Multi-Step Fraction Word Problems with Detailed Solutions

Master multi-step fraction word problems through comprehensive examples, rules, and step-by-step solutions for 4th grade students.

Solution: Exercises 1 to 3
1 Recipe Problem
Exercise 1
A recipe calls for 3/4 cup of sugar. Maya wants to make half of the recipe, then she decides to make 2/3 of that reduced amount. How much sugar does Maya need?
Definition:

Multi-step problem: A problem requiring multiple operations to solve, often involving different fraction operations in sequence.

Multi-step problem-solving method:
  1. Read the entire problem carefully
  2. Identify the operations needed in order
  3. Solve step by step, keeping track of results
  4. Check if the final answer makes sense
Step 1
Half of 3/4
Step 2
2/3 of Step 1
Result
1/2 × 3/4 × 2/3
Step 1: Find half of the recipe

Half of 3/4 cup = 1/2 × 3/4 = 3/8 cup

Step 2: Find 2/3 of the reduced amount

2/3 of 3/8 cup = 2/3 × 3/8 = 6/24 = 1/4 cup

Alternative: Multiply all fractions at once

1/2 × 3/4 × 2/3 = (1×3×2)/(2×4×3) = 6/24 = 1/4 cup

Maya needs 1/4 cup of sugar
Final answer:

Maya needs 1/4 cup of sugar

Applied rules:

• Order of operations: Solve operations in the sequence described

• Fraction multiplication: Multiply numerators and denominators separately

• Simplification: Reduce fractions to lowest terms

2 Reading Problem
Exercise 2
Tom read 2/3 of his book on Monday. On Tuesday, he read 1/4 of what was left. What fraction of the book did Tom read in total?
Definition:

Fraction of remaining: To find what's left, subtract the used fraction from the whole (1).

Monday
2/3 read
Remaining
1 - 2/3 = 1/3
Tuesday
1/4 of 1/3 = 1/12
Total
2/3 + 1/12 = 9/12 = 3/4
Step 1: Find what remained after Monday

Total book = 1, Read on Monday = 2/3, Remaining = 1 - 2/3 = 3/3 - 2/3 = 1/3

Step 2: Find what Tom read on Tuesday

1/4 of remaining 1/3 = 1/4 × 1/3 = 1/12

Step 3: Find the total amount read

Monday + Tuesday = 2/3 + 1/12

Convert to common denominator: 2/3 = 8/12

8/12 + 1/12 = 9/12 = 3/4

Tom read 3/4 of the book in total
Final answer:

Tom read 3/4 of the book in total

Applied rules:

• Finding remaining: Whole - Used = Remaining

• Fraction addition: Need common denominators

• Order of operations: Subtraction first, then multiplication, then addition

3 Art Project Problem
Exercise 3
A rectangular canvas is 5/6 feet wide. The length is 3/4 of the width. Then, a frame that is 1/8 foot wide is added around the canvas. What is the total area including the frame?
Definition:

Rectangle area: Length × Width. When adding a frame, dimensions increase by twice the frame width (top and bottom, or left and right).

Width
5/6 ft
Length
3/4 × 5/6 = 15/24 = 5/8 ft
With frame
(5/6 + 1/4) × (5/8 + 1/4)
Area
4/3 × 7/8 = 7/6 ft²
Step 1: Find the length of the canvas

Length = 3/4 of width = 3/4 × 5/6 = 15/24 = 5/8 feet

Step 2: Find new dimensions with frame

Frame adds 1/8 foot on each side, so add 2 × 1/8 = 1/4 foot to each dimension

New width = 5/6 + 1/4 = 10/12 + 3/12 = 13/12 feet

New length = 5/8 + 1/4 = 5/8 + 2/8 = 7/8 feet

Step 3: Calculate the total area

Area = 13/12 × 7/8 = 91/96 square feet

Total area = 91/96 square feet
Final answer:

The total area including the frame is 91/96 square feet

Applied rules:

• Dimension increase: Frame adds to both sides, so add twice the frame width

• Fraction addition: Need common denominators

• Area calculation: Length × Width

Multi-Step Fraction Problem Solving Methods
\(\text{Step 1} \rightarrow \text{Step 2} \rightarrow \text{Step 3} \rightarrow \text{Final Answer}\)
Multi-Step Process
Method 1
\(\text{Sequential solving}\)
Solve one step at a time
Method 2
\(\text{Combined operations}\)
Write all operations at once
Method 3
\(\text{Visual modeling}\)
Draw diagrams for each step
Key definitions:

Multi-step problem: A problem requiring more than one mathematical operation to solve

Sequential operations: Operations performed in a specific order to reach the final answer

Intermediate result: The answer from one step that becomes input for the next step

Complete methodology:
  1. Read thoroughly: Understand what the problem is asking
  2. Identify steps: Break the problem into smaller operations
  3. Solve sequentially: Work through each step one at a time
  4. Track results: Keep intermediate answers organized
  5. Combine appropriately: Use previous results in subsequent steps
  6. Check reasonableness: Verify if the final answer makes sense
Tip 1: Underline or highlight key information in each step.
Tip 2: Draw pictures or models to visualize each step.
Tip 3: Always keep track of units (cups, feet, etc.) in multi-step problems.
Tip 4: Check that each intermediate answer is reasonable before proceeding.
Common errors: Performing operations out of order, forgetting to update values for subsequent steps, losing track of units.
Success strategies: Organize work clearly, label each step, double-check calculations, verify final answer makes sense.
Solution: Exercises 4 to 5
4 Juice Problem
Exercise 4
A pitcher contains 3/4 gallon of juice. Lisa pours out 1/3 of the juice, then her sister adds 1/6 gallon more juice. How much juice is now in the pitcher?
Definition:

Fraction subtraction: To subtract fractions, they must have common denominators.

Initial
3/4 gal
After pouring
3/4 - 1/3 of 3/4
After adding
Result + 1/6 gal
Step 1: Find how much Lisa poured out

1/3 of 3/4 gallon = 1/3 × 3/4 = 3/12 = 1/4 gallon poured out

Step 2: Find how much juice remains

3/4 - 1/4 = 2/4 = 1/2 gallon remaining

Step 3: Add the juice sister added

1/2 + 1/6 = 3/6 + 1/6 = 4/6 = 2/3 gallon

There are 2/3 gallons of juice in the pitcher
Final answer:

There are 2/3 gallons of juice in the pitcher

Applied rules:

• Fraction multiplication: To find part of an amount

• Fraction subtraction: Subtract amounts with common denominators

• Fraction addition: Add amounts with common denominators

5 Running Problem
Exercise 5
Jake ran 2/3 of a mile on Monday. On Tuesday, he ran 3/4 of Monday's distance. On Wednesday, he increased Tuesday's distance by 1/6 mile. How far did Jake run on Wednesday?
Definition:

Fraction addition with whole numbers: Convert whole numbers to fractions with same denominator as fractional part.

Monday
2/3 mile
Tuesday
3/4 × 2/3 = 1/2 mile
Wednesday
1/2 + 1/6 = 2/3 mile
Step 1: Find Tuesday's distance

3/4 of Monday's distance = 3/4 × 2/3 = 6/12 = 1/2 mile

Step 2: Find Wednesday's distance

Tuesday's distance + 1/6 mile = 1/2 + 1/6

Convert to common denominator: 1/2 = 3/6

3/6 + 1/6 = 4/6 = 2/3 mile

Step 3: Verify the answer

Check: 2/3 mile is more than Tuesday's 1/2 mile, which makes sense since 1/6 mile was added

Jake ran 2/3 mile on Wednesday
Final answer:

Jake ran 2/3 mile on Wednesday

Applied rules:

• Sequential multiplication: Find part of a fractional amount

• Fraction addition: Add fractions with common denominators

• Reasonableness check: Verify that the answer makes logical sense

Multi-Step Fraction Word Problems Summary
\(\text{Problem Analysis} \rightarrow \text{Step Identification} \rightarrow \text{Sequential Calculation} \rightarrow \text{Final Answer}\)
Problem-Solving Process
Key definitions:

Multi-step problem: A complex problem requiring multiple operations to solve, where the result of one step feeds into the next step.

Intermediate result: The answer from one step that serves as input for the following step.

Sequential operations: Mathematical operations performed in a specific order to reach the final answer.

Complete methodology:
  1. Read and analyze: Carefully read the entire problem to understand what's being asked
  2. Identify the steps: Break the problem into distinct mathematical operations
  3. Solve step by step: Perform each operation in the correct order
  4. Track results: Keep intermediate answers organized and accurate
  5. Combine appropriately: Use results from previous steps in subsequent calculations
  6. Check reasonableness: Verify that the final answer makes sense in context
Tip 1: Number each step clearly to keep track of your progress.
Tip 2: Circle or box your intermediate answers to easily reference them later.
Tip 3: Use scratch paper to work out each step separately if needed.
Tip 4: Always include units in your calculations and final answer.
Common errors: Performing operations out of order, miscalculating intermediate steps, failing to convert fractions to common denominators when adding/subtracting.
Success strategies: Organize work clearly, double-check each step, verify the final answer makes sense in the problem context.
Key rules and principles:

• Order matters: Perform operations in the sequence described in the problem

• Intermediate results: Use answers from previous steps as inputs for subsequent steps

• Common denominators: Required for adding or subtracting fractions

• Unit consistency: Keep track of units throughout all steps

• Verification: Check that each step produces a reasonable result

Multi-Step Fraction Problem Guide

📊
Problem-Solving Steps

1. Read entire problem

2. Identify operations needed

3. Solve step by step

4. Track intermediate results

5. Combine results appropriately

Operation Keywords

"of" → multiplication

"more than", "added" → addition

"less than", "remaining" → subtraction

"times", "product" → multiplication

Multi-Step Workflow
1
Analyze Problem
2
Plan Steps
3
Execute Step 1
4
Execute Step 2
5
Final Answer
Example Structure

Start with initial amount → Apply first operation → Get intermediate result → Apply second operation → Get final answer

Each step uses the result from the previous step

📚 Practice Sequential Thinking!

Questions & Answers

Question: I get confused about which operation to do first in multi-step problems. How do I figure out the order?

Answer: Great question! In multi-step fraction word problems, the order is usually indicated by the sequence of events in the story:

  • Read the problem from beginning to end to understand the sequence of events
  • Look for time-order words like "first", "then", "after that", "finally"
  • Follow the same order the events happen in the story

For example: "Maya used 1/2 of the paint, then she added 1/4 more" - first multiply (find 1/2 of the total), then add 1/4.

Always solve in the same order the actions occur in the problem description.

Question: My child struggles to keep track of the intermediate answers. Any suggestions?

Answer: Organization is key for multi-step problems! Here are some helpful strategies:

  • Label each step clearly: "Step 1:", "Step 2:", etc.
  • Box or circle intermediate answers so they're easy to spot
  • Use scratch paper to work out each step separately
  • Write the result of each step in the margin to reference later

For example:
Step 1: Find 1/2 of 3/4 = 3/8
Step 2: Add 1/4 to 3/8 = 5/8

This keeps the work organized and reduces errors from losing track of numbers.

Question: What if I make a mistake in Step 1? Will that mess up the whole problem?

Answer: Yes, errors in early steps do affect later steps because each step builds on the previous one. That's why checking your work is important!

  • Double-check each step before moving to the next
  • Make sure your intermediate answers make sense
  • If the final answer seems wrong, go back and check earlier steps
  • Use estimation to verify reasonableness (e.g., 1/2 of 3/4 should be close to 1/2 of 1 = 1/2)

If you realize you made an error, it's okay to start over from the point of the mistake. Learning to catch and fix errors is an important skill!