How much better (as a percentage) is $A$ than $B$?

Question

This may sound stupid, but is it correct to say

$100$ is a hundred percent better than $50$

where by better I mean higher or something like that. Similarly would it be correct to ask questions like, "How much better is $50$ than $30$?" and, "How much worse is $30$ than $50$?"

Answer 1

The words "better" and "worse" are inherently subjective, so using them in a mathematics question here will get some cringes. Also whether or not the words "better" and "worse" are correct to use in a situation depends on what the percentages represent. Certainly $100\%$ is better than $50\%$ if you are talking about exam scores, but it is certainly worse if you are talking about how much of a population is afflicted with a disease. Instead let's use objective words like "more" and "less".

From your example, $100$ is more than $50$. It is exactly twice $50$, so you could say either of the phrases:

• $100$ is $200\%$ of $50$.
• $100$ is $100\%$ more than $50\%$ (because $100\%$ of $50$ is $50$ and you need $100\%$ more to get to $100$).

For the $50$ and $30$ example, first we need to note that $50/30 \approx 1.67 = 167\%$. Similar to the previous example we can say either of the following phrases:

• $50$ is $167\%$ of $30$.
• $50$ is $67\%$ more than $30\%$ (because $100\%$ of $30$ is $30$ and you need $67\%$ more to get to $50$).

If you really wanted to, you could arrive at this "$67\%$ more" figure directly by looking at how much more than $30$ do we need to get to $50$. That is, look at $(50-30)/30 = 20/30$.

Answer 2

Perhaps I'm misinterpreting your question, but I would suggest that $\frac{A}{B}$ is your answer.

This would tell us that 100 is twice as good as 50, and that 50 is $\frac{5}{3}$ better than 30.