Difference between 'at' and 'by' for percentages

Question

Please confirm the difference between these two sentences:

Salad is the most popular dish by 43%.

Salad is the most popular dish at 43%.

Is the meaning of these two sentences same or different? (this sentence came directly from an electronic dictionary. The full text is:

In America the most popular business lunch is a salad by 43% over a meat dish at 26%.)

Does it mean that a salad is 43% more popular with a popularity of 69%? Or is the usage of the preposition by simply a mistake in this sentence?

Answer 1

If something is more popular by 43%, that means it is 43% more popular than the next most popular item (whether absolute or relative is left ambiguous).

If it's more popular at 43%, that is what it's percentage is - the next most popular item may be at 42%, for example.

Basically the difference lies in whether or not the percentage given is relative to the next most popular item. I would say that the usage in your example is mistaken - or at least very unclear

Answer 2

While what @Chromane said is correct, it changes a little with context.

In general yes, "by 43%" means 43% more, and "at 43%" means that is the flat percentage.

However, the "at" in the sentence you gave acts like a colon would. Therefore it should be read as:

Salad is the most popular business lunch in America (presumably in some kind of vote or survey, 43% of people gave this as their answer), and the next most popular is the meat dish, which 26% of people voted for.

I agree that this is not a particularly clear sentence, but the "at" given is not ungrammatical, just made even more confusing by the use of percentages.