Simplest Form of a Fraction
Imagine you cut a pizza into 8 slices and eat 4. You ate 4/8 of the pizza. But you can also say you ate 1/2 of the pizza. Both 4/8 and 1/2 mean the same thing — the fraction 1/2 is just simpler.
A fraction is in its simplest form (also called lowest terms) when the numerator and denominator have no common factor other than 1. In other words, you cannot simplify it any further.
Reducing fractions to their simplest form makes them easier to read, compare, and work with. In this chapter, we will learn how to simplify fractions using the HCF (Highest Common Factor) method.
What is Simplest Form of a Fraction - Grade 6 Maths (Fractions)?
Definition: A fraction is in its simplest form when the numerator and denominator have no common factor other than 1.
Another way to say this: the HCF of the numerator and denominator is 1.
Examples:
- 3/4 is in simplest form because the only common factor of 3 and 4 is 1.
- 6/8 is NOT in simplest form because 6 and 8 have a common factor of 2. Dividing both by 2 gives 3/4.
- 5/15 is NOT in simplest form because 5 and 15 have a common factor of 5. Dividing both by 5 gives 1/3.
Key fact: Every fraction has exactly one simplest form. No matter how you simplify, you will always reach the same answer.
Simplest Form of a Fraction Formula
Method to find the simplest form:
Simplest form = Numerator ÷ HCF / Denominator ÷ HCF
Steps:
- Find the HCF (Highest Common Factor) of the numerator and denominator.
- Divide both the numerator and denominator by the HCF.
- The result is the fraction in its simplest form.
Alternative method (repeated division):
- Check if both numerator and denominator are divisible by 2. If yes, divide both by 2.
- Check again. Keep dividing by 2 until you cannot.
- Then try dividing by 3, then by 5, then by 7, and so on.
- Stop when no common factor remains.
Derivation and Proof
Let us simplify 24/36 step by step using the HCF method:
- Find the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
- Find the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
- Common factors: 1, 2, 3, 4, 6, 12.
- HCF = 12 (the highest common factor).
- Divide numerator by 12: 24 ÷ 12 = 2.
- Divide denominator by 12: 36 ÷ 12 = 3.
- Simplest form = 2/3.
Verification using repeated division:
- 24/36 — both divisible by 2 → 12/18.
- 12/18 — both divisible by 2 → 6/9.
- 6/9 — both divisible by 3 → 2/3.
- 2 and 3 have no common factor other than 1. Done.
Both methods give the same answer: 2/3.
Types and Properties
Types of problems on simplest form:
- Type 1: Reduce to simplest form — Given a fraction like 18/24, simplify it using HCF.
- Type 2: Check if already in simplest form — Given a fraction like 7/11, check if the HCF of 7 and 11 is 1.
- Type 3: Compare fractions after simplifying — Simplify 12/20 and 15/25 to see if they are equal.
- Type 4: Simplify fractions with large numbers — Use prime factorisation to find the HCF when numbers are large.
- Type 5: Word problems — A class has 18 boys and 12 girls. What fraction of the class are boys? Simplify.
Solved Examples
Example 1: Example 1: Simplify 12/16
Problem: Reduce 12/16 to its simplest form.
Solution:
Step 1: Find HCF of 12 and 16.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 16: 1, 2, 4, 8, 16
- HCF = 4
Step 2: Divide both by HCF.
- 12 ÷ 4 = 3
- 16 ÷ 4 = 4
Answer: 12/16 = 3/4.
Example 2: Example 2: Simplify 15/45
Problem: Reduce 15/45 to its simplest form.
Solution:
- HCF of 15 and 45 = 15 (since 15 divides both exactly).
- 15 ÷ 15 = 1
- 45 ÷ 15 = 3
Answer: 15/45 = 1/3.
Example 3: Example 3: Is 7/12 in Simplest Form?
Problem: Is 7/12 already in its simplest form?
Solution:
- Factors of 7: 1, 7 (7 is a prime number).
- Factors of 12: 1, 2, 3, 4, 6, 12.
- Common factor: only 1.
- HCF = 1.
Answer: Yes, 7/12 is in its simplest form.
Example 4: Example 4: Simplify 36/48
Problem: Reduce 36/48 to its simplest form.
Solution:
Using repeated division:
- 36/48 → divide by 2 → 18/24
- 18/24 → divide by 2 → 9/12
- 9/12 → divide by 3 → 3/4
- 3 and 4 have HCF = 1. Done.
Answer: 36/48 = 3/4.
Example 5: Example 5: Simplify 25/100
Problem: Reduce 25/100 to its simplest form.
Solution:
- HCF of 25 and 100 = 25.
- 25 ÷ 25 = 1
- 100 ÷ 25 = 4
Answer: 25/100 = 1/4.
Example 6: Example 6: Are These Fractions Equal?
Problem: Are 8/12 and 10/15 equal? Simplify both to check.
Solution:
- 8/12: HCF of 8 and 12 = 4. 8/12 = 2/3.
- 10/15: HCF of 10 and 15 = 5. 10/15 = 2/3.
- Both simplify to 2/3.
Answer: Yes, 8/12 and 10/15 are equal fractions.
Example 7: Example 7: Word Problem — Class Composition
Problem: In a class of 40 students, 24 are girls. What fraction of the class are girls? Write in simplest form.
Solution:
Given:
- Total students = 40
- Girls = 24
Fraction of girls = 24/40
- HCF of 24 and 40 = 8.
- 24 ÷ 8 = 3
- 40 ÷ 8 = 5
Answer: 3/5 of the class are girls.
Example 8: Example 8: Simplify 14/21
Problem: Reduce 14/21 to its simplest form.
Solution:
- HCF of 14 and 21 = 7.
- 14 ÷ 7 = 2
- 21 ÷ 7 = 3
Answer: 14/21 = 2/3.
Example 9: Example 9: Simplify 30/42
Problem: Write 30/42 in simplest form.
Solution:
- HCF of 30 and 42 = 6.
- 30 ÷ 6 = 5
- 42 ÷ 6 = 7
Answer: 30/42 = 5/7.
Example 10: Example 10: Word Problem — Chocolates
Problem: Aman ate 6 out of 18 chocolates. What fraction did he eat? Write in simplest form.
Solution:
- Fraction eaten = 6/18.
- HCF of 6 and 18 = 6.
- 6 ÷ 6 = 1
- 18 ÷ 6 = 3
Answer: Aman ate 1/3 of the chocolates.
Real-World Applications
Where do we use simplest form?
- Comparing fractions — It is easier to compare 2/3 and 3/4 than to compare 10/15 and 15/20.
- Adding and subtracting fractions — Simplifying before finding a common denominator saves time.
- Recipes — A recipe that uses 2/4 cup of sugar is clearer as 1/2 cup.
- Test scores — If you scored 18 out of 24, expressing it as 3/4 (or 75%) is clearer.
- Ratios — A ratio like 10:15 is simpler as 2:3. This is the same process as simplifying fractions.
- Probability — Chances are always written in simplest form. 4/6 becomes 2/3.
Key Points to Remember
- A fraction is in simplest form when the HCF of numerator and denominator is 1.
- To simplify, divide both numerator and denominator by their HCF.
- Simplest form is also called lowest terms or reduced form.
- Every fraction has exactly one simplest form.
- A fraction with a prime number as numerator or denominator is often already in simplest form.
- Simplifying does not change the value of the fraction: 4/8 = 1/2.
- The repeated division method (divide by 2, then 3, then 5, etc.) also works.
- Simplifying fractions is the same idea as simplifying ratios.
- Always write your final answer in simplest form.
- Checking: after simplifying, the numerator and denominator should share no common factor other than 1.
Practice Problems
- Reduce 20/32 to its simplest form.
- Is 11/17 in simplest form? Explain why.
- Simplify 45/60.
- Are 9/12 and 12/16 equal fractions? Simplify both to check.
- In a bag of 50 marbles, 30 are red. What fraction are red? Write in simplest form.
- Simplify 56/72 using the HCF method.
- Write three fractions that are equal to 2/5.
- A student scored 16 out of 20 on a test. Express as a fraction in simplest form.
Frequently Asked Questions
Q1. What is the simplest form of a fraction?
A fraction is in simplest form when the numerator and denominator have no common factor other than 1. For example, 3/5 is in simplest form because 3 and 5 share only the factor 1.
Q2. How do I find the simplest form?
Find the HCF (Highest Common Factor) of the numerator and denominator. Divide both by the HCF. The result is the simplest form. For example, 18/24: HCF = 6, so 18/24 = 3/4.
Q3. Does simplifying change the value of a fraction?
No. Simplifying a fraction gives an equivalent fraction. 4/8 and 1/2 represent the same amount. You are dividing both parts by the same number, which is like multiplying by 1.
Q4. Can a fraction have more than one simplest form?
No. Every fraction has exactly one simplest form. No matter which method you use (HCF or repeated division), you will always get the same answer.
Q5. Is 1 always in the simplest form?
Any fraction with numerator 1 (like 1/2, 1/3, 1/7) is always in simplest form because 1 has no common factor with any number other than 1.
Q6. What if the numerator is larger than the denominator?
You can still simplify. For example, 15/10: HCF = 5, so 15/10 = 3/2. The fraction 3/2 is in simplest form (it is an improper fraction).
Q7. What is the quickest way to simplify?
Use the HCF method — it simplifies in one step. If you do not know the HCF, use repeated division by small primes (2, 3, 5, 7) until no common factor remains.
Q8. Is simplifying fractions the same as simplifying ratios?
Yes. A ratio like 12:18 can be simplified by dividing both parts by HCF (6) to get 2:3. This is the same process as simplifying the fraction 12/18 to 2/3.
Related Topics
- Equivalent Fractions
- HCF and LCM
- Comparing Fractions
- Introduction to Fractions
- Proper and Improper Fractions
- Mixed Numbers
- Like and Unlike Fractions
- Fractions on Number Line
- Addition of Fractions
- Subtraction of Fractions
- Unit Fractions
- Word Problems on Fractions
- Types of Fractions
- Addition and Subtraction of Fractions
