During 100 minutes of playing time, each of 5 teams plays each of the other teams 4 times exactly once. Only 2 teams play at any given time. If the total playing time for each team is the same, what is the total number of minutes that each team plays?
Answer: 40
No idea how to even start this. HELP
100 minutes of playing time, and 2 teams play at any time, so a total of 200 team-minutes are played.
Distributed evenly across 5 teams, this gives 40 minutes of playing time per team.
The total number of games is 10 (can you tell why?).
100min / 10 = 10min per game.
Since each team plays exactly 4 games, the total playing time for each team is 10min x 4 = 40min.
@avneety What’s this question’s source?
I misread this and got the same answer! I read that each team played the others TWICE. So I got a total of 20 games played, 5 minutes per game. Each team plays 8…5 min x 8 = 40 minutes.
So why didn’t my mistake matter? See @MITer94 's clever solution above!
Do you mean “each of 5 teams plays each of the other 4 teams exactly once?”
The problem you posted is very different, “each of 5 teams plays each of the other teams 4 times exactly once.”
How can each team play each other team “4 times” but “exactly once?”
If you mean just exactly once, the problem becomes much more simple.
What a rich question!
@curlypie99 I bet, that was a typo: the question should read
Amazingly, as @pckeller noted, it does not matter whether each of the teams played each of the other teams once or twice - the answer will be the same!
Actually, as long as each team played each of the other teams the same number of times (once, or twice, or 3 times, or whatever), the answer still will be 40 minutes.
But wait, there is more!
Both @pckeller and I arbitrarily assumed that each individual game’s time was the same number of minutes (that assumption would be okay if the question appeared on the SAT; by the way, I think it’s more like Math 2 Level 2 type of question).
Those individual games’ times can be different! (Think a tournament table and sudoku; there are just a couple of restrictions.)
That’s the power of math (and the math power of @MITer94) for ya.
^yeah, oftentimes even the hardest SAT questions have a pretty simple solution, even if it takes more effort to find.
For what it’s worth, this is pretty much a stock SAT question. It comes up often enough that it’s a good choice to make sure you understand.
@bjkmom Could you support your statement by pointing to several similar questions in some of the QAS’s? I have a colection of all of the released SAT’s going back about 13 years. (After the January SAT the value of this compendiun will drop dramatically. `: )
I’ve been teaching SAT prep for a good 20 years. Sorry, but I throw out the book each time a new edition comes out.
But every time I see a problem like this-- or like one of several other stock types-- I make a point of showing the kids the approach and tell them to take notice.