Solved Exercises on Word Problems Using Multiplication and Division in Grade 3

Master word problems using multiplication and division: real-world applications, problem-solving strategies, and step-by-step solutions through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Multiplication in groups
Exercise 1
Sarah has 4 baskets. Each basket contains 6 apples. How many apples does Sarah have in total?
Definition:

Multiplication: Repeated addition of equal groups. When we have the same number of items in multiple groups, we multiply.

Problem-solving method:
  1. Identify what is being asked (total number of apples)
  2. Find the number of groups (4 baskets)
  3. Find how many items are in each group (6 apples per basket)
  4. Multiply: Number of groups × Items per group
  5. Solve and write the answer in a complete sentence
Groups
4 baskets
Items per group
6 apples
Total
4 × 6 = 24
Step 1: Understand the problem

Sarah has 4 baskets, and each basket has 6 apples. We need to find the total number of apples.

Step 2: Identify the operation

We have 4 equal groups (baskets) with 6 items (apples) in each group. This means we need to multiply.

Step 3: Set up the equation

Number of baskets × Apples per basket = Total apples

4 × 6 = ?

Step 4: Calculate

4 × 6 = 24

Step 5: Write the answer

Sarah has 24 apples in total.

4 × 6 = 24 apples
Final answer:

Sarah has 24 apples in total.

Applied rules:

• Multiplication as repeated addition: 4 × 6 = 6 + 6 + 6 + 6 = 24

• Equal groups: When items are distributed equally among groups, use multiplication

• Problem-solving strategy: Identify groups, items per group, then multiply

2 Division for sharing
Exercise 2
Tom has 24 cookies. He wants to share them equally among 6 friends. How many cookies will each friend get?
Definition:

Division: Splitting a total amount into equal parts. When we share equally, we divide.

Total items
24 cookies
Number of people
6 friends
Each gets
24 ÷ 6 = 4
Step 1: Understand the problem

Tom has 24 cookies and wants to share them equally among 6 friends. We need to find how many cookies each friend gets.

Step 2: Identify the operation

We need to split 24 cookies equally among 6 friends. This means we need to divide.

Step 3: Set up the equation

Total cookies ÷ Number of friends = Cookies per friend

24 ÷ 6 = ?

Step 4: Calculate

24 ÷ 6 = 4

Step 5: Write the answer

Each friend will get 4 cookies.

24 ÷ 6 = 4 cookies per friend
Final answer:

Each friend will get 4 cookies.

Applied rules:

• Division as equal sharing: Splitting a total into equal parts

• Division relationship: 24 ÷ 6 = 4 because 4 × 6 = 24

• Check your work: Multiply the answer by the divisor to get the dividend

3 Multiplication for area
Exercise 3
A rectangular garden has 5 rows of flowers. Each row has 7 flowers. How many flowers are there in the garden?
Definition:

Area model: A rectangle with rows and columns represents multiplication. Rows × Columns = Total items.

Rows
5 rows
Columns
7 flowers
Total
5 × 7 = 35
Step 1: Understand the problem

The garden has 5 rows with 7 flowers in each row. We need to find the total number of flowers.

Step 2: Visualize the problem

Think of a rectangle with 5 rows and 7 columns. Each intersection represents a flower.

Step 3: Identify the operation

We have 5 equal groups (rows) with 7 items (flowers) in each group. This means we need to multiply.

Step 4: Set up the equation

Number of rows × Flowers per row = Total flowers

5 × 7 = ?

Step 5: Calculate

5 × 7 = 35

Step 6: Write the answer

There are 35 flowers in the garden.

5 × 7 = 35 flowers
Final answer:

There are 35 flowers in the garden.

Applied rules:

• Area model: Rectangular arrays represent multiplication

• Commutative property: 5 × 7 = 7 × 5 = 35

• Visualization: Drawing or imagining the array helps solve the problem

Rules and methods, laws,...
\(a \times b = \text{total in } a \text{ groups of } b\)
Multiplication Concept
Multiplication
Groups × Items = Total
Repeated Addition
Division
Total ÷ People = Each Gets
Equal Sharing
Area Model
Rows × Columns = Total
Rectangular Array
Key definitions:

Multiplication: Adding equal groups together (e.g., 3 groups of 4 = 3 × 4 = 12)

Division: Splitting into equal parts (e.g., 12 items shared among 3 people = 12 ÷ 3 = 4 each)

Word problems: Real-life situations that require mathematical operations to solve

Complete methodology:
  1. Read carefully: Understand what the problem is asking
  2. Identify key information: Find numbers and what they represent
  3. Choose the operation: Decide whether to multiply or divide
  4. Set up the equation: Write the mathematical expression
  5. Solve: Calculate the answer
  6. Check: Make sure the answer makes sense
Tip 1: Look for clue words: "each," "per," "groups of" suggest multiplication; "shared equally," "split," "divided" suggest division.
Tip 2: Draw pictures or make arrays to visualize the problem.
Tip 3: Always check your answer by doing the opposite operation.
Tip 4: Write your answer in a complete sentence.
Common errors: Confusing multiplication and division, forgetting units, not reading the question carefully.
Key connections: Multiplication and division are inverse operations. 3 × 4 = 12 and 12 ÷ 4 = 3.
Formulas to know by heart:

• Groups × Items per group = Total items

• Total items ÷ Number of groups = Items per group

• Rows × Columns = Total in rectangular array

• Check: Divisor × Quotient = Dividend

Solution: Exercises 4 to 5
4 Multiplication for comparison
Exercise 4
A store has 8 boxes of crayons. Each box has 5 times as many crayons as a small pack. If a small pack has 3 crayons, how many crayons are in all the boxes?
Definition:

Comparison multiplication: When one quantity is a multiple of another (e.g., 5 times as many).

Small pack
3 crayons
Multiplier
5 times
Per box
3 × 5 = 15
Total
8 × 15 = 120
Step 1: Understand the problem

There are 8 boxes. Each box has 5 times as many crayons as a small pack. A small pack has 3 crayons. Find total crayons in all boxes.

Step 2: Find crayons per box

Each box has 5 times as many crayons as a small pack.

Small pack = 3 crayons

Crayons per box = 5 × 3 = 15 crayons

Step 3: Find total crayons

There are 8 boxes, each with 15 crayons.

Total crayons = 8 × 15 = 120 crayons

Step 4: Verify the result

Check: 120 ÷ 8 = 15 (crayons per box) ✓

And 15 ÷ 3 = 5 (5 times as many as small pack) ✓

8 × (5 × 3) = 8 × 15 = 120 crayons
Final answer:

There are 120 crayons in all the boxes.

Applied rules:

• Comparison multiplication: "Times as many" means multiply

• Order of operations: Solve parentheses first (5 × 3), then multiply by number of boxes

• Multiple steps: Break complex problems into simpler parts

5 Division for grouping
Exercise 5
At a party, there are 36 slices of pizza. If each person eats 4 slices, how many people can be served? If 2 more people come, how many slices will each person get if they share equally?
Definition:

Grouping division: Finding how many equal groups can be made (36 ÷ 4 = 9 groups).

Total slices
36
Slices per person
4
People served
36 ÷ 4 = 9
New people
9 + 2 = 11
New share
36 ÷ 11 ≈ 3.27
Step 1: Find how many people can be served initially

Total slices = 36, Slices per person = 4

People served = 36 ÷ 4 = 9 people

Step 2: Find new number of people

Original people = 9, Additional people = 2

New total people = 9 + 2 = 11 people

Step 3: Find new share per person

Total slices = 36, New number of people = 11

Slices per person = 36 ÷ 11 ≈ 3.27

Since we can't have fractional slices in this context, each person gets 3 slices with 3 left over.

Step 4: Answer both parts of the question

Part 1: 9 people can be served initially.

Part 2: With 11 people, each person gets 3 slices, with 3 slices remaining.

Initial: 36 ÷ 4 = 9 people
With extra people: 36 ÷ 11 = 3 slices each (remainder 3)
Final answer:

Initially, 9 people can be served. If 2 more people come, each of the 11 people will get 3 slices with 3 slices remaining.

Applied rules:

• Grouping division: How many groups of a certain size can be made

• Remainders: In real-world problems, remainders may need interpretation

• Multi-step problems: Solve one part at a time, keeping track of all information

Complete Guide: Laws, Methods, Rules, Definitions
\(\text{Groups} \times \text{Items per group} = \text{Total}\)
Multiplication Model
Key definitions:

Multiplication: Adding equal groups together (e.g., 3 groups of 4 = 3 × 4 = 12)

Division: Splitting into equal parts or finding how many equal groups (e.g., 12 ÷ 3 = 4)

Word problems: Mathematical problems presented in a real-world context

Complete methodology:
  1. Read the problem carefully: Identify what is being asked
  2. Find important numbers: Circle or underline key information
  3. Determine the operation: Look for clue words and situation
  4. Draw a picture: Visualize the problem if needed
  5. Write an equation: Translate words into math
  6. Solve the equation: Calculate the answer
  7. Check your answer: Does it make sense in the context?
Tip 1: Clue words for multiplication: "each," "per," "times," "product," "groups of."
Tip 2: Clue words for division: "shared equally," "divided," "split," "quotient," "how many in each."
Tip 3: Always write your final answer as a complete sentence that addresses the question.
Tip 4: Check your work by using the opposite operation (multiply to check division, divide to check multiplication).
Common errors: Reading too quickly, misidentifying the operation, forgetting units, calculation mistakes.
Key connections: Multiplication and division are inverse operations. They undo each other.
Formulas to know by heart:

• Total items = Number of groups × Items per group

• Items per group = Total items ÷ Number of groups

• Number of groups = Total items ÷ Items per group

• Check: Divisor × Quotient + Remainder = Dividend

Visual Problem-Solving: Multiplication and Division Concepts
Exercise 6: Understanding Relationships
Compare these scenarios:
Scenario A: 4 bags with 6 candies each
Scenario B: 24 candies divided among 4 friends
Scenario C: A 5×7 rectangular grid

Analysis: These scenarios demonstrate different applications of multiplication and division.

  • Scenario A: 4 × 6 = 24 (multiplication - finding total)
  • Scenario B: 24 ÷ 4 = 6 (division - sharing equally)
  • Scenario C: 5 × 7 = 35 (area model - rows × columns)

Questions & Answers

Question: I sometimes get confused about when to multiply and when to divide in word problems. How can I tell which one to use?

Answer: Great question! Here's how to tell the difference:

  • Use multiplication when: You have equal groups and want to find the total. Look for phrases like "each," "per," "times," "groups of," or "rows of."
  • Use division when: You have a total amount and want to share it equally or find how many groups you can make. Look for phrases like "shared equally," "divided among," "split between," or "how many in each."

Think of it this way: Multiplication makes things bigger (combining groups), while division makes things smaller (splitting apart).

Example: "There are 5 boxes with 4 apples each" → 5 × 4 = 20 apples (multiplication)
Example: "20 apples shared among 5 friends" → 20 ÷ 5 = 4 apples each (division)

Question: When I draw pictures to help solve multiplication problems, should I always make arrays or can I just draw circles?

Answer: Both methods work well! The choice depends on what helps you visualize the problem best:

  • Arrays: Great for showing rows and columns (like the garden problem). They help you see the structure of multiplication.
  • Groups of circles: Good for showing separate equal groups (like bags of apples). Each circle represents an item.
  • Tally marks: Useful for smaller numbers, where you can group them easily.

The key is that your drawing clearly shows the equal groups. For example, if you have 3 bags with 4 apples each, draw 3 circles with 4 dots inside each circle, OR draw a 3×4 array.

Try both methods and see which one helps you understand the problem better!

Question: What should I do when I get a decimal answer in a division word problem, like 36 ÷ 5 = 7.2?

Answer: This is an excellent observation! When you get a decimal in a word problem, think about what makes sense in the real world:

  • Context matters: If you're dividing 36 cookies among 5 people, each person gets 7 whole cookies with 1 cookie left over (since 5 × 7 = 35, remainder 1).
  • Remainders: In many problems, you'll express the answer as "7 cookies each with 1 left over" or "7 R1".
  • Fractional parts: Sometimes you can divide the remainder equally (7.2 cookies each), but this isn't always practical.

Always consider what the problem is asking and what makes sense in that situation. If you're sharing physical objects, you often deal with whole numbers and remainders.

You can also check: 7.2 × 5 = 36, confirming your division was correct!

Question: How can I check if my answer to a word problem is correct?

Answer: There are several ways to check your answer:

  1. Use the opposite operation: If you multiplied, divide to check. If you divided, multiply to check.
  2. Estimate first: Round numbers to get an approximate answer. Your exact answer should be close to your estimate.
  3. Plug back in: Put your answer back into the original problem to see if it works.
  4. Does it make sense?: Ask yourself if your answer is reasonable given the problem situation.

Example: If you found that 4 boxes with 6 apples each equals 24 apples, check: 24 ÷ 4 = 6 ✓ or 24 ÷ 6 = 4 ✓

This verification step helps catch errors and builds confidence in your answers!

Question: Are there any tricks to remember multiplication facts faster?

Answer: Yes, here are some helpful strategies:

  • Patterns: Any number × 10 ends in 0, any number × 5 ends in 0 or 5
  • Commutative property: 4 × 6 = 6 × 4, so you only need to learn half the facts
  • Double doubles: For ×4, double the number twice (7 × 4: double 7 is 14, double 14 is 28)
  • Add a group: For 6 × 7, think 5 × 7 = 35, then add one more group: 35 + 7 = 42
  • Skip counting: Practice counting by 2s, 3s, 4s, etc.

Practice regularly with flashcards, games, or worksheets. The more you practice, the more automatic these facts become!

Remember, it's okay to use strategies at first - eventually, you'll memorize them naturally!