“Carrot” (a vegetable) sounds like “karat” (a unit of gold), and there’s a pun:
 
Q: How do you turn soup into gold?
A: Add 24 carrots.
 
The joke has been cited in print since at least 1980 (when it was “14 carrots”).
 
   
Wikipedia: Fineness
The karat (not carat as a unit of mass) (symbol: K or kt) (US) or carat (symbol: C) is a unit of purity for gold alloys.
(...)
Therefore, 24-karat gold is pure (also includes down to 99.95% gold by mass), 18-karat gold is 18 parts gold, 6 parts another metal (forming an alloy with 75% gold), 12-karat gold is 12 parts gold (12 parts another metal), and so forth.
   
12 October 1980, Newsday (Long Island, NY), “Smiles,” Kidsday, pg. L9, col. 3:
How do you turn soup into gold?
Add 14 carrots.
—Kenny Myuna, no address
 
11 July 2001, Los Angeles (CA)

, “The Kid’s Reading Room; Reading by 9; Jokes & Riddles,” pg. E8:
How do you turn soup into gold? Add 24 carrots! (Raquel, 9, Pressman Academy, Los Angeles)
     
Google Groups: alt.humor.puns
Puns of the Weak 11-14-03 Part 2 The Puns
Stan Kegel
11/15/03
(...)
How do you make soup into gold?
Put 24 carrots in it! (Dana, 14)
 
Google Books
The Ultimate Kid’s Joke Book
By Peter Coup
London: Arcturus Publishing Limited
2008
Pg. ?:
How do you turn soup into gold?
Put 18 carrots in it!
 
Twitter
John Erik Metcalf
‏@johnerik
@ksm718 hmmm. I like one. Haha. Here’s one I like: how do u turn soup into gold?  ... Add 14 carrots. muhahha
4:31 PM - 23 May 2008
 
Twitter
Laffy Taffy
‏@laffytaffyjokes
How do you turn soup into gold?—Add 24 carrots!
8:15 AM - 19 Apr 2009
   
reddit
How do you turn soup into gold? (self.cleanjokes)
submitted June 2, 2013 by Theorize
You add 24 carats!
 
reddit
How do you make soup into gold? Add 24 carrots! (imgur.com)
submitted April 25, 2015 by Blizzy_Nuggs
 
Google Books
1001 Kid’s Jokes
By Kay Barnham
London: Arcturus Publishing Limited
2016
Pg. ?:
How do you turn soup into gold?
Add 24 carrots.