A GMAT question that is wrong

[thecalifornialife]

I was just thumbing through GMAT for Dummies and I came across a question that I believe is wrong. It goes as follows:

If 1/2 of the air is removed from a balloon every 10 seconds, what fraction of the air has been removed from a balloon after 30 seconds?

(a) 1/8
(b)1/6
(c)1/4
(d)5/8
(e)7/8


They stated that the correct answer is E.

I believe that is wrong. The math is sound when you ask someone to solve the equation (1/2) x 3. The distance from 1 - 1/8 is 7/8. However, what this question supposes is that the balloon will not reach empty. The reason I come to this conclusion is that if they asked the question "If 1/2 of the air is removed from a balloon every 10 seconds, what fraction of the air has been removed from a balloon after 3000 seconds?" That would mean that 1/150 of the air is remaining. 3000 goes into 10, 300 times. Then you times (1/2) x 300. You will end up with 1/150. 3000 seconds is 50 minutes. After 50 minutes this balloon is still not empty?

The correct answer should be that the balloon is empty after 20 seconds.

[Jim@StratusPrep]

Half of the remaining air leaves the balloon.

So, in the first ten seconds, half leaves the balloon is half full.

Then, half of one half or a quarter leaves.

1/8 leaves the balloon in the final ten.

The answer is correct.

[thecalifornialife]

Half of the remaining air leaves the balloon.

So, in the first ten seconds, half leaves the balloon is half full.

Then, half of one half or a quarter leaves.

1/8 leaves the balloon in the final ten.

The answer is correct.

But like I said with my adjusted problem, 50 minutes later, or 3000 seconds later, 1/150 of the air is left in the balloon? - No. This question logically and mathematically doesn't make sense. What this question assumes is that the balloon is logarithmic. I'll pose the question another way to prove it doesn't make sense. How long will it take for the balloon to be empty?

[GMATGuruNY]

How long will it take for the balloon to be empty?

If we keep removing 1/2 of the air, the balloon will NEVER be empty.
The reason is that 1/2 of the air will always REMAIN in the balloon.
If the balloon starts with 8 units of air, and we keep removing half, the following series of amounts will be left in the balloon:
8
4
2
1
1/2
1/4
1/8
1/16
1/32
1/64
1/128
And so on.
The amount of air left in the balloon will get infinitely small, but there will always be SOME air left in the balloon.

[Jim@StratusPrep]

Agreed. It will never empty....


As for the Gmat take away: don't over complicate problems, start with the basics and build from there.

[thecalifornialife]

That isn't what the question was asking. It wasn't saying that the volume decreases at a rate of 1/2 of the remaining volume every 10 seconds. If that was the case then their answer is correct. It's like asking the question "You spend half of your money on your first date then half on your second date, how much do you have left?" You would have none! I think the best defense of my analysis is the logical proof that I added earlier. It doesn't make sense to depreciate the value of the original principle each time since after 50 minutes the balloon still has air. This doesn't logically make sense.


Can I assume this is a badly worded question by "GMAT for Dummies?"

[Tommy Wallach]

Hey California,

Unfortunately, this is not a badly worded question. I think you're misreading it. If you take away half of something multiple times, you cannot reset it to its original value. In other words:

You have $10. Take away half.

Now you have $5. Take away half.

At this point, you do not still have $10. You can't reset the meaning of "half" back to where you started.

-t

[thecalifornialife]

Hey California,

Unfortunately, this is not a badly worded question. I think you're misreading it. If you take away half of something multiple times, you cannot reset it to its original value. In other words:

You have $10. Take away half.

Now you have $5. Take away half.

At this point, you do not still have $10. You can't reset the meaning of "half" back to where you started.

-t

There are two types of depreciation. Straight-line depreciation and amortization. Let's say you have a car and it depreciates 2% straight-line every month. That means that the principle that the depreciation that the 2% is calculated off of stays the same every month. So if it is a 20000 car and it depreciates 2% straight-line every month, that is 400 every month. It isn't 20000 - 2%. Then 19600 - 2%. Amortization is depreciation off the original principle. That is the same for this question.

If they asked, "Usain Bolt ran 1/2 the lap in 10 seconds, how much of the lap did he run in 20 seconds?" or "The water was 1/2 boiled after 5 minutes. How much of the water would be boiled after 10 minutes?" The answer would be the whole lap and the whole water. It is the same thing for this question.

The question wants us to assume the balloon is logarithmic in that it never reaches 0. That doesn't logically make sense.

If the question asked was asked properly, it would say "If the volume of a balloon decreases at a rate of 1/2 of the remaining volume every 10 seconds. How much of the air is gone after 30 seconds?"

That is properly written logarithmic question.

[ceilidh.erickson]

Here's something that I ask my GMAT students all the time: do you want to be right, or do you want to get a good score?

Of course you're right - balloons in the real world do not behave that way. Nor do populations of bacteria double exactly in size precisely once per hour. The point is not to analyze whether these situations make sense. The point is to apply basic mathematical principles to the given situations.

Here, the test writer wanted to test the idea of successive proportion change in a word problem format. There aren't many real-world situations that would exactly fit the kind of relationship they're describing here, so the writer made one up. I agree that the question could probably be written in a more straightforward way, but it never helps to be too nit-picky.

Your examples aren't necessarily showing the same principle. "Half the lap" in this instance would be half the total each time, whereas "half the air" in the balloon question can be understood to mean half the remaining air. The first is not successive; the second is. This question couldn't be an amortization question unless it specified "half of the original air" or actually used the word amortization.

You'll run across a lot of GMAT problems that seem poorly worded, or not real-world true (such as apples costing $0.05, or rooms only taking 2 hrs to paint!). If you obsess over those details, you miss the bigger picture. Start by asking - what concept is the test writer getting at here? If you can identify the test-writer's objective, it can help prevent over-thinking.