I may be wrong, but I think the issue is: what does it mean to divide x by 1/2? It means to multiply it by 2. We can easily understand that dividing by 2 mean

I may be wrong, but I think the issue is: what does it mean to divide x by 1/2? It means to multiply it by 2. We can easily understand that dividing by 2 means to split something into 2 pieces. What does it mean to split something into 1/2 pieces? Why does dividing by 1/2 mean to multiply it by 2 and does it always work that way? Can you prove it? Summation of fractions is much easier to visualise, even if it is harder to calculate.