Solved Exercises on Multiplication and Division Word Problems Common Core in Grade 4

Master Common Core aligned multiplication and division word problems: multi-digit calculations, problem-solving strategies, and mathematical reasoning through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Multi-digit multiplication
Exercise 1
The school library has 32 shelves. Each shelf holds 24 books. How many books can the library store in total? Use the standard algorithm to solve and verify your answer.
Definition:

Common Core Standard 4.NBT.B.5: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations.

Problem-solving method:
  1. Identify what you're looking for (total number of books)
  2. Find the number of groups (32 shelves)
  3. Find how many in each group (24 books per shelf)
  4. Set up multiplication: 32 × 24
  5. Use standard algorithm with place value understanding
Problem
32 shelves × 24 books per shelf
Standard Algorithm
32 × 24
Result
768
Step 1: Set up the standard algorithm

32 × 24

24

× 32

Step 2: Multiply by ones place

24 × 2 = 48

This represents 2 groups of 24

Step 3: Multiply by tens place

24 × 30 = 720

This represents 30 groups of 24 (the 3 tens in 32)

Step 4: Add partial products

48 + 720 = 768

32 × 24 = 768
Final answer:

The library can store 768 books in total.

Applied Common Core Standards:

• 4.NBT.B.5: Using place value understanding in multiplication

• 4.OA.A.3: Solving multistep word problems with whole numbers

• MP.1: Making sense of problems and persevering in solving them

2 Multi-digit division
Exercise 2
A bakery made 432 muffins for a school event. They want to package the muffins into boxes that hold 8 muffins each. How many boxes will they need? Explain your thinking using the relationship between multiplication and division.
Definition:

Common Core Standard 4.NBT.B.6: Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.

Problem
432 ÷ 8
Division
54
Step 1: Identify the division problem

Total muffins ÷ Muffins per box = Number of boxes

432 ÷ 8 = ?

Step 2: Use relationship between multiplication and division

I need to find how many groups of 8 are in 432

8 × ? = 432

8 × 50 = 400, so 432 - 400 = 32 remaining

8 × 4 = 32, so 50 + 4 = 54

Step 3: Verify using multiplication

54 × 8 = 432 ✓

432 ÷ 8 = 54
Final answer:

They will need 54 boxes.

Applied Common Core Standards:

• 4.NBT.B.6: Using relationship between multiplication and division

• 4.OA.A.3: Solving word problems with whole numbers

• MP.2: Reasoning abstractly and quantitatively

3 Mixed operations
Exercise 3
Mrs. Johnson bought 15 packs of markers for her class. Each pack costs $3. She also bought 24 glue sticks that cost $2 each. If she paid with a $100 bill, how much change did she receive? Show your work using equations with a letter standing for the unknown quantity.
Definition:

Common Core Standard 4.OA.A.3: Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity.

Cost of markers
15 × $3 = $45
Cost of glue sticks
24 × $2 = $48
Total cost
$45 + $48 = $93
Change
$100 - $93 = $7
Step 1: Calculate cost of markers

Number of packs × Cost per pack = Total cost for markers

15 × $3 = $45

Step 2: Calculate cost of glue sticks

Number of glue sticks × Cost per stick = Total cost for glue

24 × $2 = $48

Step 3: Calculate total cost

Cost of markers + Cost of glue = Total cost

$45 + $48 = $93

Step 4: Calculate change

Amount paid - Total cost = Change

$100 - $93 = $7

Let c = change received
c = $100 - (15 × $3 + 24 × $2)
c = $100 - ($45 + $48)
c = $100 - $93 = $7
Final answer:

Mrs. Johnson received $7 in change.

Applied Common Core Standards:

• 4.OA.A.3: Using equations with letters for unknowns

• 4.MD.A.2: Solving word problems involving money

• MP.4: Modeling with mathematics

Common Core Standards and Methods
\(a \times b = \text{total when you have } a \text{ groups of } b\)
Common Core Multiplication
4.NBT.B.5
Multiply up to 4-digit by 1-digit numbers
Using place value and properties of operations
4.NBT.B.6
Divide up to 4-digit by 1-digit numbers
Using place value and relationship between multiplication and division
4.OA.A.3
Multistep word problems
Using equations with letters for unknowns
Key Common Core Definitions:

4.NBT.B.5: Multiply using strategies based on place value and properties of operations

4.NBT.B.6: Divide using strategies based on place value and relationship between multiplication and division

4.OA.A.3: Solve multistep problems using equations with letters for unknowns

Mathematical Practices: Problem solving, reasoning, modeling, precision

Common Core Problem-Solving Method:
  1. Make sense of the problem (MP.1): Understand what is being asked
  2. Reason quantitatively (MP.2): Think about the relationships between numbers
  3. Model with mathematics (MP.4): Use equations to represent the problem
  4. Use appropriate tools strategically (MP.5): Choose efficient methods
  5. Attend to precision (MP.6): Calculate accurately and label answers
Tip 1: Always represent problems with equations using letters for unknowns as required by 4.OA.A.3.
Tip 2: Use place value understanding when multiplying and dividing large numbers.
Tip 3: Connect multiplication and division - they are inverse operations.
Common Core Focus: Conceptual understanding, procedural fluency, and application.
Assessment Emphasis: Showing work, explaining reasoning, and connecting concepts.
Common Core Formulas:

• Multiplication: Total = Groups × Items per group

• Division: Items per group = Total ÷ Number of groups

• Relationship: Dividend = (Divisor × Quotient) + Remainder

• Equation representation: Use letters for unknowns in multistep problems

Solution: Exercises 4 to 5
4 Multi-step problem
Exercise 4
A factory produces 18 crates of toys each day. Each crate contains 25 toys. At the end of 12 days, the toys are packed into boxes that hold 9 toys each. How many boxes are needed to pack all the toys? Use equations to represent each step and explain your reasoning.
Definition:

Common Core Standard 4.OA.A.3: Solving multistep word problems with whole numbers and interpreting remainders when necessary.

Step 1
Toys per day × Days = Total toys
Step 2
Total toys ÷ Toys per box = Boxes needed
Result
600
Step 1: Calculate total number of toys produced

Let t = total toys

t = 18 crates/day × 25 toys/crate × 12 days

t = 18 × 25 × 12

t = 450 × 12 = 5,400 toys

Step 2: Calculate number of boxes needed

Let b = number of boxes

b = Total toys ÷ Toys per box

b = 5,400 ÷ 9

b = 600 boxes

Step 3: Verify the calculation

600 × 9 = 5,400 ✓

t = 18 × 25 × 12 = 5,400
b = 5,400 ÷ 9 = 600
Final answer:

600 boxes are needed to pack all the toys.

Applied Common Core Standards:

• 4.OA.A.3: Using equations with letters for unknowns in multistep problems

• 4.NBT.B.5: Multiplying multi-digit numbers

• MP.4: Modeling with mathematics

5 Division with remainder interpretation
Exercise 5
A school is organizing a field trip for 250 students. Buses can hold 35 students each. How many buses are needed? Explain how you handled the remainder and why your answer makes sense in the real-world context.
Definition:

Common Core Standard 4.OA.A.3: Interpreting remainders in word problems within the context of the real-world situation.

Division
250 ÷ 35
Quotient
7
Remainder
5
Buses needed
8
Step 1: Set up the division problem

Let b = number of buses needed

b = 250 ÷ 35

Step 2: Use long division

35 goes into 250 seven times (35 × 7 = 245)

Remainder: 250 - 245 = 5

Step 3: Interpret the remainder in context

7 buses can hold 245 students

5 students remain who need transportation

Therefore, an additional bus is needed

Step 4: State the final answer

Total buses needed = 7 + 1 = 8 buses

250 ÷ 35 = 7 R 5
Buses needed = 8
Final answer:

8 buses are needed for the field trip.

Applied Common Core Standards:

• 4.OA.A.3: Interpreting remainders in real-world contexts

• 4.NBT.B.6: Finding whole-number quotients and remainders

• MP.2: Reasoning abstractly and quantitatively

Detailed Summary: Common Core Multiplication and Division Word Problems
\(\text{Total} = \text{Groups} \times \text{Items per group}\)
Common Core Multiplication Relationship
Key Common Core Definitions:

4.NBT.B.5 (Number & Operations in Base Ten): Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations.

4.NBT.B.6 (Number & Operations in Base Ten): Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.

4.OA.A.3 (Operations & Algebraic Thinking): Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity.

Common Core Problem-Solving Process:
  1. Understand the problem (MP.1): Read carefully and identify what is being asked
  2. Represent with equations (4.OA.A.3): Use letters for unknowns
  3. Choose appropriate operations: Based on the situation described
  4. Apply computation skills (4.NBT.B.5/B.6): Use place value understanding
  5. Interpret results (4.OA.A.3): Consider real-world context
  6. Check reasonableness (MP.2): Does the answer make sense?
Tip 1: Always represent multistep problems with equations using letters for unknowns as required by Common Core.
Tip 2: Pay attention to remainders in division problems - they often require interpretation in real-world contexts.
Tip 3: Use place value understanding when multiplying and dividing large numbers.
Tip 4: Remember that multiplication and division are inverse operations.
Common Core Emphasis: Conceptual understanding alongside procedural fluency and application to real-world situations.
Assessment Focus: Students must show their work, explain their reasoning, and connect mathematical concepts to practical applications.
Essential Common Core Formulas:

• Multiplication: Total = Groups × Items per group

• Division: Items per group = Total ÷ Number of groups

• Division Algorithm: Dividend = (Divisor × Quotient) + Remainder

• Equation Representation: Use variables (letters) for unknown quantities in multistep problems

Common Core Standards Visualization
Understanding Common Core Standards
Compare these Common Core standards:
• 4.NBT.B.5: Multiplication with place value
• 4.NBT.B.6: Division with place value
• 4.OA.A.3: Multistep word problems

Analysis: The chart shows the focus areas of each Common Core standard for multiplication and division word problems.

  • 4.NBT.B.5: Emphasizes place value and properties of operations in multiplication
  • 4.NBT.B.6: Emphasizes place value and relationship between multiplication and division
  • 4.OA.A.3: Emphasizes multistep problems and equation representation

Questions & Answers

Question: Why does Common Core require us to use equations with letters for unknowns? Isn't that too complicated for 4th grade?

Answer: That's a great question! The Common Core requirement to use letters for unknowns serves several important purposes:

  • Preparation for algebra: It introduces algebraic thinking early, making the transition to middle school math smoother
  • Clarity in problem-solving: Writing equations helps organize your thinking and makes your solution process clear
  • Communication: It allows you to explain your mathematical reasoning more precisely
  • Understanding relationships: Equations help you see the connections between numbers and operations

For example, instead of just calculating "32 × 24 = 768", writing "t = 32 × 24" where t represents the total number of books makes it clear what the equation represents and helps others understand your thinking.

It might seem complicated at first, but it actually makes complex problems easier to solve!

Question: How do I know when I need to interpret remainders differently in word problems?

Answer: The Common Core standard 4.OA.A.3 specifically addresses interpreting remainders, and the interpretation depends on the context:

  1. Round up: When you need complete groups (like buses for students, rooms for people)
  2. Ignore remainder: When only complete groups matter (like how many full packages)
  3. Use remainder: When the question asks about what's left over
  4. Express as fraction: When dealing with measurements

Examples:

  • "How many buses needed?" → Round up (remainder needs another bus)
  • "How many full bags?" → Ignore remainder
  • "How many left over?" → Use remainder value

Always consider: "What does the remainder mean in this specific real-world situation?"

Question: What does "using place value understanding" mean in the Common Core standards?

Answer: "Using place value understanding" means recognizing that each digit in a number has a specific value based on its position:

  • Breaking apart numbers: 324 = 300 + 20 + 4
  • Multiplying by place value: 24 × 32 = 24 × (30 + 2) = (24 × 30) + (24 × 2)
  • Understanding regrouping: Knowing that 10 ones make 1 ten, 10 tens make 1 hundred, etc.
  • Estimating: Using place value to determine if answers are reasonable

For example, when multiplying 24 × 32, place value understanding helps you think: "24 times 30 is 720, and 24 times 2 is 48, so the total is 768."

This builds number sense and helps you understand WHY the algorithms work, not just HOW to do them.