SAT math question of the day

Discus: SAT/ACT Tests and Test Preparation: May 2003 Archive: SAT math question of the day
 By Testtaker (Testtaker) on Thursday, May 01, 2003 - 05:09 pm: Edit

They say its medium but I say its hard!
For some reason I disagree with the answer even after reading the explanation:

If half the people in a room leave at the end of every five-minute interval and at the end of twenty minutes the next to the last person leaves, how many people were in the room to start with? (Assume that no one enters the room once the process begins.)

a. 32
b. 28
c. 16
d. 12
e. 8

 By Xiggi (Xiggi) on Thursday, May 01, 2003 - 05:20 pm: Edit

Okies, first, the expression "next to the last" means that after 20 minutes, there will be ONE person left.

This problem is easily solved by pluggin-in an answer and backsolving.

If 16 people were in the room to begin with, this is what would happen:

Beginning 16
After 5 m (8) leave and 8 remain in room
After 10 m (4) leave and 4 remain
After 15 m (2) leave and two people remain in room
After 20 min, (1) leaves and NOBODY is left.

Well, that is obviously incorrect.

So try a bigger number, the number 32 (being exactly double of 16) immediately comes to mind because the progression would be

32>16>8>4>2 and 1.

PS You start with the middle one because you can then evaluate if your answer needed to be bigger or smaller. That cuts down 2 answers and possible waste of time.

Hope this helps

X

 By Apguy (Apguy) on Thursday, May 01, 2003 - 05:21 pm: Edit

Is it just me or was that question phrased in an unusually confusing manner? I am having more trouble reading it that doing the actual math!

 By Testtaker (Testtaker) on Thursday, May 01, 2003 - 05:30 pm: Edit

Xiggi, I completely agrea with your explanation and that is the reason I am very confused. The correct answer is c or 16.

 By Expataznamerica (Expataznamerica) on Thursday, May 01, 2003 - 05:42 pm: Edit

umm, Xiggi, I think the answer it 16.

Think of it this way, with your reasoning. At the 20 minute mark, the next to the last person leaves. That means there's 1 person left at the 20th minute. So...it would look like this:

20 Minute: 1 Person (1 person left)
15 Minuth: 2 Persons (2 persons left)
10 Minute: 4 Persons (4 persons left
5 Minute: 8 Persons (8 persons left)
0 Minute (beginning): 16 Persons

See, at the 20th minute, the second to last person leaves, which leaves 1 person in the room. With your explanation, the "next to last" person left at the 15 minute mark.

The question asked "at the end of twenty minutes, the next to the last person leaves"

This phrase means that only the next to last person left, which still leaves the last person still in the room.

 By Apguy (Apguy) on Thursday, May 01, 2003 - 05:50 pm: Edit

Yep, the collegeboard answer is 16... This one was hard IMO.

http://www.collegeboard.com/apps/qotd/question

 By Testtaker (Testtaker) on Thursday, May 01, 2003 - 06:04 pm: Edit

I think me and xiggi were misinterpreting the question. I think we were thinking that at the end of twenty minutes an additional person would leave and that would be the second to last person. This question was confusing!

 By Brd (Brd) on Thursday, May 01, 2003 - 06:26 pm: Edit

"I think me and xiggi were misinterpreting the question."

I think Ziggi had the right idea, he just made silly mistake, here:

"After 15 m (2) leave and two people remain in room
After 20 min, (1) leaves and NOBODY is left."

If 2 people remain in the room, and then 1 of them leaves, that leaves 1 person, not "nobody".

For what it's worth, this type of problem illustrates what is called a recurrence relation (since it's discrete -- a continuous case would be a differential equation for "exponential decay") and it is possible to solve them in a systematic way. Though for something this simple it's probably just quicker to guess until you get it right.

 By Testtaker (Testtaker) on Thursday, May 01, 2003 - 06:37 pm: Edit

The question states

"If half the people in a room leave at the end of every five-minute interval and **** at the end of twenty minutes the next to the last person leaves****"

What I was thinking and suspect xiggi was too, is that at the end of twenty minutes, ANOTHER person would leave and this person would be the second to last person in the room. Interpreting the question this way, would give 32 as the correct answer as xiggi illustrated. This is why this question seems unfair.

 By Coles (Coles) on Thursday, May 01, 2003 - 06:40 pm: Edit

I worked it and got 16, i just drew a diagram.

20-----------------------------
The number between 15 and 20 must be 2 since one person leaves at 20 minutes (and that 1 person must be half of the room)
2 PEOPLE

15-----------------------------
For the number of people b/w 10 and 15, double the number of people b/w 15 and 20.
4 PEOPLE

10-----------------------------
People b/w 5 and 10. Double again.
8 PEOPLE

5-----------------------------
People b/w 0 and 5, this is the number we want because when time=0 the problem is just starting. Just double the number of people you have between 5 and 10 minutes.
START-------------------------

Sorry if that was overly confusing.

 By Expataznamerica (Expataznamerica) on Thursday, May 01, 2003 - 07:19 pm: Edit

Re-interpret last phrase from "at the end of twenty minutes, the next to the last person leaves" to:

"At the end of 20 minutes, the 2nd to last person left," thus leaving one person.

 By Brd (Brd) on Thursday, May 01, 2003 - 08:40 pm: Edit

Testtaker, interpretation was not xiggi's error. His error was that in the last step he calculated 2 - 1 and got zero. His analysis was completely correct up to that point, and he would have gotten the correct answer, but for the silly mistake.

 By Xiggi (Xiggi) on Thursday, May 01, 2003 - 08:57 pm: Edit

XIGGI IS A DUMBASS!

That will teach me to type answers while solving the problems and being online. Indeed, I made the silly error. It should have been written like this

Okies, first, the expression "next to the last" means that after 20 minutes, there will be ONE person left.

This problem is easily solved by pluggin-in an answer and backsolving.

If 16 people were in the room to begin with, this is what would happen:

Beginning 16
After 5 m (8) leave and 8 remain in room
After 10 m (4) leave and 4 remain
After 15 m (2) leave and two people remain in room
After 20 min, (1) leaves and 1 is left.

Well, that is the correct answer!

As Brd said, I lost focus and typed it without checking the simple math.

C was the anwer. Sorry for the carelessness. On the oher hand, that is sooooo SAT Test taking like. We all see the problem, understand it, do all correctly but then get the wrong answer because being careless.

That is the lesson of the day!