IQ Question That Stumped Me?

Question

I came across the following question on an IQ test, the answer to which I was left guessing on, after several minutes trying to solve it (non-mathematically):

Water is dripping into an empty glass. Each minute, the number of drops doubles. The glass is full in one hour. How many minutes does it take to fill the glass up to half?

a. 30 minutes
b. 40 minutes
c. 59 minutes
d. 1 minute
e. 23 minutes

Answer 1

Hmm...I don't think this question is well-worded. What does "each minute, the number of drops doubles" mean? Does it mean that each minute, the total number of drops in the glass doubles? Or does it mean that after each minute, the water suddenly begins dripping twice as fast as it had before? I'll answer both interpretations.

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If the first intepretation is correct, then the answer is easy--since the number of drops in the glass doubled from the 59th minute to the 60th minute, then there must have been half as much water in the glass after 59 minutes then there were after 60 minutes. So the answer is (c).

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If the other intepretation is correct, then the problem is much more difficult, but the best answer is still (c), 59 minutes. (The exact answer is very, very slightly more than 59 minutes.)

Suppose that the water starts dripping at n drips per minute. Then

In Minute 1, n drips fall in
In Minute 2, 2n drips fall in
In Minute 3, 4n drips fall in
In Minute 4, 8n drips fall in
...

Mathematically, the pattern is that during Minute k, [2^(k - 1)] * n drips fall in. So the total number of drips (which fill the glass) is

n + 2n + 4n + 8n + ... + (2^59) * n

Pulling out an n, the total number of drips to fill the glass is

n(1 + 2 + 4 + 8 + ... + 2^59)

Now, this expression in the parentheses sums to (2^60 - 1). If you don't know this, you can probably convince yourself by looking at the pattern:

1 + 2 = 3 = (2^2) - 1
1 + 2 + 4 = 7 = (2^3) - 1
1 + 2 + 4 + 8 = 15 = (2^4) - 1
1 + 2 + 4 + 8 + 16 = 31 = (2^5) - 1
...
1 + 2 + 4 + ... + 2^59 = (2^60) - 1 [If the pattern were to continue in the same way, which in fact it does]

So, in total, we have ((2^60) - 1) * n drips. The glass, then , will be half full when we have ((2^60) - 1) * n/2 drips; this simplifies to ((2^59) - 1/2)n drips.

After 59 minutes, by the same pattern as above, we have ((2^59) - 1) * n drips. This is almost exactly half of the final amount, so it must be the answer we seek.

(The exact answer is 59 + 2^(-60) minutes, which is about 59.0000000000000000008673617 minutes. Close enough, I think.)

Answer 2

Answer is C because in 60 minutes the glass will be full. So in 59 minutes the glass is in a half way to full.

Answer 3

The posters answering c are correct. I'm just responding to comment about how unrealistic this question is. The number of drops in the last minute is 2^59 times the number of drops in the first minute. There isn't enough water in the ocean to fill this glass. :)

Answer 4

The answer is (almost exactly) 59 minutes.

After 1 minute, there is a tiny amount of water in the glass. We will call it "x". During the second minute, 2x is added, bringing our total to 3x. During the third minute, 4x is added, bringing our total to 7x. During the fourth minute, 8x is added, bringing our total to 15x. During the fifth minute, 16x is added, bringing our total to 31x.

Notice the pattern forming. After t minutes, the amount of water is (2^t - 1)*x, where x is the amount of water added in the first minute.

When t becomes large (such as 59), the "-1" in the statement is very small, so it can effectively be ignored. When t = 60, the glass is full. The amount of water added during the 60th minute is almost exactly the same as the amount that was already in the glass at the 59th minute. So, the glass was almost exactly half-full after 59 minutes.

Answer 5

The answer is c, the best way to find the answer is to work backwards. If at 60 minutes the glass is full, then at 59 minutes it is half-full. Because it doubles every minute, the minute before last it will be exactly half-way full.

Answer 6

Lets try this one
m=minute
V=Volume

V in 60 min = SUM(2^(m-1) ) from m=1 to m=60 fills the glass

program in excel and V in 60 min +2 X V in 59 min

Answer 7

59 minutes.

Answer 8

It's c...59 minutes...because if it's doubling in full-ness for every minute it gains, that means if you go backwards it's halving in fullness every minute you go back...so if it is full after 60 minutes, then one minute before that it will be half full.

Answer 9

logically, i believe it has to be 40. due to drops doubling, it cannot be half of an hour or less. plus 59 is too close to 1 hr. so yea.

Answer 10

c