Divide Unit Fractions and Whole Numbers

Fifth Grade Math - Complete Guide

📌 What is a Unit Fraction?

A unit fraction is a fraction where the numerator (top number) is always 1.

Examples: \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), \(\frac{1}{5}\), \(\frac{1}{6}\), \(\frac{1}{8}\), \(\frac{1}{10}\)

1️⃣ Divide Unit Fractions by Whole Numbers

📐 Formula

\[\frac{1}{a} \div b = \frac{1}{a} \times \frac{1}{b} = \frac{1}{a \times b}\]

📝 Steps to Solve

Keep the first fraction (unit fraction) as it is
Change the division sign (÷) to multiplication (×)
Flip the whole number to its reciprocal (Example: 5 becomes \(\frac{1}{5}\))
Multiply the numerators together and denominators together
Simplify if needed

💡 Examples

Example 1: \(\frac{1}{3} \div 5\)

Step 1: Keep → \(\frac{1}{3}\)
Step 2: Change ÷ to × → \(\frac{1}{3} \times\)
Step 3: Flip 5 → \(\frac{1}{3} \times \frac{1}{5}\)
Step 4: Multiply → \(\frac{1 \times 1}{3 \times 5} = \frac{1}{15}\)

✓ Answer: \(\frac{1}{15}\)

Example 2: \(\frac{1}{6} \div 4\)

Step 1: \(\frac{1}{6} \times \frac{1}{4}\)
Step 2: \(\frac{1 \times 1}{6 \times 4} = \frac{1}{24}\)

✓ Answer: \(\frac{1}{24}\)

2️⃣ Divide Whole Numbers by Unit Fractions

📐 Formula

\[a \div \frac{1}{b} = a \times b\]

Simply multiply the whole number by the denominator of the unit fraction!

📝 Steps to Solve

Keep the whole number as it is
Change the division sign (÷) to multiplication (×)
Flip the unit fraction (Example: \(\frac{1}{4}\) becomes 4)
Multiply the whole number by the denominator

💡 Examples

Example 1: \(3 \div \frac{1}{4}\)

Step 1: Flip the fraction → \(3 \times 4\)
Step 2: Multiply → \(3 \times 4 = 12\)

✓ Answer: 12

📌 This means: "How many fourths are in 3 wholes?" Answer: 12 fourths

Example 2: \(2 \div \frac{1}{3}\)

Step 1: \(2 \times 3\)
Step 2: \(2 \times 3 = 6\)

✓ Answer: 6

📌 This means: "How many thirds are in 2 wholes?" Answer: 6 thirds

⚠️ Important Rules to Remember

Rule 1: About Quotients

• When dividing a unit fraction by a whole number greater than 1, the answer is smaller than the original fraction
Example: \(\frac{1}{3} \div 5 = \frac{1}{15}\) (and \(\frac{1}{15}\) is smaller than \(\frac{1}{3}\))

Rule 2: About Reciprocals

• Two numbers are reciprocals if their product equals 1
• The reciprocal of a whole number: Turn it into a fraction with 1 on top
Example: Reciprocal of 5 is \(\frac{1}{5}\) (because \(5 \times \frac{1}{5} = 1\))

Rule 3: Greater or Smaller?

• When dividing a whole number by a unit fraction, the answer is greater than the original whole number
Example: \(3 \div \frac{1}{4} = 12\) (and 12 is greater than 3)

3️⃣ Mixed Practice: Identify and Solve

🎯 Quick Reference Table

Problem Type	What to Do	Example
\(\frac{1}{a} \div b\)	Multiply denominators	\(\frac{1}{5} \div 3 = \frac{1}{15}\)
\(a \div \frac{1}{b}\)	Multiply whole × denominator	\(6 \div \frac{1}{2} = 12\)

💪 Practice Problems

1. \(\frac{1}{8} \div 2 = ?\)

Answer: \(\frac{1}{16}\)

2. \(4 \div \frac{1}{5} = ?\)

Answer: 20

3. \(\frac{1}{10} \div 5 = ?\)

Answer: \(\frac{1}{50}\)

4. \(8 \div \frac{1}{3} = ?\)

Answer: 24

4️⃣ Word Problems with Unit Fractions

📖 How to Solve Word Problems

Read the problem carefully
Identify what you're dividing (unit fraction or whole number)
Write the division equation
Solve using the correct method
Check if your answer makes sense

Problem 1: Dividing a Unit Fraction by a Whole Number

Sarah has \(\frac{1}{4}\) gallon of paint. She wants to divide it equally into 3 containers. How much paint will be in each container?

Step 1: Write the equation → \(\frac{1}{4} \div 3\)
Step 2: Multiply by reciprocal → \(\frac{1}{4} \times \frac{1}{3} = \frac{1}{12}\)
Step 3: Answer → Each container will have \(\frac{1}{12}\) gallon of paint

✓ Final Answer: \(\frac{1}{12}\) gallon per container

Problem 2: Dividing a Whole Number by a Unit Fraction

A ribbon is 6 feet long. How many \(\frac{1}{2}\)-foot pieces can be cut from it?

Step 1: Write the equation → \(6 \div \frac{1}{2}\)
Step 2: Multiply by denominator → \(6 \times 2 = 12\)
Step 3: Answer → 12 pieces can be cut

✓ Final Answer: 12 pieces

Problem 3: Real-World Application

Tim has \(\frac{1}{3}\) of a pizza. He wants to share it equally among 4 friends. What fraction of the whole pizza does each friend get?

Step 1: Write the equation → \(\frac{1}{3} \div 4\)
Step 2: Multiply by reciprocal → \(\frac{1}{3} \times \frac{1}{4} = \frac{1}{12}\)
Step 3: Answer → Each friend gets \(\frac{1}{12}\) of the whole pizza

✓ Final Answer: \(\frac{1}{12}\) of the pizza per friend