Fraction Word Problems: Multiplication and Division with Detailed Solutions

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

Solution: Exercises 1 to 3
1 Sharing Problem
Exercise 1
A baker has 3/4 of a pound of flour. She wants to divide it equally among 3 containers. How much flour will go in each container? Then, if she uses 1/2 of what's in each container for a recipe, how much flour does she use per recipe?
Definition:

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

Multiplication and division method:
  1. Identify whether to multiply or divide based on the problem context
  2. Division by a whole number: multiply by its reciprocal (1/n)
  3. Multiplication: multiply numerators and denominators
  4. Simplify the result if possible
Step 1
3/4 ÷ 3
Step 2
1/4 × 1/2
Result
1/8 lb
Step 1: Divide flour among containers

3/4 ÷ 3 = 3/4 × 1/3 = 3/12 = 1/4 pound per container

Step 2: Find flour for each recipe

1/2 of what's in each container = 1/2 × 1/4 = 1/8 pound per recipe

Step 3: Verify the answer

If each recipe uses 1/8 lb and there are 3 containers, total used = 3 × 1/8 = 3/8 lb

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

Each recipe uses 1/8 pound of flour
Final answer:

Each recipe uses 1/8 pound of flour

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 Problem
Exercise 2
Emma reads 2/3 of a book in 4 days. If she reads the same amount each day, what fraction of the book does she read each day? Then, if she continues at this rate, how many days will it take her to read the entire book?
Definition:

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

Daily rate
2/3 ÷ 4 = 1/6
Total days
1 ÷ 1/6 = 6
Step 1: Find daily reading rate

2/3 of the book in 4 days = 2/3 ÷ 4 = 2/3 × 1/4 = 2/12 = 1/6 of the book per day

Step 2: Find total days to finish book

Entire book (1) ÷ daily rate (1/6) = 1 ÷ 1/6 = 1 × 6/1 = 6 days

Step 3: Verify the answer

At 1/6 per day for 6 days = 6 × 1/6 = 1 whole book ✓

Emma reads 1/6 of the book each day; 6 days total
Final answer:

Emma reads 1/6 of the book each day, and it takes 6 days to read the entire book

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 Supplies Problem
Exercise 3
A teacher has 5/6 of a bottle of paint. She divides it equally among 5 students. Then, each student uses 3/4 of their share. How much of the original bottle does each student use?
Definition:

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

Step 1
5/6 ÷ 5 = 1/6
Step 2
1/6 × 3/4 = 1/8
Step 1: Divide paint among students

5/6 ÷ 5 = 5/6 × 1/5 = 5/30 = 1/6 of the bottle per student

Step 2: Find how much each student uses

3/4 of their share = 3/4 × 1/6 = 3/24 = 1/8 of the original bottle

Step 3: Verify the answer

If each of 5 students uses 1/8 of the bottle, total used = 5 × 1/8 = 5/8

They started with 5/6, so remaining = 5/6 - 5/8 = 20/24 - 15/24 = 5/24

Each student uses 1/8 of the original bottle
Final answer:

Each student uses 1/8 of the original bottle of paint

Applied rules:

• Division by whole number: Multiply by reciprocal

• Fraction multiplication: Multiply numerators and denominators separately

• Sequential operations: Perform division first, then multiplication

Fraction Multiplication and Division Rules
\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)
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

Fraction division: Multiply by the reciprocal of the divisor

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

Complete methodology:
  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
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.
Key concepts: Division makes fractions smaller when dividing by numbers > 1; multiplication makes fractions smaller when multiplying by fractions < 1.
Solution: Exercises 4 to 5
4 Juice Sharing Problem
Exercise 4
A pitcher contains 2/3 gallon of juice. If each glass holds 1/6 gallon, how many glasses can be filled? If 3/4 of the filled glasses are drunk, how much juice was consumed?
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
2/3 ÷ 1/6 = 4
Step 2
4 × 1/6 × 3/4 = 1/2
Step 1: Find number of glasses that can be filled

2/3 ÷ 1/6 = 2/3 × 6/1 = 12/3 = 4 glasses

Step 2: Find how many glasses were drunk

3/4 of 4 glasses = 3/4 × 4 = 12/4 = 3 glasses

Step 3: Find total juice consumed

3 glasses × 1/6 gallon per glass = 3/6 = 1/2 gallon

4 glasses can be filled; 1/2 gallon consumed
Final answer:

4 glasses can be filled, and 1/2 gallon of juice 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 Running Distance Problem
Exercise 5
A runner plans to run 3/4 of a mile each day for 6 days. After 4 days, she realizes she wants to run the same total distance but spread it over 8 days. How much should she run 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
6 × 3/4 = 4.5 miles
Completed
4 × 3/4 = 3 miles
Remaining
4.5 - 3 = 1.5 miles
New daily
1.5 ÷ 4 = 3/8 mile
Step 1: Find total planned distance

6 days × 3/4 mile/day = 18/4 = 4.5 miles (or 9/2 miles)

Step 2: Find distance already run

4 days × 3/4 mile/day = 12/4 = 3 miles

Step 3: Find remaining distance

Total - Completed = 9/2 - 3 = 9/2 - 6/2 = 3/2 miles remaining

Step 4: Find remaining days

New plan is 8 days total, 4 already completed = 4 remaining days

Step 5: Find new daily distance

Remaining distance ÷ Remaining days = 3/2 ÷ 4 = 3/2 × 1/4 = 3/8 mile per day

Runner should run 3/8 mile each day for the remaining 4 days
Final answer:

The runner should run 3/8 mile each day for the remaining 4 days

Applied rules:

• Multiplication for totals: Rate × Time = Total

• Subtraction for remaining: Total - Completed = Remaining

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

Fraction Multiplication and Division Summary
\(\text{Multiplication: } \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
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.

Complete 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.
Success strategies: Practice keyword identification, master the "keep, change, flip" method for division, learn to estimate reasonable answers.
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

Fraction Multiplication vs Division 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
Word Problem Keywords
×
Multiplication
÷
Division
?
Context Clues
"of", "times", "product"
"divide", "share", "each", "per"
Equal groups vs part of part
Real-World Examples

Multiplication: "I ate 1/2 of 3/4 of a pizza" = 1/2 × 3/4

Division: "I have 3/4 of a pizza to share equally among 3 people" = 3/4 ÷ 3

📚 Practice Keyword Recognition!

Questions & Answers

Question: I don't understand when to multiply and when to divide in word problems. They seem similar sometimes.

Answer: Great question! Here's how to tell the difference:

  • Multiplication: Use when finding "part of a part" or scaling up/down. Look for words like "of", "times", "product". Example: "What is 1/2 of 3/4?" means 1/2 × 3/4
  • Division: Use when sharing equally or finding how many times one quantity fits into another. Look for words like "divide", "share", "each", "per", "into". Example: "Share 3/4 equally among 3 people" means 3/4 ÷ 3

Think of it this way: multiplication makes parts smaller (when multiplying by fractions less than 1), while division can make parts bigger (when dividing by fractions less than 1).

Question: Why do we "flip" the second fraction when dividing? It seems backwards.

Answer: The "flip" or "reciprocal" rule is based on the fundamental principle that dividing by a number is the same as multiplying by its reciprocal.

Think of it with whole numbers first: 12 ÷ 3 = 12 × 1/3 = 4

For fractions: 3/4 ÷ 2/3 is asking "how many 2/3 are in 3/4?"

This equals 3/4 × 3/2 (flipped version of 2/3) = 9/8

The reciprocal (flipped fraction) represents the "inverse" operation - if multiplying by 2/3 shrinks something, multiplying by its reciprocal 3/2 grows it back.

Question: When I divide fractions, sometimes my answer is bigger than the original numbers. Is that normal?

Answer: Yes, that's completely normal! When you divide by a fraction that's less than 1, your answer gets bigger.

Think of it this way: If you have 1/2 of a pizza and you want to know how many 1/4-sized pieces you can get, you can get 2 pieces because 1/2 ÷ 1/4 = 1/2 × 4/1 = 2.

Here's the pattern:
• Dividing by a fraction < 1 makes the answer bigger
• Dividing by a fraction > 1 makes the answer smaller
• Dividing by 1 leaves the answer unchanged

This happens because dividing by a small fraction is like asking "how many tiny pieces fit into this amount?" which results in a larger count.