Compare Non-Equivalent Fractions

How many equal-sized pieces the whole is divided into.

How many equal-sized pieces we have.

[ggfrac]4/6[/ggfrac], [ggfrac]6/9[/ggfrac], [ggfrac]8/12[/ggfrac]. I can find equivalent fractions by multiplying both the numerator and denominator in [ggfrac]2/3[/ggfrac] by any number, as long as it is the same number.

Since 6 × 4 is 24, I also need to multiply 5 by 4 to get an equivalent fraction. The equivalent fraction is [ggfrac]20/24[/ggfrac].

20 is a multiple of 5, so I can multiply the denominator, 5, by 4 to get 20. [ggfrac]2x4/5x4[/ggfrac] = [ggfrac]8/20[/ggfrac]. The fractions are equivalent.

I can use [ggfrac]1/2[/ggfrac] as a benchmark fraction. [ggfrac]1/2[/ggfrac] = [ggfrac]2/4[/ggfrac], so [ggfrac]3/4[/ggfrac] is more than [ggfrac]1/2[/ggfrac]. Since [ggfrac]1/3[/ggfrac] is less than [ggfrac]1/2[/ggfrac], we know that [ggfrac]3/4[/ggfrac] is greater than [ggfrac]1/3[/ggfrac].

[ggfrac]1/2[/ggfrac] is not very useful because both [ggfrac]4/5[/ggfrac] and [ggfrac]6/7[/ggfrac] are greater than [ggfrac]1/2[/ggfrac].

Denominators tell the size of the unit, so if I have two different denominators, it is like comparing different numbers of different-sized slices.

I can find equivalent fractions for each fraction until I find two that have the same denominator.

I could cut the slice in the first lasagna into three equal-sized pieces, and cut each of the slices in the second lasagna into two equal-sized pieces. Now each of the lasagnas can be expressed using 6 as a denominator.