Solved Exercises on Multiplying Multi-Digit Numbers by Two-Digit Numbers in Grade 4

Master multiplying multi-digit numbers by two-digit numbers: step-by-step solutions, rules, methods, and practice exercises.

Solution: Exercises 1 to 3
1 Basic multiplication
Exercise 1
Calculate: 342 × 23
Definition:

Multi-Digit Number: A number with three or more digits (e.g., 342)

Standard Algorithm Method:
  1. Write the numbers vertically, aligning by place value
  2. Multiply the top number by the ones digit of the bottom number
  3. Multiply the top number by the tens digit of the bottom number
  4. Shift the second partial product one place to the left
  5. Add the partial products together
Problem
342 × 23
Setup
342 × 23
Solution
7,866
Step 1: Set up vertically

342 × 23

Step 2: Multiply by ones digit (3)

342 × 3 = 1,026

Step 3: Multiply by tens digit (20)

342 × 20 = 6,840

Step 4: Add partial products

1,026 + 6,840 = 7,866

342 × 23 = 7,866
Final answer:

342 × 23 = 7,866

Applied rules:

• Place Value: 2 in 23 represents 20, so shift one place left

• Carrying: When products exceed 9, carry to next column

• Alignment: Properly align digits for addition

2 With zeros
Exercise 2
Calculate: 508 × 47
Definition:

Zero Placeholders: Zeros in the middle of numbers affect place value but are multiplied normally.

Problem
508 × 47
Setup
508 × 47
Solution
23,876
Step 1: Set up vertically

508 × 47

Step 2: Multiply by ones digit (7)

508 × 7 = 3,556

Step 3: Multiply by tens digit (40)

508 × 40 = 20,320

Step 4: Add partial products

3,556 + 20,320 = 23,876

508 × 47 = 23,876
Final answer:

508 × 47 = 23,876

Applied rules:

• Zero Handling: Multiply zeros like other digits, but they may result in zero products

• Carrying with Zeros: Even with zeros, carry when necessary

• Place Value: Maintain proper place value alignment

3 Larger numbers
Exercise 3
Calculate: 1,245 × 36
Definition:

Four-Digit Number: A number with four places: thousands, hundreds, tens, and ones.

Problem
1,245 × 36
Setup
1,245 × 36
Solution
44,820
Step 1: Set up vertically

1,245 × 36

Step 2: Multiply by ones digit (6)

1,245 × 6 = 7,470

Step 3: Multiply by tens digit (30)

1,245 × 30 = 37,350

Step 4: Add partial products

7,470 + 37,350 = 44,820

1,245 × 36 = 44,820
Final answer:

1,245 × 36 = 44,820

Applied rules:

• Scale Independence: Same method works for any size multi-digit number

• Multiple Carries: May need to carry multiple times in one multiplication

• Accuracy: Extra care needed with larger numbers

Multiplication Rules and Methods
\(abc \times de = (abc \times e) + (abc \times d0)\)
Multi-Digit by Two-Digit Formula
Commutative Property
\(a \times b = b \times a\)
Order doesn't matter
Associative Property
\((a \times b) \times c = a \times (b \times c)\)
Grouping doesn't matter
Distributive Property
\(a \times (b + c) = a \times b + a \times c\)
Break apart numbers
Key definitions:

Multi-Digit Number: A number with 3 or more digits (100, 1,234, etc.)

Two-Digit Number: A number with tens and ones place (10-99)

Partial Product: Result of multiplying by one digit of the multiplier

Standard Algorithm Steps:
  1. Write numbers vertically, aligning by place value
  2. Multiply top number by ones digit of bottom number
  3. Write result below, aligning by place value
  4. Multiply top number by tens digit of bottom number
  5. Write result shifted one place left
  6. Add the partial products together
Tip 1: Always estimate first (round numbers) to check reasonableness.
Tip 2: Use zeros as placeholders when shifting for tens digit.
Tip 3: Line up digits carefully to avoid addition errors.
Tip 4: Check by estimating: 1,245 × 36 ≈ 1,200 × 40 = 48,000

Solution: Exercises 4 to 5
4 Word problem
Exercise 4
A school has 24 classrooms. Each classroom has 28 desks. How many desks are there in total?
Definition:

Word Problem Strategy: Identify key information and set up appropriate multiplication.

Problem
24 classrooms × 28 desks
Setup
24 × 28
Solution
672 desks
Step 1: Set up the multiplication

24 × 28

Step 2: Multiply by ones digit (8)

24 × 8 = 192

Step 3: Multiply by tens digit (20)

24 × 20 = 480

Step 4: Add partial products

192 + 480 = 672

24 × 28 = 672 desks
Final answer:

There are 672 desks in total.

Applied rules:

• Context Recognition: Identify "equal groups" situation

• Units: Include appropriate units in final answer

• Verification: Estimate to check reasonableness

5 Real-world application
Exercise 5
A factory produces 1,573 widgets each day. How many widgets does it produce in 35 days?
Definition:

Rate Problem: Finding total amount given a rate and time period.

Problem
1,573 widgets × 35 days
Setup
1,573 × 35
Solution
55,055 widgets
Step 1: Set up the multiplication

1,573 × 35

Step 2: Multiply by ones digit (5)

1,573 × 5 = 7,865

Step 3: Multiply by tens digit (30)

1,573 × 30 = 47,190

Step 4: Add partial products

7,865 + 47,190 = 55,055

1,573 × 35 = 55,055 widgets
Final answer:

The factory produces 55,055 widgets in 35 days.

Applied rules:

• Rate × Time = Total: Standard pattern for rate problems

• Large Number Handling: Same algorithm works for any size

• Real-World Context: Interpret results in problem context

Detailed Summary: Multiplying Multi-Digit by Two-Digit Numbers
\(ABC \times DE = (ABC \times E) + (ABC \times D0)\)
Core Formula
Key definitions:

Multi-Digit Number: Any number with three or more digits (e.g., 123, 4,567, 89,012)

Two-Digit Number: Any number from 10 to 99 (e.g., 12, 45, 89)

Partial Product: The result of multiplying the top number by one digit of the bottom number

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

Complete methodology:
  1. Align numbers vertically: Match digits by place value
  2. Multiply by ones digit: Multiply top number by bottom number's ones digit
  3. Record first partial product: Write result aligned under ones place
  4. Multiply by tens digit: Multiply top number by bottom number's tens digit
  5. Shift and record: Write result one place to the left (representing tens)
  6. Add partial products: Sum both partial products to get final answer
Tip 1: Always write neatly and align digits properly to avoid addition errors.
Tip 2: When multiplying by tens digit, remember the result represents tens, so shift left.
Tip 3: Use estimation to check if your answer makes sense (round numbers and multiply).
Tip 4: If you get stuck, try breaking the problem into smaller parts.
Common errors: Misaligning digits, forgetting to shift for tens digit, incorrect carrying.
Key strategies: Estimate first, check work, use placeholders for zeros.
Properties to remember:

• Commutative: Order doesn't change the product (a × b = b × a)

• Associative: Grouping doesn't change the product ((a × b) × c = a × (b × c))

• Distributive: Helps break apart larger numbers (a × (b + c) = a × b + a × c)

• Identity: Any number × 1 = itself

• Zero Property: Any number × 0 = 0

Multiplication Patterns and Mental Math
Mental Math Strategies
Useful patterns for multiplying multi-digit by two-digit numbers:
• Round and adjust: 124 × 19 = 124 × 20 - 124
• Break apart: 124 × 15 = 124 × (10 + 5) = 1,240 + 620
• Friendly numbers: Look for multiples of 10, 25, 50

Pattern Recognition: Notice how numbers relate to make calculations easier

  • Near 100: 124 × 98 = 124 × (100 - 2) = 12,400 - 248
  • Ending in 5: Numbers ending in 5 often have predictable patterns
  • Round numbers: 124 × 30 is easier than 124 × 29

Questions & Answers

Question: I get confused about when to shift the second partial product to the left. Why do we do that?

Answer: Great question! We shift because of place value. When you multiply by the tens digit, you're really multiplying by that digit times 10.

  • In 342 × 23, the 2 in 23 represents 20 (not just 2)
  • So 342 × 20 means 342 × 2 × 10
  • That extra factor of 10 shifts everything one place to the left
  • Think of it as: 342 × 20 = (342 × 2) × 10 = 684 × 10 = 6,840

Memory trick: "Tens digit means tens place - shift one place left!"

Example: 456 × 78
456 × 8 = 3,648 (ones place)
456 × 70 = 31,920 (tens place - shifted left)
Total: 3,648 + 31,920 = 35,568

Question: My child makes many mistakes with carrying. How can I help them be more accurate?

Answer: Carrying mistakes are very common! Try these strategies:

  • Circle carries: Have your child circle the number being carried to keep track
  • Small print: Write carries small above the next column
  • Slow down: Take time with each multiplication step
  • Check work: Go back and verify each partial product

Practice with smaller numbers first: 12 × 34 before moving to 123 × 45

Encourage your child to say the process out loud: "4 times 7 is 28, write 8, carry 2..."

Estimation helps catch major errors: if 123 × 45 should be around 5,000, an answer like 500 or 50,000 signals a mistake.

Question: Are there faster ways to multiply larger numbers?

Answer: Yes! Here are some efficient strategies:

  • Round and adjust: For 245 × 19, think 245 × 20 = 4,900, then subtract 245 = 4,655
  • Break apart: 245 × 16 = 245 × (10 + 6) = 2,450 + 1,470 = 3,920
  • Factor pairs: Look for numbers that are easier to work with
  • Patterns: Numbers ending in 5 or 0 often have shortcuts

However, master the standard algorithm first - it's reliable for ALL problems!

Example: 375 × 24
Think: 375 × 25 = 9,375 (because 25 = 100 ÷ 4, so 375 × 100 = 37,500, then ÷ 4 = 9,375)
Then subtract 375: 9,375 - 375 = 9,000
But 375 × 24 = 375 × (25 - 1) = 9,375 - 375 = 9,000