, Tricks, and Common Pitfalls

  • Tip 1: Remember: multiplication makes numbers bigger, division makes them smaller
  • Tip 2: Use place value charts to visualize digit movement
  • Tip 3: Always verify with the inverse operation
  • Common Pitfall: Confusing the direction of digit movement between multiplication and division
  • Common Mistake: Forgetting to remove zeros when dividing by 10

Key Notes for Memorization

  • Rhyme: "To multiply by 10, add a zero at the end!"
  • Rhyme: "To divide by 10, take a zero off the end!"
  • Memory Aid: Multiplication moves LEFT, Division moves RIGHT
  • Pattern: These operations are opposites that undo each other

Explanation for Students

When you multiply by 10, you're making the number 10 times bigger. When you divide by 10, you're making it 10 times smaller. These operations are like opposites - if you multiply by 10 and then divide by 10, you get back to where you started! Think of it like going forward and then backward - you end up in the same place. The key is to remember that multiplying adds zeros (makes numbers bigger) and dividing removes zeros (makes numbers smaller).