A rectangular solid has length, width, and height of L cm, W cm, and H

Hola amigos

length = \(L\),
width = \(W\),
height = \(H\),

length increased by \(x\) % = \(L(1+\frac{x}{100})\)
width increased by \(y\) % = \(W(1+\frac{y}{100})\)
height increased by \(z\) % = \(H(1+\frac{z}{100})\)

What is the percentage increase in the total surface area?

Total surface area before increase = \(2LW + 2WH + 2LH\)
Total surface area after increase = \(2LW(1+\frac{x}{100})(1+\frac{y}{100}) + 2WH(1+\frac{y}{100})(1+\frac{z}{100}) + 2LH(1+\frac{x}{100})(1+\frac{z}{100})\)

Percentage increase in the total surface area = \(\frac{Total surface area AFTER increase - Total surface area BEFORE increase}{Total surface area BEFORE increase}*100\)

1. \(L, W,\) and \(H\) are in the ratios of \(5:3:4\).
\(L = 5k\)
\(W = 3k\)
\(H = 4k\)
We know nothing about \(x, y,\) and \(z\) to calculate the percentage increase in the total surface area.
Insufficient

2. \(x = 5\), \(y = 10\), \(z = 20\)
We know nothing about \(L, W,\) and \(H\) to calculate the percentage increase in the total surface area.
Insufficient

1+2. Now we have all required variables for calculation. We need to understand abovementioned formulas to exactly know at least which variables should be IDENTIFIED for the formula to work. In this case ALL of them are required. If the VOLUME were asked, knowing only \(x, y,\) and \(z\) would be enough and that was the trap most people fall into choosing B under time pressure.
Sufficient

The answer is C.

Hi All,

We're told that a rectangular solid has length, width, and height of L cm, W cm, and H cm, respectively. We're asked if these dimensions are increased by X%, Y%, and Z%, respectively, what would be the PERCENTAGE INCREASE in the total SURFACE AREA of the solid. This question is based around a couple of specific math formulas and can be solved with a mix of Arithmetic and TESTing VALUES. There are clearly a lot of variables in this question, so we'll need a lot of information to define the percentage increase in the total surface area.

To start, Total Surface area is SA = 2(L)(W) + 2(L)(H) + 2(W)(H) and the Percentage Change Formula = (New - Old)/(Old) = (Difference)/(Original).

(1) L, W, and H are in the ratios of 5:3:4

Fact 1 defines the relationships between the three dimensions (for example, the width is 3/4 of the height), but tells us nothing about the percent increase in any of the 3 dimensions, so there's clearly no way to define the percentage increase in surface area.
Fact 1 is INSUFFICIENT

(2) X = 5, Y = 10, Z = 20

Fact 2 gives us the exact percent increase in each dimension, but without any information on the original dimensions of the rectangular solid, we have no way to define the 'impact' that each increase would have on the total surface area.
Fact 2 is INSUFFICIENT

Combined, we know...
L, W, and H are in the ratios of 5:3:4
X = 5, Y = 10, Z = 20

With the ratio in Fact 1, we can refer to the three dimensions as Length = 5X, Width = 3X and Height = 4X, so whatever "X" actually is, the increase or decrease in the side lengths will be proportional. This means that the impact on the Original Surface Area and New Surface Area will always be the same in the above calculation and we will ALWAYS end up with the exact same answer (the math would look a bit 'ugly', so I'm going to refrain from presenting it here).
Combined, SUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

\(L\), \(W\), and \(H\) are the dimensions of a rectangular solid, in cm. The original question: \(\Delta SA\%=?\)

1) We know the ratio of the dimensions of this rectangular solid, but no information is given about the actual percent increases of its dimensions. Thus, we can't get a unique value to answer the original question. \(\implies\) Insufficient

2) We know the actual percent increases of the dimensions of this rectangular solid. To determine the percent increase of its surface area, we try to determine the growth factor for its surface area.

\(\frac{2(1.05\cdot1.1LW+1.05\cdot1.2LH+1.1\cdot1.2WH)}{2(LW+LH+WH)}=\)

\(=\frac{1.05\cdot1.1+1.05\cdot1.2\frac{H}{W}+1.1\cdot1.2\frac{H}{L}}{1+\frac{H}{W}+\frac{H}{L}}\)

Clearly, the above growth factor depends on the ratio of the dimensions, about which we don't have any information. Thus, we can't get a unique value to answer the original question. \(\implies\) Insufficient

1&2) We have the information to determine the percent increase of the surface area. Thus, we could get a unique value to answer the original question. \(\implies\) Sufficient

Answer: C

Hi gmatjindi,

Your goal when dealing with any individual Quant question should NOT be to try to answer it in under 2 minutes (that is bad advice that will likely cost you some points on Test Day) - a number of the questions that you will face are designed to take up to 3 minutes to solve (and that's if you know what you are doing). Your actual goal is to be 'efficient' with your work, meaning that you don't waste time and you choose the most efficient method for answering that question that's in front of you. Most GMAT questions are actually written so that you can approach them in more than one way - so as you continue to study, you might consider reviewing more than just the questions that you got wrong. There might be much faster ways to deal with questions that you have correctly answered (and you have to learn and practice those methods if you want to properly take advantage of them on Test Day).

GMAT assassins aren't born, they're made,
Rich

TSA ; 2(lw+wh+lh)
#1
LWH given but % variation is missing; insufficient
#2
x = 5, y = 10, z = 20
dimensions not given
insufficeint
from 1 & 2
we can determine % increase for l,w,h
sufficient
IMO C

Hello Gmaters !

Can you help me please?

I attached 2 notes, and they refer my process of thinking.

However, I have a doubt about 1)&2) part.. Can I just directly judge the answer in that way without the verification?

Thank you in advance!!

Hello Bijaykru,
For a moment, let’s forget that this is a question on calculating volume.

Let’s assume that we are discussing different sources of YOUR income.
Let’s say you have 3 sources of income – Salary, income from rent of a house that you have let out AND income from interest on a Deposit you have in a bank. So, effectively, there are three variables that have an effect on your total income, isn’t it?

Let’s assume two cases – your total income 5 years ago and your total income NOW. Clearly, most of these variables would have changed – your salary would have increased, you may have increased the rent of your tenants and the bank might be giving you a higher rate of interest now compared to 5 years ago.

One fine day, your buddy suddenly asks “ Hey Bijaykru, what’s the percentage increase in your total income mate, compared to 5 years ago?”. You do not want to answer this question directly, so you tell him “Look here, I’ll give you some information; based on that, you can calculate the percentage increase yourself”.

You give him information about your salary, rent and interest, 5 years ago in statement I. Can he calculate the percentage increase?
You give him information about how much your salary, rent and interest have increased in statement II. For example, you say “ Hey, my salary increased by 5%, rent by 10% and interest by 20%”. Will he be able to calculate the percentage increase?

Your answers to these questions will tell you why the correct answer is C on the original question. Do let me know if my analogy helped you understand this better.

Hope this helps!
_________________

Hi Ryanh7788,

In DS questions, a Fact is only SUFFICIENT if the answer to the given question (in this case: "what is the percentage increase in the total surface area of the solid?") stays the SAME. If the answer to the question changes, then the information in the Fact is INSUFFICIENT.

With the information in Fact 2, you know the percent increase of each side, but you do not know the exact 'base values' of what you are increasing. Here's a simple example of how the results can change:

With the information in Fact 2, what if the Length = 1, Width = 1 and Height = 100.... what would the percentage increase in the total surface area be? Now, what if you made the Length = 100 and the Height = 1 instead?

GMAT assassins aren't born, they're made,
Rich

Solution:

Question Stem Analysis:

We need to determine the percentage increase in the total surface area of the solid given that the length, width, and height of the solid are L cm, W cm, and H cm, respectively and these dimensions are increased by x%, y%, and z%, respectively,

Statement One Alone:

Without knowing the values of x, y, and z, we can’t determine the percentage increase in the total surface area of the solid. Statement one alone are not sufficient.

Statement Two Alone:

Without knowing the values of L, W, and H, we can’t determine the percentage increase in the total surface area of the solid. Statement two alone are not sufficient.

Statements One and Two Together:

Although we don’t know the exact values of L, W, and H, we can let them be 5s, 3s, and 4s, respectively since their ratio is 5:3:4. Therefore, the original surface area of the solid is 2(5s * 3s + 5s * 4s + 3s * 4s) = 2(47s) = 94s.

The new values of L, W, and H, in terms of s, are 5.25s, 3.3s, and 4.8s. Therefore, the new surface area of the solid is 2(5.25s * 3.3s + 5.25s * 4.8s + 3.3s * 4.8s) = 2(58.365s) = 116.73s.

Therefore, the percent increase in surface area is (116.73s - 94s) / (94s) * 100%. We see that variable s will cancel out, leaving us a unique value for the percent increase. Both statements together are sufficient.

Answer: C
_________________

Solution

Steps 1 & 2: Understand Question and Draw Inferences
In this question, we are given

• A rectangular solid has length, width, and height of L cm, W cm, and H cm, respectively.

We need to determine

• If the length, width, and height are increased by x%, y% and z% respectively, then the corresponding percentage increase in the total surface area of the solid.

The total surface area of the rectangular solid, before increase = 2(LW + LH + WH)
To increase the percentage of the total surface area, we need to know

• The values of L, W, H, before increase (or the ratio of L, W, H)
• The values of x, y, and z.

With this understanding, let us now analyse the individual statements.

Step 3: Analyse Statement 1
As per the information given in statement 1, L, W, and H are in the ratios of 5: 3: 4.

• However, from this statement, we cannot determine the values of x, y, and z.

Hence, statement 1 is not sufficient to answer the question.

Step 4: Analyse Statement 2
As per the information given in statement 2, x = 5, y = 10, z = 20.

• However, from this statement, we cannot determine the values of L, W, and H (or their ratios).

Hence, statement 2 is not sufficient to answer the question.

Step 5: Combine Both Statements Together (If Needed)
Combining information from both statements, we get

• L: W: H = 5: 3: 4 and x = 5, y = 10, z = 20.

As we have all the necessary information to calculate the percentage increase, we can say the correct answer is option C.


_________________

Can someone explain why not knowing the L,W,H in statement 2 is insufficient? It seems like as long as you know that L,W,H is being increased by the respective %, the overall increase will always be the same, regardless of the actual dimensions.

ie. if L,W,H = 1,1,1 or 2,3,5, and we know the % increase is 5%,10%,20%, the resulting increase from original to new will always be the same.

Can someone show it with actual examples? I figure I must be performing the question wrong.

Hi cchen679,

Fact 2 doesn't define any of the original dimensions - and increasing different measurements by 5%, 10% or 20% will lead to different overall increases in overall surface areas (try a couple of different examples and you'll see).

GMAT assassins aren't born, they're made,
Rich

Hi cchen679,

In your examples, you're calculating the volume of the solid - you're supposed to be calculating the SURFACE AREA though.

Surface Area = (2)(L)(W) + (2)(L)(H) + (2)(W)(H)

You'll find that the answer varies when you change the initial Surface Area.

GMAT assassins aren't born, they're made,
Rich

I have a quick question. In case the question focused on asking the difference between the new surface area and the original surface area (the actual number and not the percentage increase). Would the answer in that case be E, since we do not know the constant of the ratio?

Posted from my mobile device

Hi gauranggarg,

Assuming that everything else in the prompt remained the same - and the question was changed to ask "what is the difference between the new surface area and the original surface area of the solid?" - then the answer would be E (since we have no actual measurements to work with).

It's worth noting that NOTHING about a GMAT question is ever "random" - the wording, numbers, etc. are all specifically chosen for a reason: to 'test' you on your understanding of certain concepts (and in many cases, whether you are able to take advantage of a built-in 'shortcut' or not). With this prompt, the question asks about 'percent increase' to see how well you understand how ratios impact that type of work. A question on the GMAT that asked for the actual difference between two surface areas would likely give you different information to work with.

GMAT assassins aren't born, they're made,
Rich

There's no issue with your thought process. Just know that we are dealing in ratios and we are required to find out percentage which is another way of writing ratios.
So, basically we need not to have any specific value of any of the sides. This is what you have nicely shown in your work.
_________________

Can someone explain why the ratio part 1 is even needed? to solve for Part 2?

Even if you use L = X, W = Y, H = Z
Does it matter?

i.e. Original SA: 2(xy + xz + yz)
example increase for each length, width height = 150%

2[(1.5x)(1.5y) + (1.5x)(1.5z) + (1.5y)(1.5z)] - 2xy + 2xz + 2yz
____________________________________________________

2xy + 2xz + 2yz


= Reduce:

2.5 (xy + xz + yz)
_______________
2 (xy + xz + yz)

Answer: 2.5/2 = 5/4 = 1.25 = 125% increase


I still don't understand

Hi Ryanh7788,

In your example, you chose to increase each dimension by 50% of its original value (which is fine) - and you ended up with a specific result (a 125% increase in Total Surface Area). What happens IF you increase each dimension by something OTHER than 50% though? What happens if you increase each of the three dimensions by a different percent? Will you still end up with a 125% overall increase in Total Surface Area?

GMAT assassins aren't born, they're made,
Rich

Answer: Option C

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