### Brain Teasers

# Non-Fat Frankfurters

"I'm telling you, Kurt," Rod said to his best friend, "these Rnkfurers are the greatest thing since whatever was before sliced bread! They're low-calorie, high-protein, fortified with vitamins B-1, B-36, and F-22, and come with three months of basic cable!"

Kurt finally gave in, and agreed to go in with Rod on a minimum special order of the multi-colored treats. When the box arrived, they divided the bounty as originally planned.

"Interesting," Kurt noted, "that the number of Rnkfurers you got is the same integer as my percentage."

Rod did a little mental math. "You're right," he noted, "and I think that's the lowest pair of integers we could do that with."

"Maybe; the next largest set is trivial. 50 Rnkfurers each, and then we'd each have the percentage the other guy got: 50% and 50 Rnkfurers each."

Kurt unwrapped a Rnkfurer and took his first bite. He glowed with pleasure for half a minute, and then spoke quietly.

"Rod, how many of these in a case?"

"Fifty."

"I think we'd better order a few. And this time, it's my turn to get more than you do."

"Well, we can still arrange it so the number I get is your percentage of the total."

Kurt did a little mental math this time. "Yeah, that's right. I can't fit more than 100 in my desk at school, so this will do the trick."

How many did each boy get each time?

Kurt finally gave in, and agreed to go in with Rod on a minimum special order of the multi-colored treats. When the box arrived, they divided the bounty as originally planned.

"Interesting," Kurt noted, "that the number of Rnkfurers you got is the same integer as my percentage."

Rod did a little mental math. "You're right," he noted, "and I think that's the lowest pair of integers we could do that with."

"Maybe; the next largest set is trivial. 50 Rnkfurers each, and then we'd each have the percentage the other guy got: 50% and 50 Rnkfurers each."

Kurt unwrapped a Rnkfurer and took his first bite. He glowed with pleasure for half a minute, and then spoke quietly.

"Rod, how many of these in a case?"

"Fifty."

"I think we'd better order a few. And this time, it's my turn to get more than you do."

"Well, we can still arrange it so the number I get is your percentage of the total."

Kurt did a little mental math this time. "Yeah, that's right. I can't fit more than 100 in my desk at school, so this will do the trick."

How many did each boy get each time?

### Hint

Although there are a dozen pairs of numbers that satisfy the percentage condition, the three used in the story are the only ones in which each boy gets fewer than 100 Rnkfurers.### Answer

1st order: Rod 20, Kurt 5. Kurt has 20% of the total.2nd order: Rod 60, Kurt 90. Kurt has 60% of the total.

The equation to satisfy is

R = 100*K / (R+K)

where R and K are both integers.

Another way to look at this is

K = R*R/(100-R)

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