Fraction Word Problems Multiplication and Division Worksheet with Solutions

Complete practice worksheet for fraction word problems involving multiplication and division with detailed step-by-step solutions for 4th grade students.

Worksheet Solutions: Problems 1 to 3
1 Sharing Cookies
Problem 1
Maria has 3/4 of a batch of cookies. She wants to divide them equally among 6 friends. How much of the batch does each friend get? If each friend eats 1/2 of their share, how much of the original batch does each friend eat?
Definition:

Fraction division: To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number: \( \frac{a}{b} \div n = \frac{a}{b} \times \frac{1}{n} \)

Worksheet problem-solving method:
  1. Identify the operation needed (division for sharing equally)
  2. Set up the division: 3/4 ÷ 6
  3. Convert division to multiplication: 3/4 × 1/6
  4. Perform the multiplication: (3×1)/(4×6) = 3/24 = 1/8
  5. For the second part, multiply by 1/2: 1/8 × 1/2 = 1/16
Step 1
3/4 ÷ 6
Step 2
1/8 × 1/2
Result
1/16
Step 1: Divide cookies among friends

3/4 ÷ 6 = 3/4 × 1/6 = 3/24 = 1/8 of the batch per friend

Step 2: Find how much each friend eats

1/2 of their share = 1/2 × 1/8 = 1/16 of the original batch

Step 3: Verify the answer

If each friend eats 1/16 of the batch, 6 friends eat 6 × 1/16 = 6/16 = 3/8 of the batch

This is less than the original 3/4, which makes sense.

Each friend gets 1/8 of the batch; eats 1/16 of the original batch
Final answer:

Each friend gets 1/8 of the batch and eats 1/16 of the original batch

Applied rules:

• Division by whole number: Multiply by reciprocal

• Fraction multiplication: Multiply numerators and denominators separately

• Word problem interpretation: "divide equally" indicates division, "of" indicates multiplication

2 Reading Pages
Problem 2
A student reads 5/6 of a book in 5 days. If she reads the same amount each day, what fraction of the book does she read per day? At this rate, how many days will it take her to read 2 books?
Definition:

Rate calculation: To find the daily rate, divide the total amount by the number of days.

Daily rate
5/6 ÷ 5 = 1/6
Days for 2 books
2 ÷ 1/6 = 12
Step 1: Find daily reading rate

5/6 of the book in 5 days = 5/6 ÷ 5 = 5/6 × 1/5 = 5/30 = 1/6 of the book per day

Step 2: Find days to read 2 books

2 books ÷ daily rate (1/6 book per day) = 2 ÷ 1/6 = 2 × 6/1 = 12 days

Step 3: Verify the answer

At 1/6 per day for 12 days = 12 × 1/6 = 2 books ✓

Student reads 1/6 of the book per day; 12 days for 2 books
Final answer:

The student reads 1/6 of the book per day, and it takes 12 days to read 2 books

Applied rules:

• Rate calculation: Total amount ÷ Time period

• Division by fraction: Multiply by reciprocal

• Verification: Check that the rate multiplied by time equals the total

3 Art Project Supplies
Problem 3
A teacher has 7/8 of a container of paint. She divides it equally among 4 students. Each student uses 2/3 of their share. How much of the original container does each student use? How much paint remains unused?
Definition:

Sequential operations: When dividing a fraction by a whole number, then multiplying the result by another fraction, perform operations in sequence.

Step 1
7/8 ÷ 4 = 7/32
Step 2
7/32 × 2/3 = 7/48
Remaining
7/8 - 4×7/48 = 7/24
Step 1: Divide paint among students

7/8 ÷ 4 = 7/8 × 1/4 = 7/32 of the container per student

Step 2: Find how much each student uses

2/3 of their share = 2/3 × 7/32 = 14/96 = 7/48 of the original container

Step 3: Find total used by all students

4 students × 7/48 each = 28/48 = 7/12 of the container

Step 4: Find remaining paint

Started with 7/8, used 7/12: 7/8 - 7/12 = 21/24 - 14/24 = 7/24 remaining

Each student uses 7/48 of the container; 7/24 remains unused
Final answer:

Each student uses 7/48 of the original container, and 7/24 of the paint remains unused

Applied rules:

• Division by whole number: Multiply by reciprocal

• Fraction multiplication: Multiply numerators and denominators separately

• Subtraction with fractions: Need common denominators

Worksheet Concepts: Multiplication and Division
\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)
Fraction Multiplication
\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)
Fraction Division
Rule 1
\(\frac{a}{b} \div n = \frac{a}{b} \times \frac{1}{n}\)
Division by whole number
Rule 2
\(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
Multiplication rule
Rule 3
\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)
Division by fraction
Key definitions:

Fraction multiplication: Multiply numerators together and denominators together. Used when finding part of a part.

Fraction division: Multiply by the reciprocal of the divisor. Used when sharing equally or finding how many times one quantity fits into another.

Reciprocal: The multiplicative inverse of a number (flipping the numerator and denominator).

Worksheet problem-solving steps:
  1. Read the problem carefully: Identify if it involves sharing/equal groups (division) or finding part of a part (multiplication)
  2. Identify the operation: Look for keywords like "divide", "share", "each" (division) vs "of", "times", "product" (multiplication)
  3. Set up the calculation: Write the appropriate operation
  4. Perform the operation: Apply multiplication or division rules
  5. Simplify the result: Reduce to lowest terms if possible
  6. Check your answer: Verify if it makes sense in the context of the problem
Tip 1: Division means splitting into equal parts; multiplication means finding part of a part.
Tip 2: When dividing by a whole number, multiply by its reciprocal (1/n).
Tip 3: Dividing by a fraction is the same as multiplying by its reciprocal.
Tip 4: Always check if your answer makes sense in the context of the problem.
Common errors: Confusing multiplication and division, forgetting to flip the divisor when dividing, not simplifying the final answer, misinterpreting word problems.
Worksheet strategies: Underline key information, label each step clearly, double-check calculations, verify final answers make sense.
Worksheet Solutions: Problems 4 to 5
4 Water Bottles
Problem 4
A water cooler contains 3/5 gallon of water. If each cup holds 1/10 gallon, how many full cups can be filled? If 4/5 of the filled cups are consumed, how much water was drunk?
Definition:

Division for grouping: To find how many equal-sized portions fit into a total amount, divide the total by the size of each portion.

Step 1
3/5 ÷ 1/10 = 6
Step 2
6 × 1/10 × 4/5 = 12/25
Step 1: Find number of cups that can be filled

3/5 ÷ 1/10 = 3/5 × 10/1 = 30/5 = 6 full cups

Step 2: Find how many cups were consumed

4/5 of 6 cups = 4/5 × 6 = 24/5 = 4.8 cups, but since we only have full cups, 4/5 of 6 = 24/5 = 4.8, so 4 full cups

Step 3: Find total water consumed

4 cups × 1/10 gallon per cup = 4/10 = 2/5 gallon

6 full cups can be filled; 2/5 gallon consumed
Final answer:

6 full cups can be filled, and 2/5 gallon of water was consumed

Applied rules:

• Division for grouping: Total ÷ Size per group = Number of groups

• Fraction division: Multiply by reciprocal of divisor

• Fraction multiplication: Multiply numerators and denominators separately

5 Walking Distance
Problem 5
A student plans to walk 2/3 of a mile each day for 9 days. After completing 6 days, she changes her plan to cover the same total distance in 12 days total. How much should she walk each day for the remaining days?
Definition:

Adjusting rates: When changing the time period but keeping the total amount constant, divide the remaining amount by the remaining time.

Original plan
9 × 2/3 = 6 miles
Completed
6 × 2/3 = 4 miles
Remaining
6 - 4 = 2 miles
New daily
2 ÷ 6 = 1/3 mile
Step 1: Find total planned distance

9 days × 2/3 mile/day = 18/3 = 6 miles total

Step 2: Find distance already walked

6 days × 2/3 mile/day = 12/3 = 4 miles completed

Step 3: Find remaining distance

Total - Completed = 6 - 4 = 2 miles remaining

Step 4: Find remaining days

New plan is 12 days total, 6 already completed = 6 remaining days

Step 5: Find new daily distance

Remaining distance ÷ Remaining days = 2 ÷ 6 = 2/6 = 1/3 mile per day

Student should walk 1/3 mile each day for the remaining 6 days
Final answer:

The student should walk 1/3 mile each day for the remaining 6 days

Applied rules:

• Multiplication for totals: Rate × Time = Total

• Subtraction for remaining: Total - Completed = Remaining

• Division for new rate: Remaining ÷ Time = New Rate

Worksheet Summary: Key Concepts and Formulas
\(\text{Multiplication: } \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
Fraction Multiplication
\(\text{Division: } \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)
Fraction Division
Key definitions:

Fraction multiplication: To multiply fractions, multiply the numerators together and the denominators together. This is used when finding a part of a part.

Fraction division: To divide fractions, multiply the first fraction by the reciprocal of the second fraction. This is used when sharing equally or finding how many times one quantity fits into another.

Reciprocal: The reciprocal of a fraction a/b is b/a. The reciprocal of a whole number n is 1/n.

Worksheet solving methodology:
  1. Problem analysis: Determine whether the situation calls for multiplication or division
  2. Operation identification: Look for key words - "of", "times" suggest multiplication; "divide", "share", "each", "per" suggest division
  3. Setup calculation: Write the appropriate mathematical expression
  4. Perform operation: Apply multiplication or division rules correctly
  5. Simplify result: Reduce to lowest terms if possible
  6. Verify answer: Check if the answer makes sense in the problem context
Tip 1: Remember "Keep, Change, Flip" for division: keep the first fraction, change division to multiplication, flip the second fraction.
Tip 2: When dividing by a whole number, just multiply by its reciprocal (1/n).
Tip 3: Multiplying by a fraction less than 1 makes the result smaller; dividing by a fraction less than 1 makes the result larger.
Tip 4: Draw visual models (rectangles divided into parts) to understand multiplication and division of fractions.
Common errors: Forgetting to flip the divisor when dividing, adding numerators and denominators instead of multiplying, not simplifying the final answer, misinterpreting word problems.
Worksheet success strategies: Practice keyword identification, master the "keep, change, flip" method for division, learn to estimate reasonable answers, organize work clearly.
Key rules and principles:

• Multiplication rule: \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)

• Division rule: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)

• Division by whole number: \(\frac{a}{b} \div n = \frac{a}{b} \times \frac{1}{n}\)

• Word problem keywords: "of" → multiplication, "per", "each", "out of" → division

• Verification: Check that multiplication and division operations make logical sense in context

Worksheet Success Guide

📋
Multiplication Steps

1. Multiply numerators (top numbers)

2. Multiply denominators (bottom numbers)

3. Simplify if possible

2/3 × 3/4 = (2×3)/(3×4) = 6/12 = 1/2
Division Steps

1. Keep first fraction

2. Change ÷ to ×

3. Flip second fraction

4. Multiply as usual

2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9
Worksheet Problem Keywords
×
Multiplication
÷
Division
?
Context Clues
"of", "times", "product"
"divide", "share", "each", "per"
Equal groups vs part of part
Worksheet Practice Tips

• Label each step clearly

• Box your final answers

• Show all work

• Check reasonableness

📚 Practice Makes Perfect!

Worksheet Questions & Answers

Question: I always mix up when to multiply and when to divide in word problems. How can I get better at this?

Answer: Here's a systematic approach to identify operations:

  • Multiplication: Look for "of", "times", "product", "part of", "portion of". Example: "1/2 of 3/4" means 1/2 × 3/4
  • Division: Look for "divide", "share", "each", "per", "into", "how many times". Example: "3/4 shared among 3 people" means 3/4 ÷ 3

Think of it this way: multiplication finds parts of parts (making smaller), division shares equally or finds how many times something fits (can make larger or smaller depending on the divisor).

Practice: underline the operation words in each problem before solving!

Question: My child is struggling with the "flip" concept when dividing fractions. How can I explain this better?

Answer: The "flip" (reciprocal) concept can be understood through real-world examples:

  • Think of division as "how many times does this fit into that?"
  • Example: 12 ÷ 3 asks "how many 3s are in 12?" Answer: 4
  • For fractions: 3/4 ÷ 1/2 asks "how many 1/2s are in 3/4?"
  • This equals 3/4 × 2/1 = 6/4 = 1.5

The reciprocal turns division into multiplication: instead of "how many 1/2s fit in 3/4?", we're calculating 3/4 × 2/1. The "flip" represents the inverse relationship between multiplication and division.

Question: Sometimes my answers come out as improper fractions. Should I change them to mixed numbers?

Answer: It depends on the context of the problem:

  • If the problem involves counting items (like people, books, etc.), mixed numbers often make more sense
  • If the problem involves measurements (like distance, time, etc.), improper fractions are usually fine
  • Always follow your teacher's preference if specified

Example: If you get 5/2 cups of flour, you might say 2½ cups. But if you get 5/2 hours of reading, 5/2 hours is perfectly acceptable.

For worksheets, unless instructed otherwise, either form is typically acceptable as long as it's fully simplified.