Multi-Step Problem Solving Strategy

Complete guide with exercises on determining what to solve for first in multi-step problems: step-by-step solutions and detailed explanations.

Problem Solving Strategy 1 to 3
1 Identifying the First Step
Problem 1
"Sarah has 24 stickers. She gets 15 more stickers from her friend. Then she gives 8 stickers to her sister. How many stickers does Sarah have now? What will you solve for first?"
Definition:

First step identification: Determining the initial calculation needed to solve a multi-step problem

Solution method:
  1. Read the problem completely to understand the sequence of events
  2. Identify what happens first (gets 15 stickers)
  3. Calculate the result of the first action (24 + 15 = 39)
  4. Continue with subsequent steps using this result
First step
24 + 15
Second step
- 8
Result
31
Step 1: Identify the first action

Sarah starts with 24 stickers and gets 15 more

First calculation: 24 + 15 = 39 stickers

Step 2: Continue with next action

Now she has 39 stickers and gives away 8

Next calculation: 39 - 8 = 31 stickers

Step 3: State the final answer

Sarah has 31 stickers now

Step 4: Verify the answer

Check: 31 + 8 = 39, and 39 - 15 = 24 ✓

First solve: 24 + 15 = 39, then 39 - 8 = 31
Final answer:

You should solve for 24 + 15 = 39 first. Sarah has 31 stickers now.

Applied rules:

• Sequence identification: Follow the order of events in the problem

• Step-by-step approach: Solve one operation at a time

• Context preservation: Keep track of the changing quantities

2 Shopping Problem First Step
Problem 2
"A store has 120 apples in 6 boxes. They sell 2 boxes of apples. Then they buy 3 times as many apples as they sold. What should you solve for first?"
Definition:

Multi-step problem sequence: Identifying the foundational calculation needed to solve subsequent parts

First step
120 ÷ 6
Second step
× 2
Third step
× 3
Result
180
Step 1: Identify the first calculation needed

First, find how many apples are in each box: 120 ÷ 6 = 20 apples per box

Step 2: Calculate apples sold

They sold 2 boxes: 2 × 20 = 40 apples sold

Step 3: Calculate apples bought

They bought 3 times as many as sold: 3 × 40 = 120 apples bought

Step 4: Find final count

Started with 120, sold 40, bought 120: 120 - 40 + 120 = 200 apples

First solve: 120 ÷ 6 = 20 (apples per box)
Final answer:

You should solve for 120 ÷ 6 = 20 first (apples per box). The store now has 200 apples.

Applied rules:

• Foundational calculation: Find the unit rate first (per box)

• Sequential dependency: Later steps depend on earlier results

• Multiplicative reasoning: Use the unit rate to find totals

3 Weight Problem First Step
Problem 3
"A truck weighs 4 tons when empty. It carries 3 loads of 1,200 pounds each. Then it unloads 2,000 pounds. What should you solve for first?"
Definition:

Unit conversion priority: Identifying when to convert units before performing calculations

First step
4 tons to lbs
Second step
3 × 1,200
Third step
- 2,000
Result
8,600
Step 1: Convert units to match

Convert 4 tons to pounds: 4 × 2,000 = 8,000 pounds

Step 2: Calculate cargo loaded

3 loads × 1,200 pounds each = 3,600 pounds

Step 3: Calculate total weight

Empty truck + cargo: 8,000 + 3,600 = 11,600 pounds

Step 4: Subtract unloaded weight

11,600 - 2,000 = 9,600 pounds

First solve: 4 × 2,000 = 8,000 (convert tons to pounds)
Final answer:

You should solve for 4 × 2,000 = 8,000 first (convert tons to pounds). The truck now weighs 9,600 pounds.

Applied rules:

• Unit consistency: Convert to matching units before adding/subtracting

• Foundation first: Establish base measurement system

• Sequential operations: Follow the order of operations and events

Problem Analysis Methods
Step Identification Process 🧠
Step 1: Read Carefully
Understand the sequence of events and what's being asked.
Step 2: Identify Dependencies
Find which calculations must be done before others.
Step 3: Find Foundation
Locate the calculation that other steps depend on.
Step 4: Verify Sequence
Ensure the order makes logical sense.
Dependency Recognition 📊
Find Unit Rate First
Boxes → Items per box
Convert Units First
Different units → Same units
Calculate Totals First
Groups → Individual amounts
Solve Known Values First
Known → Unknown
Visual Problem Sequence 📐
1
Identify
→
2
Plan
→
3
Solve
→
4
Check
120 ÷ 6
Per Box
→
× 2
Sold
→
× 3
Bought
→
200
Total
Advanced Problem Sequence Analysis 4 to 6
4 Multi-Operation First Step
Problem 4
"A baker made 180 cupcakes. She packed 8 boxes with 12 cupcakes each. Then she divided the remaining cupcakes equally among 6 baskets. What should you solve for first?"
Definition:

Multi-operation sequence: Identifying the foundational calculation in problems with multiple operations

Solution method:
  1. First, calculate how many cupcakes were packed in boxes: 8 × 12
  2. Then, find remaining cupcakes: 180 - (8 × 12)
  3. Finally, divide remaining among baskets: Remaining ÷ 6
  4. The first step is the multiplication
First step
8 × 12
Second step
180 - 96
Third step
÷ 6
Result
14
Step 1: Calculate cupcakes in boxes

8 boxes × 12 cupcakes per box = 96 cupcakes in boxes

Step 2: Find remaining cupcakes

180 - 96 = 84 cupcakes remaining

Step 3: Divide among baskets

84 ÷ 6 = 14 cupcakes per basket

Step 4: Interpret result

Each basket gets 14 cupcakes

First solve: 8 × 12 = 96 (cupcakes in boxes)
Final answer:

You should solve for 8 × 12 = 96 first (cupcakes in boxes). Each basket gets 14 cupcakes.

Applied rules:

• Foundational operation: Perform multiplication before subtraction

• Dependency chain: Later steps depend on earlier results

• Context awareness: Understand the real-world sequence

5 Financial Problem First Step
Problem 5
"Emma earned $125 from her job. Her brother earned $48 less than Emma. Their sister earned twice as much as their brother. What should you solve for first?"
Definition:

Financial sequence analysis: Determining the order of calculations in money-related problems

First step
125 - 48
Second step
× 2
Third step
Add all
Result
379
Step 1: Calculate brother's earnings

Emma's earnings - $48 = $125 - $48 = $77

Step 2: Calculate sister's earnings

2 × brother's earnings = 2 × $77 = $154

Step 3: Calculate total earnings

$125 + $77 + $154 = $356

Step 4: Identify first step

The first step is calculating the brother's earnings

First solve: 125 - 48 = 77 (brother's earnings)
Final answer:

You should solve for 125 - 48 = 77 first (brother's earnings). Total family earnings are $356.

Applied rules:

• Dependency analysis: Sister's earnings depend on brother's earnings

• Sequence logic: Solve for intermediate values first

• Arithmetic operations: Follow order based on relationships

6 Measurement Problem First Step
Problem 6
"A piece of ribbon is 8 feet long. Sarah cuts off 15 inches. Then she attaches 2 pieces of ribbon, each 8 inches long. What should you solve for first?"
Definition:

Measurement conversion priority: Identifying when unit conversion is the first necessary step

First step
8 ft to inches
Second step
- 15
Third step
+ (2 × 8)
Result
96 + 16 - 15 = 97
Step 1: Convert to matching units

8 feet × 12 inches per foot = 96 inches

Step 2: Subtract cut piece

96 - 15 = 81 inches

Step 3: Add attached pieces

81 + (2 × 8) = 81 + 16 = 97 inches

Step 4: Interpret result

The ribbon is now 97 inches long

First solve: 8 × 12 = 96 (convert feet to inches)
Final answer:

You should solve for 8 × 12 = 96 first (convert feet to inches). The ribbon is now 97 inches long.

Applied rules:

• Unit consistency: Convert to same units before operating

• Foundation first: Establish measurement baseline

• Sequential operations: Follow the order of events

Detailed Summary & Study Guide
Multi-Step Problem Solving Strategy - Complete Summary
Definition of Multi-Step Problems:
Multi-step problems are mathematical problems that require more than one operation to solve. They involve a sequence of calculations where the result of one step becomes the input for the next step. Identifying the correct first step is crucial for solving these problems efficiently.
First Step Identification:
The process of determining which calculation must be performed first in a multi-step problem. This often involves finding foundational information that other calculations depend on, such as unit conversions, rate calculations, or intermediate values.
Core Identification Rules:

1. Read Completely: Understand the full sequence of events before deciding where to start

2. Look for Dependencies: Identify which calculations require results from other calculations

3. Find Foundations: Locate the base information needed for subsequent steps

4. Check Units: Identify if unit conversion is necessary before other operations

Common First Step Categories:

Unit Conversions: When problems mix different measurement units

Rate Calculations: Finding per-unit amounts (per box, per hour, per person)

Intermediate Values: Calculating amounts needed for later operations

Known Quantities: Establishing baseline values before transformations

Simple First Step Example:

Problem: "Sarah has 10 apples, gets 5 more, then gives 3 away."

First step: 10 + 5 = 15 (calculate total after receiving more)

Next step: 15 - 3 = 12 (subtract given away)

Complex First Step Example:

Problem: "A store has 200 items in 10 boxes, sells 3 boxes, buys 2 times as many as sold."

First step: 200 ÷ 10 = 20 (items per box)

Next steps depend on knowing 20 items per box

Key Tips for Success:

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

2. Underline Actions: Mark the operations that occur in sequence

3. Sketch the Flow: Draw arrows showing how calculations connect

4. Ask "What do I need to know first?": Identify the foundational information

5. Check Dependencies: Verify that later steps can use earlier results

Common Mistakes to Avoid:

1. Jumping Ahead: Starting with later steps instead of foundational ones

2. Ignoring Units: Not converting to matching units first

3. Misreading Sequence: Performing operations in the wrong order

4. Missing Dependencies: Not recognizing that some calculations depend on others

5. No Verification: Not checking if the sequence makes logical sense

Problem-Solving Strategy:

1. Read Thoroughly: Understand the entire problem and sequence of events

2. Identify the Goal: Determine what the problem is asking for

3. Map Dependencies: Find which calculations rely on others

4. Locate Foundation: Identify the calculation that starts the chain

5. Solve Sequentially: Work through steps in the correct order

Verification Techniques:

Logical Check: Does the first step make sense in the problem context?

Dependency Check: Can subsequent steps use the result of the first step?

Unit Check: Are all measurements in compatible units?

Completeness Check: Does the first step provide the needed foundation?

Questions & Answers

Question: How do I know which calculation to do first in a multi-step problem?

Answer: Look for the calculation that provides information needed for other steps. Usually, this involves:

• Finding unit rates (per box, per hour, per person)

• Converting units to match (feet to inches, pounds to ounces)

• Calculating intermediate values that other operations depend on

Ask yourself: "What do I need to know before I can do the next step?" That's usually your first step!

Question: What if I start with the wrong step? Will I still get the right answer?

Answer: If you start with the wrong step, you likely won't get the right answer. Multi-step problems have a logical sequence, and starting in the middle often leads to incorrect results.

However, recognizing that your approach isn't working is a valuable skill! If your answer doesn't make sense or you can't continue, go back and reconsider which calculation should come first.

Always read the entire problem to understand the sequence before starting your calculations.

Question: What if there are two things I could solve for first?

Answer: Sometimes there are multiple valid ways to approach a problem! Both paths may lead to the correct answer.

However, one path is usually more efficient. Look for the approach that requires fewer steps or simpler calculations. The "best" first step is the one that makes the rest of the problem easier to solve.

If you're unsure, start with the calculation that seems most straightforward or that converts units to match.