Herman provides ketchup, relish, and mustard for the customers at his hot dog stand. During the lunch rush one day, he recorded how many customers had each of these three condiments. He observed the following:
The total number of customers was \(125\), and each ordered a single hot dog.
\(82\) customers had ketchup, \(47\) had relish, and \(80\) had mustard.
\(32\) customers had mustard and ketchup, but not relish.
\(5\) customers had mustard and relish, but not ketchup.
The number of customers who had all three of the condiments was double the number of customers who had none of the condiments.
The number of customers who had exactly two of the condiments was double the number of customers who had all three of the condiments.
How many customers had ketchup, relish, and mustard on their hot dog?
Let \(x\) be the number of customers who had none of the condiments. Then \(2x\) customers had all three of the condiments, and \(4x\) customers had exactly two of the condiments.
From the given information, since \(32\) customers had mustard and ketchup but not relish, and \(5\) customers had mustard and relish but not ketchup, it follows that \(4x-32-5=4x-37\) customers had ketchup and relish, but not mustard.
Also, since \(82\) customers in total had ketchup, then \(82-32-2x-(4x-37)=87-6x\) customers had only ketchup. Similarly, since \(47\) customers in total had relish, then \(47-5-2x-(4x-37)=79-6x\) customers had only relish. As well, since \(80\) customers in total had mustard, then \(80-32-2x-5=43-2x\) customers had only mustard. We summarize this information in the following Venn diagram.
Since there were \(125\) customers in total, \[\begin{aligned} 125 &= (87-6x) + (4x-37) + 2x + 32 + (79-6x) + 5 + (43-2x) + x\\ 125 &= 209 - 7x\\ 7x &= 84\\ x &= 12 \end{aligned}\] Therefore, \(2x=24\) customers had ketchup, relish, and mustard on their hot dog.