The source of this image is an article in NY Times Magazine, the text of which (about a quarter down the page) is quoted exactly.
Kevin Drum, a blogger for MotherJones, tried to track down the source of this anecdote. He found a tweet by the article's author, Elizabeth Green, saying that she got it from "Threshold Resistance", the memoirs of Alfred Taubman, then-owner of A&W.
Reproducing the quote from that book that's on the MJ site:
Well, it turned out that customers preferred the taste of our fresh beef over traditional fast-food hockey pucks. Hands down, we had a better product. But there was a serious problem. More than half of the participants in the Yankelovich focus groups questioned the price of our burger. "Why," they asked, "should we pay the same amount for a third of a pound of meat as we do for a quarter-pound of meat at McDonald's? You're overcharging us." Honestly. People thought a third of a pound was less than a quarter of a pound. After all, three is less than four!
Since any actual data on the study performed is probably private data of A&W and the Yankelovich research group, we'll probably have to take Taubman at his word here.