Solving Multi-Step Problems with Fractions and Mixed Numbers

Master multi-step problems involving fractions and mixed numbers through 3 detailed exercises with step-by-step solutions.

Solutions: Multi-Step Problems with Fractions
1 Pizza Party Problem
Exercise 1
At a party, there were 2 pizzas. Each pizza was cut into 4 equal slices. After the party, 3 slices were eaten. What fraction of a pizza was left?
Definition:

Fraction: A number that represents part of a whole. A fraction has a numerator (top number) and a denominator (bottom number).

Mixed number: A number made up of a whole number and a fraction (like 1 1/2).

Step-by-step method:
  1. Find the total number of slices (multiply pizzas by slices per pizza)
  2. Find how many slices are left (subtract eaten slices from total)
  3. Express the remaining slices as a fraction of one pizza
Step 1
2 × 4 = 8
Step 2
8 - 3 = 5
Step 3
5/4 = 1 1/4
Step 1: Find total slices

There are 2 pizzas, each cut into 4 slices

2 × 4 = 8 slices total

Step 2: Find remaining slices

3 slices were eaten from 8 total

8 - 3 = 5 slices remaining

Step 3: Express as fraction of one pizza

One pizza has 4 slices, so 5 slices = 5/4 of a pizza

5/4 = 1 1/4 pizzas

1 1/4 pizzas were left
Final answer:

1 1/4 pizzas were left after the party

Applied rules:

• Multiplication: Finding total in equal groups

• Subtraction: Taking away part of the whole

• Fractions: Representing parts of a whole

2 Apple Sharing Problem
Exercise 2
Maria has 3 1/2 apples. She gives 1 apple to her brother and eats 1/2 apple herself. How many apples does Maria have left?
Definition:

Mixed number: A combination of a whole number and a fraction (like 3 1/2 = 3 + 1/2).

Step 1
3 1/2 = 7/2
Step 2
7/2 - 1 - 1/2
Step 3
7/2 - 2/2 - 1/2 = 4/2 = 2
Step 1: Convert mixed number to improper fraction

3 1/2 = (3 × 2 + 1)/2 = 7/2

Step 2: Subtract the amounts given away

Maria gives 1 apple and eats 1/2 apple

7/2 - 1 - 1/2 = 7/2 - 2/2 - 1/2

Step 3: Perform the subtraction

7/2 - 2/2 - 1/2 = (7-2-1)/2 = 4/2 = 2

Step 4: Verification

If Maria has 2 apples left and gave away 1 1/2: 2 + 1 1/2 = 3 1/2 ✓

Maria has 2 apples left
Final answer:

Maria has 2 apples left

Applied rules:

• Converting mixed to improper: Whole × Denominator + Numerator

• Subtraction with fractions: Need common denominators

• Simplification: Reduce fractions to lowest terms

3 Ribbon Cutting Problem
Exercise 3
A craft project needs 2 pieces of ribbon that are each 3/4 yard long. Another piece of 1 1/2 yards is also needed. How much total ribbon is needed for the project?
Definition:

Adding fractions: To add fractions, they need the same denominator (bottom number).

Step 1
2 × 3/4 = 6/4
Step 2
6/4 + 1 1/2
Step 3
6/4 + 3/2 = 6/4 + 6/4 = 12/4 = 3
Step 1: Find total length for 2 pieces

2 pieces × 3/4 yard each = 2 × 3/4 = 6/4 yards

Step 2: Convert mixed number to improper fraction

1 1/2 = (1 × 2 + 1)/2 = 3/2

Step 3: Add the fractions

6/4 + 3/2 = 6/4 + 6/4 = 12/4 = 3 yards

Step 4: Verification

6/4 = 1 1/2, so 1 1/2 + 1 1/2 = 3 yards ✓

3 yards of ribbon are needed
Final answer:

3 yards of ribbon are needed for the project

Applied rules:

• Multiplication of fractions: Whole × Numerator / Denominator

• Converting mixed numbers: (Whole × Denominator) + Numerator

• Adding fractions: Need common denominators

Fractions Concepts & Methods
Numerator / Denominator
Fraction Structure
Mixed Number
2 1/3
2 + 1/3 = 7/3
Improper Fraction
7/3
Top > Bottom
Equivalent
1/2 = 2/4
Same value
Key definitions:

Fraction: A number representing part of a whole (numerator/denominator).

Proper fraction: Numerator < denominator (like 1/2).

Improper fraction: Numerator ≥ denominator (like 5/3).

Mixed number: Whole number + fraction (like 1 1/2).

Fraction conversion methods:
  1. Mixed to Improper: (Whole × Denominator) + Numerator = New Numerator
  2. Improper to Mixed: Divide numerator by denominator
  3. Adding/Subtracting: Find common denominators first
  4. Multiplying: Multiply numerators and denominators separately
Detailed Summary: Multi-Step Problems with Fractions
(Whole × Denominator) + Numerator
Mixed to Improper Conversion
Key Definitions:

Multi-step problem: A math problem that requires more than one operation to solve. When working with fractions, it might involve addition, subtraction, multiplication, or conversion between mixed numbers and improper fractions.

Fraction: A number that represents part of a whole. It has a numerator (top number) and a denominator (bottom number).

Mixed number: A number made up of a whole number and a fraction (like 1 1/2).

Improper fraction: A fraction where the numerator is greater than or equal to the denominator (like 5/3).

Core Rules:

• Convert mixed numbers to improper fractions before performing operations

• Find common denominators when adding or subtracting fractions

• Multiply numerators together and denominators together

• Reduce fractions to simplest form when possible

• Check your work by estimating if the answer makes sense

Step-by-Step Method for Multi-Step Fraction Problems:
  1. Read the problem carefully - Understand what is being asked
  2. Identify the operations needed - Addition, subtraction, multiplication?
  3. Convert mixed numbers to improper fractions if necessary
  4. Perform operations step by step - One operation at a time
  5. Simplify your answer - Reduce fractions and convert back to mixed numbers if needed
  6. Check your answer - Does it make sense in the context of the problem?
Tip 1: Draw pictures to visualize fractions when solving problems.
Tip 2: Always convert mixed numbers to improper fractions before operating.
Tip 3: When adding/subtracting, find the least common denominator.
Tip 4: Estimate your answer to check if it's reasonable.
Common mistakes: Forgetting to find common denominators, mixing up numerators and denominators, not converting mixed numbers properly.
Success strategies: Take your time with conversions, use visual models, check each step carefully.
More Examples

Simple example: 1/2 + 1/4 = 2/4 + 1/4 = 3/4

Medium example: 2 1/3 + 1/3 = 7/3 + 1/3 = 8/3 = 2 2/3

Advanced example: 3 × 1 1/2 = 3 × 3/2 = 9/2 = 4 1/2

Questions & Answers

Question: I get confused when changing mixed numbers to improper fractions. Can you explain the process again?

Answer: Sure! Converting mixed numbers to improper fractions follows this pattern:

  • Multiply the whole number by the denominator
  • Add the numerator to that product
  • Keep the same denominator

For example, with 2 3/4:

  • Whole number (2) × Denominator (4) = 8
  • Add numerator (3): 8 + 3 = 11
  • Keep denominator (4): 11/4

So 2 3/4 = 11/4. Remember: "Multiply, add, keep the bottom!"

Question: How can I help my child understand when to add versus multiply fractions in word problems?

Answer: Focus on the language and context in the problem:

  • Addition: Look for words like "total," "altogether," "combined," "left after taking away," "how many more."
  • Multiplication: Look for words like "of," "times," "each," "product," "groups of."

For example: "3 groups of 1/4" means 3 × 1/4 = 3/4, while "1/4 plus 1/2" means 1/4 + 1/2 = 3/4.

Encourage your child to visualize the problem with drawings or manipulatives.

Question: Why do we need to find common denominators when adding fractions?

Answer: Think of fractions like slices of different-sized pies. You can't add 1/2 of a small pie to 1/3 of a large pie directly because they're different sizes!

Finding a common denominator is like cutting both pies into the same size slices. Once they're the same size, you can count the total number of slices.

For example, to add 1/2 + 1/3, we change them to 3/6 + 2/6 (same size slices), then add: 3/6 + 2/6 = 5/6.

It's like saying "3 sixths plus 2 sixths equals 5 sixths" - all the same size pieces!