To compare the relative sizes of fracitons, you are going to want to get them with a common denominator. In this case, the denominators of our fractions are 3 and 4, meaning that we are going to need to find a number that is a common multiple of both numbers.
To do this, simplify each number through prime factorization:
[tex]3 = 3[/tex] (it's prime)
[tex]4 = 2^2[/tex]
Now, the LCM (least common multiple), or the number we are trying to find for, is going to be the product of the each unique base with the lowest exponent. That can sound a little confusing, so let's see how we could do it with this case and that may help to simplify the concept.
Finding the LCM
First, we see that we have two unique bases, 2 and 3. 2 is to the second power and 3 is to the first power. Since the two numbers that we simplified share no common numbers, the LCM is simply going to be the product of the two numbers:
[tex]3 \cdot 4 = 12[/tex]
The LCM is 12.
Now, let's get our fractions with a denominator of 12.
[tex]\dfrac{3 \cdot 3}{4 \cdot 3} = \dfrac{9}{12}[/tex]
[tex]\dfrac{2 \cdot 4}{3 \cdot 4} = \dfrac{8}{12}[/tex]
By simplifying, we can see that 3/4 is larger because it has a greater numerator when the fractions have common denominators.
