Learn how different fractions can represent the same amount
What we'll learn:
✓ What equivalent fractions are
✓ How to find equivalent fractions
✓ Multiple methods for creating equivalents
✓ Lots of practice problems!
Equivalent Fractions: Fractions that look different but have the same value.
They represent the same amount or the same part of a whole.
Imagine you have a pizza cut into 4 slices. You eat 2 slices.
Now imagine the same pizza cut into 8 slices. You eat 4 slices.
Did you eat the same amount? YES!
2/4 = 4/8
These are equivalent fractions!
1/2 (One Half)
2/4 (Two Fourths)
Both show HALF of the whole!
1/2 = 2/4
2/3 (Two Thirds)
4/6 (Four Sixths)
Both show the same amount shaded!
2/3 = 4/6
3/4 (Three Fourths)
6/8 (Six Eighths)
Same amount of the whole!
3/4 = 6/8
1/2 = 2/4
1 × 2 = 2 2 × 2 = 4
2/3 = 4/6
2 × 2 = 4 3 × 2 = 6
3/4 = 6/8
3 × 2 = 6 4 × 2 = 8
We multiplied BOTH the numerator and denominator by the SAME NUMBER!
To find an equivalent fraction:
Multiply the numerator and denominator by the SAME NUMBER.
Important: You MUST use the same number for both top and bottom!
We multiply by a fraction equal to 1 (n/n = 1)
Let's multiply by 3!
Step 1: Write the original fraction
2/5
Step 2: Multiply the numerator by 3
2 × 3 = 6
Step 3: Multiply the denominator by 3
5 × 3 = 15
Step 4: Write the new fraction
6/15
2/5 = 6/15
Let's multiply by 4!
Step 1: Write the original fraction
3/7
Step 2: Multiply the numerator by 4
3 × 4 = 12
Step 3: Multiply the denominator by 4
7 × 4 = 28
Step 4: Write the new fraction
12/28
3/7 = 12/28
Remember: Any number divided by itself equals 1
2/2 = 1 3/3 = 1 4/4 = 1 5/5 = 1
5 × 1 = 5
100 × 1 = 100
When we multiply both the numerator and denominator by the same number, we're really just multiplying the fraction by 1 in disguise!
2/5 × 3/3 = 6/15
Since 3/3 = 1, we haven't changed the value!
Problem 1: Find an equivalent fraction for 1/3 by multiplying by 4
Solution:
Multiply numerator: 1 × 4 = 4
Multiply denominator: 3 × 4 = 12
Answer: 1/3 = 4/12
Problem 2: Find an equivalent fraction for 2/7 by multiplying by 3
Solution:
Multiply numerator: 2 × 3 = 6
Multiply denominator: 7 × 3 = 21
Answer: 2/7 = 6/21
Problem 3: Find an equivalent fraction for 3/5 by multiplying by 2
Solution:
Multiply numerator: 3 × 2 = 6
Multiply denominator: 5 × 2 = 10
Answer: 3/5 = 6/10
Problem 4: Find an equivalent fraction for 4/9 by multiplying by 5
Solution:
Multiply numerator: 4 × 5 = 20
Multiply denominator: 9 × 5 = 45
Answer: 4/9 = 20/45
Problem 5: Find an equivalent fraction for 5/8 by multiplying by 3
Solution:
Multiply numerator: 5 × 3 = 15
Multiply denominator: 8 × 3 = 24
Answer: 5/8 = 15/24
Problem 6: Find an equivalent fraction for 1/6 by multiplying by 7
Solution:
Multiply numerator: 1 × 7 = 7
Multiply denominator: 6 × 7 = 42
Answer: 1/6 = 7/42
We can multiply by any number to get an equivalent!
1/2 × 2/2 = 2/4
1/2 × 3/3 = 3/6
1/2 × 4/4 = 4/8
1/2 × 5/5 = 5/10
1/2 × 10/10 = 10/20
All of these fractions are equivalent to 1/2!
There are infinite equivalent fractions for any fraction!
To simplify a fraction to an equivalent:
Divide the numerator and denominator by the SAME NUMBER.
Important: This only works if both numbers divide evenly by that number!
If multiplication makes fractions "bigger" (more pieces), then division makes them "simpler" (fewer pieces).
6/8 ÷ 2/2 = 3/4
We divided both top and bottom by 2!
We need to find a number that divides evenly into both 4 and 12.
Step 1: Find a common factor
Both 4 and 12 can be divided by 4
Step 2: Divide the numerator by 4
4 ÷ 4 = 1
Step 3: Divide the denominator by 4
12 ÷ 4 = 3
Step 4: Write the simplified fraction
1/3
4/12 = 1/3
We need to find a number that divides evenly into both 10 and 15.
Step 1: Find a common factor
Both 10 and 15 can be divided by 5
Step 2: Divide the numerator by 5
10 ÷ 5 = 2
Step 3: Divide the denominator by 5
15 ÷ 5 = 3
Step 4: Write the simplified fraction
2/3
10/15 = 2/3
Problem 7: Simplify 6/9
Solution:
Step 1: Both 6 and 9 are divisible by 3
Step 2: Divide numerator: 6 ÷ 3 = 2
Step 3: Divide denominator: 9 ÷ 3 = 3
Answer: 6/9 = 2/3
Problem 8: Simplify 8/12
Solution:
Step 1: Both 8 and 12 are divisible by 4
Step 2: Divide numerator: 8 ÷ 4 = 2
Step 3: Divide denominator: 12 ÷ 4 = 3
Answer: 8/12 = 2/3
Problem 9: Simplify 15/20
Solution:
Step 1: Both 15 and 20 are divisible by 5
Step 2: Divide numerator: 15 ÷ 5 = 3
Step 3: Divide denominator: 20 ÷ 5 = 4
Answer: 15/20 = 3/4
Problem 10: Simplify 14/21
Solution:
Step 1: Both 14 and 21 are divisible by 7
Step 2: Divide numerator: 14 ÷ 7 = 2
Step 3: Divide denominator: 21 ÷ 7 = 3
Answer: 14/21 = 2/3
Problem 11: Simplify 12/18
Solution:
Step 1: Both 12 and 18 are divisible by 6
Step 2: Divide numerator: 12 ÷ 6 = 2
Step 3: Divide denominator: 18 ÷ 6 = 3
Answer: 12/18 = 2/3
Problem 12: Simplify 20/25
Solution:
Step 1: Both 20 and 25 are divisible by 5
Step 2: Divide numerator: 20 ÷ 5 = 4
Step 3: Divide denominator: 25 ÷ 5 = 5
Answer: 20/25 = 4/5
To find a missing number in equivalent fractions:
Step 1: Look at the denominators: 5 and 15
Step 2: What do we multiply 5 by to get 15? Answer: 3 (5 × 3 = 15)
Step 3: Multiply the numerator by the same number: 2 × 3 = 6
2/5 = 6/15
Step 1: Look at the numerators: 3 and 9
Step 2: What do we multiply 3 by to get 9? Answer: 3 (3 × 3 = 9)
Step 3: Multiply the denominator by the same number: 4 × 3 = 12
3/4 = 9/12
Look at the numbers you DO have. Figure out the pattern, then apply it to find what's missing!
Problem 13: 1/4 = ?/12
Solution:
Step 1: Look at denominators: 4 and 12
Step 2: 4 × ? = 12 → 4 × 3 = 12
Step 3: Multiply numerator by 3: 1 × 3 = 3
Answer: 1/4 = 3/12
Problem 14: 2/3 = 8/?
Solution:
Step 1: Look at numerators: 2 and 8
Step 2: 2 × ? = 8 → 2 × 4 = 8
Step 3: Multiply denominator by 4: 3 × 4 = 12
Answer: 2/3 = 8/12
Problem 15: 3/5 = ?/25
Solution:
Step 1: Look at denominators: 5 and 25
Step 2: 5 × ? = 25 → 5 × 5 = 25
Step 3: Multiply numerator by 5: 3 × 5 = 15
Answer: 3/5 = 15/25
Problem 16: 4/7 = 12/?
Solution:
Step 1: Look at numerators: 4 and 12
Step 2: 4 × ? = 12 → 4 × 3 = 12
Step 3: Multiply denominator by 3: 7 × 3 = 21
Answer: 4/7 = 12/21
Problem 17: 5/6 = ?/18
Solution:
Step 1: Look at denominators: 6 and 18
Step 2: 6 × ? = 18 → 6 × 3 = 18
Step 3: Multiply numerator by 3: 5 × 3 = 15
Answer: 5/6 = 15/18
Problem 18: 2/9 = 6/?
Solution:
Step 1: Look at numerators: 2 and 6
Step 2: 2 × ? = 6 → 2 × 3 = 6
Step 3: Multiply denominator by 3: 9 × 3 = 27
Answer: 2/9 = 6/27
These problems require you to think carefully and use what you've learned!
Challenge 1: Are 3/4 and 9/12 equivalent? Prove it!
Solution:
Method 1 - Multiply:
Start with 3/4
3 × 3 = 9 and 4 × 3 = 12
So 3/4 = 9/12 ✓
Method 2 - Simplify:
Start with 9/12
9 ÷ 3 = 3 and 12 ÷ 3 = 4
So 9/12 = 3/4 ✓
Yes! They are equivalent!
Challenge 2: Find THREE equivalent fractions for 2/5
Solution:
Multiply by 2/2: 2/5 = 4/10
Multiply by 3/3: 2/5 = 6/15
Multiply by 4/4: 2/5 = 8/20
Answer: 4/10, 6/15, and 8/20
(Note: Many other answers are possible!)
Challenge 3: Complete the pattern: 1/3 = 2/6 = ?/9 = 4/?
Solution:
For ?/9:
Denominator went from 3 to 9, so we multiplied by 3
1 × 3 = 3
Answer: 3/9
For 4/?:
Numerator went from 1 to 4, so we multiplied by 4
3 × 4 = 12
Answer: 4/12
Complete Pattern: 1/3 = 2/6 = 3/9 = 4/12
Challenge 4: Which fraction is NOT equivalent to 1/2?
A) 2/4 B) 3/6 C) 4/8 D) 5/9
Solution:
Let's check each one:
A) 2/4: 1 × 2 = 2 and 2 × 2 = 4 ✓ Equivalent
B) 3/6: 1 × 3 = 3 and 2 × 3 = 6 ✓ Equivalent
C) 4/8: 1 × 4 = 4 and 2 × 4 = 8 ✓ Equivalent
D) 5/9: We can't multiply 1 and 2 by the same number to get 5 and 9 ✗
Answer: D) 5/9 is NOT equivalent to 1/2
WRONG: 2/3 = 4/6 because I added 2 to both
❌ You must MULTIPLY (or divide), not add or subtract!
✅ CORRECT: 2/3 = 4/6 because I multiplied both by 2
WRONG: 1/2 = 3/2 (only multiplied numerator)
❌ You must multiply BOTH numerator AND denominator!
✅ CORRECT: 1/2 = 3/6 (multiplied both by 3)
WRONG: 2/5 = 6/10 (multiplied top by 3, bottom by 2)
❌ The multiplier must be THE SAME for both!
✅ CORRECT: 2/5 = 4/10 (multiplied both by 2)
A recipe calls for 1/2 cup of milk, but you only have a 1/4 cup measuring cup.
Solution: 1/2 = 2/4, so use the 1/4 cup twice!
You need to pay 3/4 of a dollar.
Solution: 3/4 = 75/100, which is 75 cents!
Which is more: 2/3 of a pizza or 5/8 of a pizza?
Solution: Convert to equivalent fractions with the same denominator to compare!
A board is 3/4 inch thick. Your ruler shows eighths.
Solution: 3/4 = 6/8 inches!
Whatever you do to the TOP, you must do to the BOTTOM!
You mastered:
✓ Understanding what equivalent fractions are
✓ Multiplying to create equivalent fractions
✓ Dividing to simplify fractions
✓ Finding missing numbers
✓ Solving challenge problems
Coming Next:
Using equivalent fractions to add and subtract!
Finding common denominators
And even more fraction adventures!
Keep practicing and you'll be a fraction master! 🌟