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## No-Fat Frankfurters, Partition Deux

Math brain teasers require computations to solve.

 Puzzle ID: #30519 Fun: (2.72) Difficulty: (2.85) Category: Math Submitted By: norcekri Corrected By: lessthanjake789

Kurt & Rod were hanging around the window at school munching the last of their Rnkfurers hoard. They spotted their neighbor lounging in the enclosed courtyard, looking hungry.
"Patty O'Furniture! Good to see you! Want a Rnkfurter?"
"A what?"
Kurt handed her the last he had. She bit in, and her face lit up like a newly-coined metaphor.
"Where can I get more of these?"
"Rod and I were about to go order some more. But there's a catch: we always order so one of us gets the quantity that the other has for a percentage." Patty's blank look bade him explain. "In our first batch, we got a total of 25 -- 20 for me, and 5 for Rod. I had 20, and he had 20% of the total."
"And you think you can do that for three people? You get my percentage, and I get Rod's? I don't think we can do it."
Rod had been thinking about it, and rattled off three numbers to her.
"Those aren't right. The last one isn't the right percentage; there's an added fraction."
"So? My mom says they're the same on her old phonograph speeds."
Patty looked at Kurt, each waiting for the other to give an opinion. Finally, she gestured at the empty wrapper in his hand.
"I'll round off a little for a week's supply of these. Let's go!"

Since Rnkfurters come in cases of 25, 50, and 100, they had to leave 10 with the storekeeper to make the numbers work, but they got their munchies.

How many did each student get this time?

27 for Kurt, 30 for Patty, and 33 for Rod.
That's 30% for Kurt and 33-1/3% for Patty.

Method:

Let K, P, and R stand for the quantity each of them had. Let T be the total, K+P+R.
Now, since Kurt's percentage equals Patty's quantity, we get
100*K/T = P.
Similarly,
100*P/T = R plus a fraction. Now, look at the common factor, 100/T. Let's call this F, the factor relating one quantity to the next.
K*F = P and P*F = R (plus a fraction),
and K*F^2 = R (plus a fraction).
The clue about the phonograph leaves us looking for a triple with 33 Rnkfurters for Rod, with Patty getting 33-1/3% (one-third) of the total.
P = T/3
P = (K+P+33)/3 -- remember, R=33
Now, combine this with 100*K/T = P, and a bit of algebra gives us the required answer.
-------------
Alternately, there are 11 triples that fit the factor requirement (courtesy of a short computer program), ranging from (19, 25, 32) to (2376, 96, 3). The ones that total 10 less than a multiple of 25 are
27, 30, 33
63, 45, 32
891, 90, 9

Given the phonograph clue, I expect you can recognize that the first triple above is the answer.

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