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Question

Please explain the best way to solve.

theCodeToGMAT · Answer

Measured = 200 * 200 * 300 Error : 1 cm Possible ranges --> 199-201, 199-201, 299-301 Differences=> 1*(200)*(300) + (200)*1*(300) + (200)(200)(1) ==> 60000 + 60000 + 40000 = 160000

lunarpower · Answer

I don't know what thecodetogmat is doing with "differences". Looks like some sort of calculus thing.

If you don't know that, you can still do this:
Imagine the shape of the parts that would be added to the box (if you increased each dimension by 1 cm) or cut off of it (if you decreased each dimension by that amount). Let's say the box is originally 200cm x 200cm long and wide, and 300cm high.
* You'd be adding another cm of height. That extra increment would measure 200cm x 200cm x 1cm, or 40,000 cubic cm.
* You'd be adding another cm of width. That extra increment would measure 200cm x 300cm x 1cm, or 60,000 cubic cm.
* You'd be adding another cm of length. That extra increment would measure 200cm x 300cm x 1cm, or 60,000 cubic cm.

This analysis is missing the tiny slivers that you'd have to add in along the corners, but those are very small in relation to the parts described above. (The "differences" approach appears to be neglecting the same parts.)

That's a difference of 40,000 + 60,000 + 60,000 = 160,000 cubic cm.

Ron has been teaching various standardized tests for 20 years.

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lunarpower · Answer

And then, of course, there's the "I'm 11 years old again" method. You can just multiply out 200 x 200 x 300 and 201 x 201 x 301, and then subtract them.

If you are thinking "That might take too long" -- don't think that. Do not EVER tell yourself "I think ______ will take too long". Just start trying.
In the (extremely unlikely) event that something really is taking a long time, you can just quit and try to think of something else.

I'm pretty slow and error-prone when it comes to arithmetic, but I was able to do (201 x 201 x 301) - (200 x 200 x 300) = 12160701 - 12000000 = 160701 in about one minute.
One minute!

And, I'm pretty sure, just about everyone on this board is way, way faster at arithmetic than I am. You know what that means.

Ron has been teaching various standardized tests for 20 years.

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ngalinh · Answer

So what does this question actually test? Thinking practically? Any tool that can kill the beast is useful. None discrimination between a sophisticated tool and a hand-made tool?

Uva@90 · Answer

theCodeToGMAT wrote:Measured = 200 * 200 * 300
Error : 1 cm

Possible ranges --> 199-201, 199-201, 299-301

Differences=> 1*(200)*(300) + (200)*1*(300) + (200)(200)(1) ==> 60000 + 60000 + 40000 = 160000

Rahul,
Sorry, I couldn't get your method. Could you please explain me in detail.\

Thanks in advance.

Regards,
Uva.

Uva@90 · Answer

lunarpower wrote:
I'm pretty slow and error-prone when it comes to arithmetic, but I was able to do (201 x 201 x 301) - (200 x 200 x 300) = 12160701 - 12000000 = 160701 in about one minute.
One minute!

Ron,
I have doubt..
In question they mentioned that error could be of at-most 1cm.
So, Box measurement could be of 201*201*301 or 199*199*299 only right ?

and hence MAXIMUM possible difference should be of (201*201*301) - (199*199*299) only right ?

Please help me why/where I am wrong.

Thanks in advance.

Regards,
Uva.

ngalinh · Answer

Uva,

I think the question asks the difference between “the actual capacity of the box” (the old box) and “the capacity computed using these measurements” (the new box with error)

In case the new box becomes biggest (added 1cm to each dimension):
--> the difference = (capacity of new box – C of old box) = 201*201*301 – 200*200*300

In case the new box becomes smallest (cut 1cm from each dimension)
--> the difference = (C of old box – C of new box) = 200*200*300 – 199*199*299

The same thing. Actually, in case 2, the difference is slightly smaller than that of case 1, but compare to the big difference between answer choices, it’s clearly C.

ngalinh · Answer

From Uva’s question, I’m wondering how could we know (201*201*301 – 200*200*300) is bigger than (200*200*300 – 199*199*299)?
(if we don’t have calculator)

Could we use an “analogy” such as: (11*11-10*10) > (10*10-9*9) ?

Uva@90 · Answer

ngalinh wrote:Uva,

I think the question asks the difference between “the actual capacity of the box” (the old box) and “the capacity computed using these measurements” (the new box with error)

Precisely. One of the two values in the subtraction must be the original capacity of the box.

Ron has been teaching various standardized tests for 20 years.

--

Pueden hacerle preguntas a Ron en castellano
Potete chiedere domande a Ron in italiano
On peut poser des questions à Ron en français
Voit esittää kysymyksiä Ron:lle myös suomeksi

--

Quand on se sent bien dans un vêtement, tout peut arriver. Un bon vêtement, c'est un passeport pour le bonheur.

Yves Saint-Laurent

--

Learn more about ron

ngalinh · Answer

yeah, buddy, I often make those mistakes too. Good luck!

Matt@VeritasPrep · Answer

ngalinh wrote:From Uva’s question, I’m wondering how could we know (201*201*301 – 200*200*300) is bigger than (200*200*300 – 199*199*299)?
(if we don’t have calculator)

Could we use an “analogy” such as: (11*11-10*10) > (10*10-9*9) ?

That analogy works well for consecutive squares, which do get further apart (1, 4, 9, 16, 25, etc.) and it could work in other equations too. I might approach it this way:

Suppose 301*201*201 - 300*200*200 > 300*200*200 - 299*199*199.
Then we have 301*201*201 + 299*199*199 > 2 * (300*200*200), which can be "verified" through some similar equation like 6*4*4 + 4*2*2 > 2 * (5*3*3) or whatever.

But I'd still try to solve this - the approach below doesn't take long (it looks longer than it really is because I've been pretty explicit about each step):

301*201*201 - 300*200*200
= (300 + 1)(201)(201) - (300)(200)(200)
= (300)(201)(201) + (1)(201)(201) - (300)(200)(200)
= 300(201*201 - 200*200) + 201*201
= 300(201+200)(201-200) + 201*201
= 300(401) + 201*201
= 300(400+1) + (200+1)(200+1)
= 120000 + 300 + 40000 + 400 + 1
= 160701

It'd be a great exercise to start a timer and see how long it takes you to do the next one (300*200*200 - 299*199*199) yourself. At first it might take a little while, but it's incredible how quickly you'll be able to do it after some practice.

ngalinh · Answer

Thanks Matt for your great response!

Matt@VeritasPrep wrote:
That analogy works well for consecutive squares, which do get further apart (1, 4, 9, 16, 25, etc.)

How do I know whether I use a right analogy in a short time if I know little about math "rules"?

My answer for my question would be: analogy is a form of testing numbers—testing numbers for numbers, not for variables. But I expect another miracle answer from math experts.

I might approach it this way:

Suppose 301*201*201 - 300*200*200 > 300*200*200 - 299*199*199.
Then we have 301*201*201 + 299*199*199 > 2 * (300*200*200), which can be "verified" through some similar equation like 6*4*4 + 4*2*2 > 2 * (5*3*3) or whatever.

Hehe, if you do that, I'll do this:
301*201*201 + 299*199*199 - 2*300*200*200 > 0?
11*11 + 9*9 - 2*10*10 > 0?
202 -200 > 0? (ah, yes!- nice game)

But I'd still try to solve this - the approach below doesn't take long (it looks longer than it really is because I've been pretty explicit about each step):

301*201*201 - 300*200*200
= (300 + 1)(201)(201) - (300)(200)(200)
= (300)(201)(201) + (1)(201)(201) - (300)(200)(200)
= 300(201*201 - 200*200) + 201*201
= 300(201+200)(201-200) + 201*201
= 300(401) + 201*201
= 300(400+1) + (200+1)(200+1)
= 120000 + 300 + 40000 + 400 + 1
= 160701

So the general rule for this manipulation is finding the common factors among terms, then using (+, -, *, /) tools to make the new one = origin?

But this way may create difficulties. Sometimes I see pattern at first, but go into a maze later. So I think about applying analogy for solving number properties questions after reading a fantastic insight of analogy from Ron's post.
(In case I can’t use other tools such as back-solving, straight calculating, or estimating)

Could you provide more information about using analogy in solving math questions if you have? If this thread is not relevant for discussing more, please give me the link in which you discuss it. Thank you!

lunarpower · Answer

ngalinh wrote:Uva,

I think the question asks the difference between “the actual capacity of the box” (the old box) and “the capacity computed using these measurements” (the new box with error)

Precisely. One of the two values in the subtraction must be the original capacity of the box.

lunarpower · Answer

From Uva’s question, I’m wondering how could we know (201*201*301 - 200*200*300) is bigger than (200*200*300 - 199*199*299)?
(if we don’t have calculator)

Could we use an “analogy” such as: (11*11-10*10) > (10*10-9*9) ?

That's how I would approach it. The problem, though, is that you should use an analogy that looks exactly like the original, but with smaller numbers.
So, using your numbers, you could check, say, (11*11*21 - 10*10*20) against (10*10*20 - 9*9*19).

The most important point is that the two differences are extremely close to each other -- so close, in fact, that it's immaterial which one is bigger. None of the other answer choices is anywhere close to either of them, so you're fine with either one.

Ron has been teaching various standardized tests for 20 years.

--

Pueden hacerle preguntas a Ron en castellano
Potete chiedere domande a Ron in italiano
On peut poser des questions à Ron en français
Voit esittää kysymyksiä Ron:lle myös suomeksi

--

Quand on se sent bien dans un vêtement, tout peut arriver. Un bon vêtement, c'est un passeport pour le bonheur.

Yves Saint-Laurent

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Learn more about ron

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