Simplifying fractions
FoundationDivide numerator and denominator by the same common factor until it cannot be simplified.
After this lesson, you can:
- Simplify = divide both by the same common factor
- Stop when nothing divides = simplest form
- Simplifying never changes the value
Before this lesson you should know: What is a fraction?
1Summary 重點摘要
Divide numerator and denominator by the same common factor until it cannot be simplified.
2Vocabulary cards Vocabulary
F
Factor因數
A number that divides another exactly, e.g. 1, 2, 4, 8 are factors of 8.
C
Common factor公因數
A factor shared by numerator and denominator.
S
Simplest form最簡分數
The fraction that cannot be simplified further, like 34.
E
Equivalent fractions等值分數
Fractions with the same value but different forms, like 68 and 34.
G
Greatest common factor最大公因數
The largest common factor, e.g. GCF of 12 and 16 is 4.
D
Divide both同除
Divide numerator and denominator by the same number; the value stays the same.
E
Even number偶數
A number divisible by 2; if both parts are even you can at least halve, e.g. 812 → 46.
R
Reduce約簡
The act of simplifying, e.g. 1015 reduces to 23.
3Step by step Steps
- Step 1: Find a common factorTo simplify 68, first find a factor shared by the top and the bottom: 6 has factors 1, 2, 3, 6 and 8 has 1, 2, 4, 8, so they share 1 and 2. Listing factors is the sure way, and small primes are worth trying first.
- Step 2: Divide both by itDivide the numerator and the denominator by the same factor 2: 6 ÷ 2 = 3 and 8 ÷ 2 = 4, giving 34. Both parts must be divided together in step; dividing only one changes the value, and then it is not simplifying.
- Step 3: Check if simplestLook at 3 and 4: they share no factor except 1, so 34 is already in simplest form and cannot be simplified further. Simplest form means the top and bottom share only the factor 1; recheck after every round.
- Step 4: Use the GCF to finish in one stepThe greatest common factor of 12 and 18 is 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3, so 1218 becomes 23 in a single move. Using the greatest common factor finishes in one step, with no repeated checking and nothing overlooked.
- Step 5: Several small steps work tooIf the greatest common factor is not obvious at once, go gradually: 1218 divided by 2 gives 69, then divided by 3 gives 23. Several small rounds reach exactly the same result, only with more steps, which suits beginners.
- Step 6: Simplifying keeps the value24 = 12: taking 2 slices from a pizza cut into 4 is exactly as much as taking 1 slice from one cut into 2. Simplifying only changes the way the fraction is written, never the amount, so there is nothing to fear.
- Step 7: Use the last digitsLet the last digits pick the divisor: 1025 ends in 0 and 5, so try 5 first: 10 ÷ 5 = 2 and 25 ÷ 5 = 5, giving 25. Endings of 0 or 5 mean a factor of 5, and even endings mean a factor of 2, which speeds up simplifying.
4Diagrams Diagrams
5Worked example Example
Example: simplify 1216 to its simplest form.
- List the factors: 12 has 1, 2, 3, 4, 6, 12; 16 has 1, 2, 4, 8, 16.
- Find common factors: both share 1, 2 and 4, so the greatest common factor is 4.
- Divide the top and bottom by 4: 12 ÷ 4 = 3 and 16 ÷ 4 = 4.
- Write the answer: 1216 = 34.
- Check: 3 and 4 share no factor except 1, so 34 is in simplest form.
- Answer: 1216 = 34. We listed the factors of 12 and 16, found the greatest common factor 4, and divided both the numerator and the denominator by it, finishing in one step. Since 3 and 4 share no factor except 1, 34 is confirmed to be in simplest form. The value of the fraction never changed; only the way it is written did.
6Common mistakes Mistakes
✗ Dividing only the numerator: 68 becomes 38.
✓ Divide numerator and denominator by the same number: 68 = 34.
✗ Stopping halfway: 1216 → 68 and stop.
✓ Keep going until simplest: 1216 → 68 → 34.
✗ Dividing by zero.
✓ Zero is never a divisor — simplify only by factors other than 1.
✗ Thinking simplifying makes the fraction smaller.
✓ Simplifying only changes the written form; the value stays the same: 36 = 12.
7Quick check Quick check
Q1Simplify 410.
Answer: 25 (divide both by 2).
Did you get it right?
Q2Simplify 912.
Answer: 34 (divide both by 3).
Did you get it right?
Q3Does 57 need simplifying?
Answer: No — 5 and 7 share no common factor except 1; it is already simplest.
Did you get it right?
Q4Simplify 1520.
Answer: The GCF is 5: 15÷5=3, 20÷5=4 → 34.
Did you get it right?
Q5Which is bigger, 68 or 34?
Answer: They are equal — 68 simplifies to 34; equivalent fractions.
Did you get it right?
8Recap Recap
- Simplify = divide both by the same common factor
- Stop when nothing divides = simplest form
- Simplifying never changes the value
- The GCF gets you to simplest form in one step
- Both even? Try 2. Ending in 05? Try 5.