Unit Fractions
A unit fraction is the simplest type of fraction. It is a fraction where the numerator is 1. Examples: 1/2, 1/3, 1/4, 1/5, 1/10, and so on.
Unit fractions represent one equal part of a whole. When you cut a cake into 5 equal pieces, each piece is 1/5 of the cake — that is a unit fraction. Every other fraction can be built from unit fractions. For example, 3/4 is just 1/4 + 1/4 + 1/4.
Understanding unit fractions helps you compare fractions, understand what fractions really mean, and build a strong foundation for all fraction operations.
What is Unit Fractions?
Definition: A unit fraction is a fraction whose numerator is 1 and whose denominator is a positive whole number.
General form:
1/n, where n is a positive whole number (n = 2, 3, 4, 5, ...)
Key facts:
- The numerator is always 1.
- The denominator tells the number of equal parts the whole is divided into.
- The larger the denominator, the smaller the unit fraction.
- Every fraction a/b can be written as the sum of a copies of 1/b.
Types and Properties
1. Comparing Unit Fractions
- The larger the denominator, the smaller the fraction.
- 1/2 > 1/3 > 1/4 > 1/5 > 1/10 > 1/100
- Why? A bigger denominator means more pieces, so each piece is smaller.
2. Unit Fractions on a Number Line
- All unit fractions lie between 0 and 1.
- 1/2 is at the midpoint. As the denominator increases, the fraction gets closer to 0.
3. Building Other Fractions
- 2/5 = 1/5 + 1/5
- 3/4 = 1/4 + 1/4 + 1/4
- 5/8 = 1/8 + 1/8 + 1/8 + 1/8 + 1/8
4. Unit Fractions and Division
- 1/n of a number = the number ÷ n.
- 1/2 of 10 = 10 ÷ 2 = 5
- 1/4 of 20 = 20 ÷ 4 = 5
Solved Examples
Example 1: Example 1: Identifying unit fractions
Problem: Which are unit fractions: 1/3, 2/5, 1/7, 3/8, 1/1, 1/100?
Solution:
- A unit fraction has numerator = 1.
- 1/3 — Yes. 2/5 — No. 1/7 — Yes. 3/8 — No. 1/1 — Yes. 1/100 — Yes.
Answer: 1/3, 1/7, 1/1, and 1/100 are unit fractions.
Example 2: Example 2: Comparing unit fractions
Problem: Arrange in ascending order: 1/6, 1/2, 1/9, 1/4.
Solution:
- Larger denominator = smaller fraction.
- Ascending order: 1/9 < 1/6 < 1/4 < 1/2.
Answer: 1/9, 1/6, 1/4, 1/2
Example 3: Example 3: Writing a fraction as unit fractions
Problem: Write 4/7 as a sum of unit fractions.
Solution:
- 4/7 = 1/7 + 1/7 + 1/7 + 1/7
Answer: 4/7 = 1/7 + 1/7 + 1/7 + 1/7
Example 4: Example 4: Finding a unit fraction of a number
Problem: Find 1/4 of 36.
Solution:
- 1/4 of 36 = 36 ÷ 4 = 9.
Answer: 1/4 of 36 = 9
Example 5: Example 5: Comparing two unit fractions
Problem: Which is larger: 1/5 or 1/8?
Solution:
- 1/5 means 5 parts — each part is bigger.
- 1/8 means 8 parts — each part is smaller.
Answer: 1/5 > 1/8
Example 6: Example 6: Unit fractions on a number line
Problem: Mark 1/3 and 1/6 on a number line between 0 and 1.
Solution:
- Divide the segment from 0 to 1 into 6 equal parts.
- 1/6 is at the first mark. 1/3 = 2/6 is at the second mark.
- 1/3 is farther from 0, confirming 1/3 > 1/6.
Answer: 1/6 is closer to 0; 1/3 is farther from 0.
Example 7: Example 7: Real-life unit fractions
Problem: A chocolate bar has 12 equal pieces. Riya eats 1 piece. What fraction did she eat?
Solution:
- Total pieces = 12. Riya ate 1.
- Fraction = 1/12.
Answer: Riya ate 1/12 of the chocolate.
Example 8: Example 8: Adding unit fractions
Problem: Find: 1/5 + 1/5 + 1/5.
Solution:
- 1/5 + 1/5 + 1/5 = 3/5.
Answer: 3/5
Example 9: Example 9: Closest to zero
Problem: Among 1/3, 1/50, 1/10, 1/1000, which is closest to 0?
Solution:
- Largest denominator = smallest fraction = closest to 0.
- Largest denominator is 1000.
Answer: 1/1000 is closest to 0.
Example 10: Example 10: Finding the whole
Problem: If 1/6 of a number is 8, what is the number?
Solution:
- The whole number = 8 × 6 = 48.
Answer: The number is 48.
Real-World Applications
Sharing Equally: When 4 friends share a pizza, each gets 1/4.
Cooking: Recipes use 1/2 cup, 1/4 teaspoon, 1/3 cup regularly.
Time: 1/4 hour = 15 minutes. 1/2 hour = 30 minutes. 1/3 hour = 20 minutes.
Money: 50 paise = 1/2 of Rs. 1. 25 paise = 1/4 of Rs. 1.
Measurement: 1/4 metre = 25 cm. 1/2 kg = 500 g. 1/10 litre = 100 ml.
Key Points to Remember
- A unit fraction has 1 as the numerator.
- The larger the denominator, the smaller the fraction.
- All unit fractions lie between 0 and 1 (except 1/1 = 1).
- Any fraction a/b = a copies of 1/b.
- Finding 1/n of a number = dividing by n.
- If 1/n of a number = k, then the number = k × n.
- 1/2 is the largest unit fraction (after 1/1).
- There is no smallest unit fraction.
Practice Problems
- List all unit fractions with denominators from 2 to 10.
- Arrange in descending order: 1/7, 1/3, 1/11, 1/5, 1/2.
- Write 5/6 as a sum of unit fractions.
- Find: (a) 1/3 of 24, (b) 1/8 of 56, (c) 1/5 of 100.
- Which is larger: 1/15 or 1/12? Explain.
- If 1/7 of a number is 9, what is the number?
- A rope is 30 metres long. What is 1/6 of the rope?
- True or false: 1/100 > 1/50. Give a reason.
Frequently Asked Questions
Q1. What is a unit fraction?
A unit fraction has 1 as the numerator. Examples: 1/2, 1/3, 1/4. The word 'unit' means 'one', so it represents one part out of equal parts.
Q2. Is 1/1 a unit fraction?
Yes, because its numerator is 1. It equals 1 (the whole).
Q3. Why is 1/3 larger than 1/5?
Dividing a cake into 3 pieces gives bigger pieces than dividing into 5 pieces. Fewer parts = larger each part.
Q4. Can a unit fraction be greater than 1?
No. The largest unit fraction is 1/1 = 1. All others are less than 1.
Q5. How do you find a unit fraction of a number?
Divide the number by the denominator. For example, 1/4 of 20 = 20 ÷ 4 = 5.
Q6. Why are unit fractions important?
They are the building blocks of all fractions. 3/5 is three copies of 1/5. Understanding unit fractions makes fraction operations easier.
Q7. What is the smallest unit fraction?
There is no smallest. As the denominator increases, the fraction gets closer to 0 but never reaches it.
Q8. Are all fractions with numerator 1 unit fractions?
Yes, as long as the denominator is a positive whole number.
Related Topics
- Introduction to Fractions
- Proper and Improper Fractions
- Comparing Fractions
- Equivalent Fractions
- Mixed Numbers
- Simplest Form of a Fraction
- Like and Unlike Fractions
- Fractions on Number Line
- Addition of Fractions
- Subtraction of Fractions
- Word Problems on Fractions
- Types of Fractions
- Addition and Subtraction of Fractions
