Solving Multi-Step Problems

Complete guide with exercises on solving multi-step problems: step-by-step solutions and detailed explanations.

Multi-Step Problems 1 to 3
1 Addition and Subtraction Multi-Step
Problem 1
Sarah has 45 stickers. She gets 23 more stickers from her friend. Then she gives 18 stickers to her sister. How many stickers does Sarah have now?
Definition:

Multi-step problem: A problem requiring multiple operations to solve (addition, then subtraction)

Solution method:
  1. Identify what happens first (gets 23 stickers)
  2. Calculate after first action (45 + 23 = 68)
  3. Calculate after second action (68 - 18 = 50)
  4. State the answer in context
Start
45
Add
+ 23
Subtract
- 18
Result
50
Step 1: Initial stickers

Sarah starts with 45 stickers

Step 2: Gets more stickers

45 + 23 = 68 stickers

Step 3: Gives some away

68 - 18 = 50 stickers

Step 4: Interpret result

Sarah has 50 stickers now

Sarah has 50 stickers now
Final answer:

Sarah has 50 stickers now.

Applied rules:

• Order of operations: Follow the sequence of events in the problem

• Context: Answer must relate back to the problem situation

• Verification: Check: 45 + 23 - 18 = 50 ✓

2 Multiplication and Addition Multi-Step
Problem 2
A store has 6 boxes of pencils. Each box contains 15 pencils. The store manager adds 25 more pencils to the stock. How many pencils are there altogether?
Definition:

Multi-step problem: A problem requiring multiple operations (multiplication, then addition)

Boxes
6 × 15
Add
+ 25
Result
115
Step 1: Calculate pencils in boxes

6 boxes × 15 pencils per box = 90 pencils

Step 2: Add additional pencils

90 + 25 = 115 pencils

Step 3: Interpret result

There are 115 pencils altogether

115 pencils altogether
Final answer:

There are 115 pencils altogether.

Applied rules:

• Order of operations: Perform multiplication before addition

• Context: Answer must relate back to the problem situation

• Verification: Check: (6 × 15) + 25 = 90 + 25 = 115 ✓

3 Division and Subtraction Multi-Step
Problem 3
A teacher has 48 crayons. She divides them equally among 6 students. Then she takes 2 crayons back for herself. How many crayons does each student get?
Definition:

Multi-step problem: A problem requiring multiple operations (division, then subtraction)

Divide
48 ÷ 6
Each gets
8
Take back
- 2
Result
6
Step 1: Divide crayons among students

48 ÷ 6 = 8 crayons per student

Step 2: Teacher takes some back

Each student loses 2 crayons: 8 - 2 = 6 crayons

Step 3: Interpret result

Each student gets 6 crayons

Each student gets 6 crayons
Final answer:

Each student gets 6 crayons.

Applied rules:

• Order of operations: Perform division before subtraction

• Context: Answer must relate back to the problem situation

• Verification: Check: (48 ÷ 6) - 2 = 8 - 2 = 6 ✓

Multi-Step Problem Solving Strategies
Problem Solving Steps 🧠
Step 1: Read Carefully
Understand what the problem is asking and identify key information.
Step 2: Plan Your Approach
Decide what operations to perform and in what order.
Step 3: Solve Step-by-Step
Work through each step carefully, keeping track of your work.
Step 4: Check Your Answer
Verify your answer makes sense in the context of the problem.
Order of Operations 🔄
Parentheses
First
Multiplication/Division
Left to Right
Addition/Subtraction
Left to Right
Visual Problem Solving 📐
1
Read
→
2
Plan
→
3
Solve
→
4
Check
45 + 23
First Step
→
68 - 18
Second Step
→
50
Answer
Advanced Multi-Step Problems 4 to 6
4 Multi-Step Fruit Problem
Problem 4
A farmer harvested 180 apples. He packed 8 boxes with 12 apples each. Then he divided the remaining apples equally among 6 baskets. How many apples were in each basket?
Definition:

Multi-step problem: A problem requiring multiple operations including division

Solution method:
  1. Calculate apples packed in boxes: 8 × 12
  2. Calculate remaining apples: 180 - (8 × 12)
  3. Divide remaining apples among baskets: Remaining ÷ 6
  4. State the answer in context
Total
180
Boxes
- 96
Baskets
÷ 6
Result
14
Step 1: Calculate apples in boxes

8 boxes × 12 apples each = 96 apples

Step 2: Calculate remaining apples

180 - 96 = 84 apples remaining

Step 3: Divide among baskets

84 ÷ 6 baskets = 14 apples per basket

Step 4: Interpret result

Each basket has 14 apples

Each basket has 14 apples
Final answer:

Each basket has 14 apples.

Applied rules:

• Order of operations: Perform multiplication before subtraction

• Division: Distribute remaining items equally

• Verification: Check: (8 × 12) + (6 × 14) = 96 + 84 = 180 ✓

5 Multi-Step Library Problem
Problem 5
A library has 15 shelves. Each shelf holds 24 books. The librarian removes 18 books from each shelf to repair them. How many books are left on all the shelves combined?
Definition:

Multi-step problem: A problem involving operations across multiple identical groups

Total books
15 × 24
Removed
- (15 × 18)
Result
90
Step 1: Calculate total books initially

15 shelves × 24 books per shelf = 360 books

Step 2: Calculate books removed

15 shelves × 18 books per shelf = 270 books removed

Step 3: Calculate remaining books

360 - 270 = 90 books

Step 4: Interpret result

90 books are left on all shelves

90 books left on all shelves
Final answer:

There are 90 books left on all the shelves combined.

Applied rules:

• Group operations: Apply operations to each group separately

• Subtraction: Subtract total removed from total initial

• Verification: Check: 90 + 270 = 360, 360 ÷ 15 = 24 ✓

6 Multi-Step Bakery Problem
Problem 6
A bakery made 210 cookies. They put 6 cookies in each bag. Then they arranged the bags into boxes, with 5 bags in each box. How many boxes did they fill completely?
Definition:

Multi-step problem: A problem requiring division in multiple steps

Cookies to bags
210 ÷ 6
Bags to boxes
÷ 5
Result
7
Step 1: Calculate number of bags

210 cookies ÷ 6 cookies per bag = 35 bags

Step 2: Calculate number of boxes

35 bags ÷ 5 bags per box = 7 boxes

Step 3: Interpret result

They filled 7 boxes completely

7 boxes filled completely
Final answer:

They filled 7 boxes completely.

Applied rules:

• Sequential division: Divide step by step through the process

• Context: Answer must relate back to the problem situation

• Verification: Check: 7 × 5 × 6 = 210 ✓

Detailed Summary & Study Guide
Solving Multi-Step Problems - Complete Summary
Definition of Multi-Step Problems:
Multi-step problems are mathematical problems that require more than one operation to solve. They often involve a sequence of operations like multiplication followed by addition, or division followed by subtraction. These problems require careful reading and planning to solve correctly.
Problem-Solving Process:
Solving multi-step problems involves a systematic approach: reading the problem carefully, identifying the operations needed, determining the order of operations, solving step by step, and checking the answer for reasonableness.
Core Problem-Solving Rules:

1. Read Carefully: Understand what the problem is asking and identify all given information

2. Plan Your Approach: Determine the sequence of operations needed to solve the problem

3. Solve Step-by-Step: Work through each operation carefully, keeping track of intermediate results

4. Check Your Answer: Verify that your answer makes sense in the context of the problem

Order of Operations:

Parentheses → Exponents → Multiplication/Division → Addition/Subtraction

When solving multi-step problems, follow this order to ensure accuracy. Operations of the same precedence are performed from left to right.

Simple Multi-Step Example:

Problem: "Sarah has 3 bags with 10 apples each. She gives 15 apples to her friend. How many apples does she have left?"

Solution: (3 × 10) - 15 = 30 - 15 = 15 apples

Complex Multi-Step Example:

Problem: "A store has 200 books. They sell 4 boxes of 12 books each, then divide the remaining books into 8 equal groups. How many books are in each group?"

Solution: (200 - (4 × 12)) ÷ 8 = (200 - 48) ÷ 8 = 152 ÷ 8 = 19 books per group

Key Tips for Success:

1. Circle Key Numbers: Highlight important numbers in the problem

2. Underline Action Words: Identify words like "total," "remaining," "each," "per" to determine operations

3. Sketch Diagrams: Draw pictures to visualize the problem when possible

4. Work Neatly: Keep your calculations organized to avoid errors

5. Estimate First: Make a rough estimate to check if your answer is reasonable

Common Mistakes to Avoid:

1. Rushing Through: Not reading the problem carefully enough

2. Wrong Order: Performing operations in the wrong sequence

3. Misinterpreting Words: Confusing "times more" with "times as many"

4. Calculation Errors: Making arithmetic mistakes in multi-step processes

5. No Verification: Not checking if the answer makes sense

Problem-Solving Strategy:

1. Read the Problem Twice: First for general understanding, second to identify details

2. Identify What You Need to Find: Clearly understand the question

3. Plan the Steps: Write down the operations needed in order

4. Solve Systematically: Complete each step before moving to the next

5. Review Your Answer: Check against the original problem

Verification Techniques:

Work Backwards: Use your answer to recreate the original conditions

Estimate Check: Compare your answer to your initial estimate

Alternative Method: Try solving using a different approach

Reasonableness Test: Ask if the answer makes sense in the real world

Questions & Answers

Question: How do I know what order to do the operations in a multi-step problem?

Answer: Follow the order of operations (PEMDAS/BODMAS): Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). In word problems, think about what makes logical sense.

For example, if you need to find how many items are in each group, you usually multiply first to find the total, then divide. Read the problem carefully to understand the sequence of events.

Always work through the problem step by step, and check if your answer makes sense.

Question: How can I help my child practice multi-step problems effectively?

Answer: Here are effective practice strategies:

  • Start with simple problems and gradually increase difficulty
  • Encourage your child to write down each step clearly
  • Remind them to always check their answer by substituting back
  • Provide positive feedback for showing work, even if the answer is incorrect
  • Practice regularly in short sessions to build confidence

Focus on building a strong foundation with basic operations before moving to complex equations.

Question: What if I make a mistake while solving a multi-step problem?

Answer: Mistakes are a normal part of learning! Here's what to do:

1. Don't panic - everyone makes mistakes

2. Use your verification step to catch errors early

3. If your answer doesn't check out, go back and review each step

4. Work the problem again from the beginning

5. Ask for help if you're stuck

The verification step (substituting your answer back into the original equation) is your best tool for catching mistakes.