Compare 3/4 and 2/3
Question
Which fraction is greater: 3/4 or 2/3?
Solution — Step by Step
These fractions have different denominators (4 and 3). Fractions with different denominators are called unlike fractions. We cannot directly compare them because the pieces are different sizes — one is split into fourths and the other into thirds.
We need to convert them so they have the same denominator before comparing. We do this using the LCM method.
Step 1: Find the LCM of the denominators.
Denominators are 4 and 3.
Multiples of 4: 4, 8, 12, 16, 20, … Multiples of 3: 3, 6, 9, 12, 15, …
LCM of 4 and 3 = 12
Step 2: Convert 3/4 to an equivalent fraction with denominator 12.
What do we multiply 4 by to get 12? → 4 × 3 = 12
Multiply both numerator and denominator by 3: 3/4 = (3 × 3)/(4 × 3) = 9/12
Step 3: Convert 2/3 to an equivalent fraction with denominator 12.
What do we multiply 3 by to get 12? → 3 × 4 = 12
Multiply both numerator and denominator by 4: 2/3 = (2 × 4)/(3 × 4) = 8/12
Step 4: Compare the two fractions now that they have the same denominator.
9/12 vs 8/12
Since the denominators are the same, compare the numerators: 9 > 8
So 9/12 > 8/12
Step 5: Write the answer in terms of the original fractions.
Since 9/12 = 3/4 and 8/12 = 2/3:
3/4 > 2/3
Answer: 3/4 is greater than 2/3.
Step 1: Find LCM of both denominators Step 2: Convert each fraction to an equivalent fraction with the LCM as denominator Step 3: Compare the new numerators Step 4: The larger numerator means the larger fraction
Why This Works
We cannot directly compare 3/4 and 2/3 because the “piece sizes” are different. Three-fourths means 3 pieces when something is cut into 4 equal parts. Two-thirds means 2 pieces when something is cut into 3 equal parts.
It’s like asking: which is more, 3 pieces of a roti cut into 4 parts, or 2 pieces of a roti cut into 3 parts? We need both rotis cut into the same number of pieces to compare fairly.
By finding the LCM (12), we imagine cutting each roti into 12 equal pieces. Three-fourths becomes 9/12, and two-thirds becomes 8/12. Now the pieces are the same size, and we can clearly see that 9 pieces is more than 8 pieces.
Quick mental check: 3/4 = 0.75 and 2/3 ≈ 0.667. So 3/4 is larger. Use this decimal check to verify your LCM answer in exams — if both methods agree, you can be confident.
Common Mistake
Wrong approach: Comparing numerators directly (3 vs 2, so 3/4 > 2/3)
This accidentally gives the right answer here, but only by coincidence. This method is wrong. For example, comparing 1/2 and 2/5: if you compare numerators (1 vs 2), you’d say 2/5 is bigger. But 1/2 = 5/10 and 2/5 = 4/10, so 1/2 is actually bigger! Always use the LCM method.
In board exams, always show the LCM calculation and all conversion steps clearly. Even if your final answer is right, you need to show the method to earn full marks.