Fraction of a Number Word Problems in Grade 4

Master fraction of a number word problems: finding parts of whole numbers, calculating fractional amounts, and applying multiplication in real-world contexts through these 5 comprehensive exercises.

Solution: Problems 1 to 3
1 Basic fraction of a number
Problem 1
What is \(\frac{2}{3}\) of 12?
Definition:

Fraction of a Number: Finding a part of a whole number by multiplying the fraction by the number

Solution method:
  1. Write the fraction and the number as a multiplication problem
  2. Multiply the numerator by the whole number
  3. Divide the result by the denominator
  4. Simplify if possible
Problem
\(\frac{2}{3} \times 12\)
Multiply
\(\frac{2 \times 12}{3} = \frac{24}{3}\)
Divide
\(24 \div 3 = 8\)
Step 1: Set up the multiplication

\(\frac{2}{3}\) of 12 means \(\frac{2}{3} \times 12\)

Step 2: Multiply the numerator by the whole number

\(2 \times 12 = 24\)

Step 3: Divide by the denominator

\(\frac{24}{3} = 24 \div 3 = 8\)

Step 4: State the answer

\(\frac{2}{3}\) of 12 equals 8.

\(\frac{2}{3}\) of 12 = 8
Final answer:

\(\frac{2}{3}\) of 12 = 8

Applied rules:

• Fraction of a number: \(\frac{a}{b}\) of \(n = \frac{a \times n}{b}\)

• Multiplication: Multiply numerator by the whole number

• Division: Divide by the denominator

2 Word problem with context
Problem 2
Sarah has 20 stickers. She gives \(\frac{3}{4}\) of them to her friend. How many stickers does she give away?
Definition:

Real-world Context: Applying fraction of a number to solve practical problems

Problem
\(\frac{3}{4}\) of 20
Multiply
\(\frac{3 \times 20}{4} = \frac{60}{4}\)
Divide
\(60 \div 4 = 15\)
Step 1: Identify what we need to find

We need to find \(\frac{3}{4}\) of 20 stickers.

Step 2: Set up the multiplication

\(\frac{3}{4} \times 20\)

Step 3: Multiply the numerator by the whole number

\(3 \times 20 = 60\)

Step 4: Divide by the denominator

\(\frac{60}{4} = 60 \div 4 = 15\)

Step 5: State the answer in context

Sarah gives away 15 stickers.

Sarah gives away 15 stickers.
Final answer:

Sarah gives away 15 stickers.

Applied rules:

• Real-world application: Translate word problems to mathematical expressions

• Fraction of a number: \(\frac{a}{b}\) of \(n = \frac{a \times n}{b}\)

• Contextual answer: Provide the answer in the context of the problem

3 Multiple fractions of a number
Problem 3
A pizza shop made 24 pizzas. \(\frac{1}{3}\) were pepperoni and \(\frac{1}{4}\) were cheese. How many pizzas were either pepperoni or cheese?
Definition:

Multiple Fractions: Finding different fractions of the same number and combining the results

Pepperoni
\(\frac{1}{3}\) of 24 = 8
Cheese
\(\frac{1}{4}\) of 24 = 6
Total
8 + 6 = 14
Step 1: Find pepperoni pizzas

\(\frac{1}{3}\) of 24 = \(\frac{1 \times 24}{3} = \frac{24}{3} = 8\)

Step 2: Find cheese pizzas

\(\frac{1}{4}\) of 24 = \(\frac{1 \times 24}{4} = \frac{24}{4} = 6\)

Step 3: Add the results

8 + 6 = 14

Step 4: State the answer

14 pizzas were either pepperoni or cheese.

14 pizzas were either pepperoni or cheese.
Final answer:

14 pizzas were either pepperoni or cheese.

Applied rules:

• Multiple fractions: Find each fraction separately

• Combination: Add the results when finding "either/or" totals

• Step-by-step: Solve each part before combining

Rules and methods for fraction of a number problems
\(\frac{a}{b} \text{ of } n = \frac{a \times n}{b}\)
Fraction of a Number Formula
Rule 1
\(\frac{a}{b} \text{ of } n = \frac{a \times n}{b}\)
Fraction of a Number
Rule 2
\(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)
Fraction Multiplication
Rule 3
\(n = \frac{n}{1}\)
Whole Number as Fraction
Key definitions:

Fraction of a Number: Finding a part of a whole number by multiplying the fraction by the number

Word Problem: A mathematical problem presented in narrative form

Real-world Context: Applying mathematical concepts to practical situations

Denominator: The bottom number in a fraction

Numerator: The top number in a fraction

Problem-solving methodology:
  1. Read carefully: Understand what the problem is asking
  2. Identify numbers: Find the fraction and the whole number
  3. Set up multiplication: Write the fraction times the whole number
  4. Perform calculation: Multiply numerator by whole number, divide by denominator
  5. Check reasonableness: Does the answer make sense?
  6. State the answer: Include units and context
Tip 1: "Of" means multiply in fraction problems.
Tip 2: Always check if your answer is reasonable in the context of the problem.
Tip 3: Draw a picture to visualize the fraction of the number.
Tip 4: Practice with simple fractions first before moving to complex ones.
Common errors: Forgetting to divide by the denominator, multiplying denominators instead of numerators, misreading the problem.
Success habits: Show all work, double-check calculations, read problems twice.
Formulas to know by heart:

• Fraction of a number: \(\frac{a}{b}\) of \(n = \frac{a \times n}{b}\)

• Fraction multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

• Whole number as fraction: \(n = \frac{n}{1}\)

• Simplification: Divide numerator and denominator by their GCD

Exercise with Visualization: Fraction of a Number
Problem 6: Fraction of Different Numbers
Consider the function: \(f(x) = \frac{1}{2} \times x\) (half of a number)
\(f(x) = \frac{1}{4} \times x\) (quarter of a number)
\(f(x) = \frac{3}{4} \times x\) (three-quarters of a number)
Where x represents the original number.

Analysis: The chart shows how different fractions affect various numbers.

  • \(f_1(x) = \frac{1}{2}x\) (increases at half rate)
  • \(f_2(x) = \frac{1}{4}x\) (increases at quarter rate)
  • \(f_3(x) = \frac{3}{4}x\) (increases at three-quarter rate)
Solution: Problems 4 to 5
4 Complex word problem
Problem 4
A bakery made 36 cupcakes. \(\frac{1}{6}\) were chocolate, \(\frac{1}{4}\) were vanilla, and the rest were strawberry. How many were strawberry?
Definition:

Remaining Amount: Finding what's left after taking specified fractions

Chocolate
\(\frac{1}{6}\) of 36 = 6
Vanilla
\(\frac{1}{4}\) of 36 = 9
Total Other
6 + 9 = 15
Strawberry
36 - 15 = 21
Step 1: Find chocolate cupcakes

\(\frac{1}{6}\) of 36 = \(\frac{1 \times 36}{6} = \frac{36}{6} = 6\)

Step 2: Find vanilla cupcakes

\(\frac{1}{4}\) of 36 = \(\frac{1 \times 36}{4} = \frac{36}{4} = 9\)

Step 3: Find total of other flavors

6 + 9 = 15

Step 4: Find strawberry cupcakes

36 - 15 = 21

Step 5: State the answer

21 cupcakes were strawberry.

21 cupcakes were strawberry.
Final answer:

21 cupcakes were strawberry.

Applied rules:

• Multiple fractions: Calculate each fraction separately

• Remaining calculation: Subtract known amounts from the total

• Verification: Check that all parts add up to the whole

5 Fraction of a fraction
Problem 5
Tom has 20 marbles. He gives \(\frac{1}{5}\) of his marbles to his sister. Then he gives \(\frac{1}{4}\) of the remaining marbles to his friend. How many marbles does he have left?
Definition:

Fraction of a Fraction: Finding a fraction of an amount that is already a fraction of something

To Sister
\(\frac{1}{5}\) of 20 = 4
Remaining
20 - 4 = 16
To Friend
\(\frac{1}{4}\) of 16 = 4
Final Amount
16 - 4 = 12
Step 1: Find marbles given to sister

\(\frac{1}{5}\) of 20 = \(\frac{1 \times 20}{5} = \frac{20}{5} = 4\)

Step 2: Find remaining marbles after giving to sister

20 - 4 = 16

Step 3: Find marbles given to friend

\(\frac{1}{4}\) of 16 = \(\frac{1 \times 16}{4} = \frac{16}{4} = 4\)

Step 4: Find final remaining marbles

16 - 4 = 12

Step 5: State the answer

Tom has 12 marbles left.

Tom has 12 marbles left.
Final answer:

Tom has 12 marbles left.

Applied rules:

• Sequential operations: Perform one fraction calculation at a time

• Remaining amounts: Update the total after each operation

• Step-by-step: Track the changing amounts carefully

Detailed Summary: Fraction of a Number Word Problems
\(\frac{a}{b} \text{ of } n = \frac{a \times n}{b}\)
Basic Fraction of a Number Formula
Key definitions:

Fraction of a Number: A mathematical operation that finds a part of a whole number by multiplying the fraction by the number

Word Problem: A mathematical problem presented in narrative form that requires translation into mathematical operations

Real-world Context: Applying mathematical concepts to practical situations

Denominator: The bottom number in a fraction, showing how many equal parts the whole is divided into

Numerator: The top number in a fraction, showing how many parts we have

Remaining Amount: What is left after some has been taken away

Complete methodology:
  1. Read the problem: Read it twice to understand the situation
  2. Identify numbers: Find the fraction and the whole number involved
  3. Set up multiplication: Write the fraction times the whole number
  4. Perform calculation: Multiply numerator by whole number, divide by denominator
  5. Check reasonableness: Verify the answer makes sense in context
  6. State the answer: Include units and context
Tip 1: The word "of" in math problems usually means multiply.
Tip 2: Draw a visual model to represent the fraction of the number.
Tip 3: Always check if your answer is reasonable compared to the original number.
Tip 4: For sequential problems, update the total after each operation.

Common errors: Forgetting to divide by the denominator, misinterpreting the word "of", not checking if the answer makes sense in context.
Memory aids: "Of means multiply" and "numerator times number, divide by denominator."
Formulas to know by heart:

• Fraction of a number: \(\frac{a}{b}\) of \(n = \frac{a \times n}{b}\)

• Fraction multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

• Whole number as fraction: \(n = \frac{n}{1}\)

• Simplification: Divide numerator and denominator by their greatest common divisor

Exercise with Visualization: Fraction of a Number
Problem 6: Fraction of Different Numbers
Consider the function: \(f(x) = \frac{1}{2} \times x\) (half of a number)
\(f(x) = \frac{1}{4} \times x\) (quarter of a number)
\(f(x) = \frac{3}{4} \times x\) (three-quarters of a number)
Where x represents the original number.

Analysis: The chart shows how different fractions affect various numbers.

  • \(f_1(x) = \frac{1}{2}x\) (increases at half rate)
  • \(f_2(x) = \frac{1}{4}x\) (increases at quarter rate)
  • \(f_3(x) = \frac{3}{4}x\) (increases at three-quarter rate)

Questions & Answers

Question: What does "fraction of a number" mean? How is it different from regular multiplication?

Answer: "Fraction of a number" is a special type of multiplication where you find a part of a whole number:

  • Fraction of a number: Finding \(\frac{2}{3}\) of 12 means taking \(\frac{2}{3}\) of the total amount
  • Regular multiplication: \(2 \times 3 = 6\) means having 2 groups of 3 items
  • Fraction of a number: \(\frac{2}{3} \times 12 = 8\) means taking 2 parts out of 3 equal parts of 12

Both involve multiplication, but fraction of a number specifically means finding a portion or part of the original number.

Question: Why do we multiply the numerator by the whole number first, then divide by the denominator?

Answer: This follows the mathematical rule for multiplying fractions by whole numbers:

  • \(\frac{a}{b} \times n = \frac{a \times n}{b}\)
  • We multiply the numerator (top number) by the whole number
  • We keep the same denominator (bottom number)
  • This is equivalent to dividing by the denominator

Example: \(\frac{2}{3} \times 12 = \frac{2 \times 12}{3} = \frac{24}{3} = 8\)

This ensures we're finding the correct proportional part of the number.

Question: How can I check if my answer is reasonable?

Answer: Here are several ways to check reasonableness:

  • Compare to the original number: If finding \(\frac{1}{2}\) of 20, the answer should be around 10
  • Estimate: Round the fraction to a simple one and estimate
  • Context check: Does the answer make sense in the story?
  • Reverse calculation: Add the parts back to see if they equal the whole

Example: If \(\frac{1}{3}\) of 24 = 8, check: 8 + 8 + 8 = 24 ✓

The answer should be smaller than the original number (unless the fraction is greater than 1).

Question: What should I do when a word problem has multiple fractions?

Answer: Handle multiple fractions systematically:

  1. Find each fraction separately: Calculate one fraction at a time
  2. Track changes: Update the total amount after each operation
  3. Combine results: Add or subtract as needed
  4. Check completeness: Ensure all parts add up to the whole

Example: If finding \(\frac{1}{4}\) and \(\frac{1}{3}\) of 12:

  • \(\frac{1}{4}\) of 12 = 3
  • \(\frac{1}{3}\) of 12 = 4
  • Total = 3 + 4 = 7

Always solve one part at a time!

Question: How do I know when to multiply and when to divide in word problems?

Answer: Look for key words and phrases:

  • "Of" usually means multiply: "\(\frac{2}{3}\) of 15" means \(\frac{2}{3} \times 15\)
  • "Per" often means divide: "miles per hour" involves division
  • "Each" can mean multiply: "3 boxes with 5 each" means \(3 \times 5\)
  • "Shared equally" means divide: "shared among 4 people"

For fraction of a number problems, the word "of" tells you to multiply.

Example: "\(\frac{3}{4}\) of 20" = \(\frac{3}{4} \times 20 = 15\)

Always read the problem carefully to understand the relationship!