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

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A rectangular solid has length, width, and height of L cm, W cm, and H cm, respectively. If these dimensions are increased by x%, y%, and z%, respectively, what is the percentage increase in the total surface area of the solid?

(1) L, W, and H are in the ratios of 5:3:4.
(2) x = 5, y = 10, z = 20

C

Source: Official Guide 2020

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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: C

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Rich

[Scott@TargetTestPrep]

A rectangular solid has length, width, and height of L cm, W cm, and H cm, respectively. If these dimensions are increased by x%, y%, and z%, respectively, what is the percentage increase in the total surface area of the solid?

(1) L, W, and H are in the ratios of 5:3:4.
(2) x = 5, y = 10, z = 20

C

Source: Official Guide 2020

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