A Solid Metallic Cube is Melted to Form Five Solid Cubes Whose Volumes GMAT Problem Solving

Question: A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1 : 1 : 8: 27: 27. The percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube is nearest to:

1. 10
2. 20
3. 30
4. 50
5. 60

Correct Answer: (D)

Solution and Explanation:
Approach Solution 1:

The problem statement informs that:

Given:

• A solid metallic cube is melted to form five solid cubes.
• The volumes of each cube are in the ratio of 1 : 1 : 8 : 27: 27.

Find out:

• The percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube.

The ratio of volumes of 5 smaller cubes and the original big cube= 1 : 1 : 8 : 27 : 27 : 64
Therefore, the ratio of sides = 1 : 1 : 2 : 3 : 3 : 4 (since as per the formula, the volume of a cube = Side^3)
Therefore, the ratio of areas = 1 : 1 : 4 : 9 : 9 : 16
The sum of the areas of the 5 smaller cubes is 24 parts while that of the big cube is 16 parts.
Hence, the sum is [(24-16)/16]*100 = 50% greater.

Approach Solution 2:

The problem statement states that:

Given:

• A solid metallic cube is melted to form five solid cubes.
• The volumes of each cube are in the ratio of 1 : 1 : 8 : 27: 27.

Find out:

• The percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube.

Let the length of each side of the original cube is “a” cm.
Therefore, the surface area of the original cube = 6 a^2 sq. cm ….(1)
The volume of the original cube = a^3 cubic cm

The ratio of the volumes of the cube is 1 : 1 : 8 : 27: 27.
Therefore, the sum of the ratios = (1+1+8+27+27) = 64

Hence, the volumes of the 5 smaller cubes will be:
a^3 / 64, a^3 / 64, 8a^3 / 64, 27a^3 / 64 and 27 a^3 / 64 cubic cm respectively.

The sides of the smaller cubes can be determined by doing the cube roots of their volumes.
Therefore, the sides of the smaller cubes will be:
a / 4, a / 4, 2a / 4, 3 a / 4 and 3a / 4 cm respectively

The surface areas of these individual smaller cubes will be:
6 a^2 / 16, 6 a^2 / 16, 6 a^2 / 4, 54 a^2 / 16 and 54 a^2 / 16 sq. cm respectively.

Therefore the sum of the surface areas of the 5 smaller cubes
= (6a^2 + 6a^2 + 24 a^2 + 54 a^2 + 54 a^2) / 16
= 144 a^2 / 16 = 9 a^2 sq. cm…. (2)

Therefore, by comparing (1) and (2), the percentage increase in surface area
=(9a^2 - 6a^2 ) * 100 / 6a^2
= 3a^2 * 100 / 6a ^2
= 50 %

Approach Solution 3:

The problem statement suggests that:

Given:

• A solid metallic cube is melted to form five solid cubes.
• The volumes of each cube are in the ratio of 1 : 1 : 8 : 27: 27.

Find out:

• The percentage increase in the sum of the surface areas of these five cubes from the surface area of the original cube.

Let’s assume the volumes of the five cubes be x, x, 8x, 27x and 27x.
Let’s assume the sides of the cubes be a, a, 2a, 3a and 3a. (where x = a^3 )
Let’s assume the side of the original cube be A.

Therefore, A^3 = x + x + 8x + 27x + 27x
= 64x
=(4a)^3

Therefore, A = 4a

Original surface area = 96a^2 (since the surface area of a cube = 6* side^2)
New surface area of the cube = 6(a^2 + a^2 + 4a^2 + 9a^2 + 9a^2)= 144a^2

Therefore, percentage increase = \(\frac{144a^2-96a^2}{96a^2} * 100\) = \(\frac{48a^2}{96a^2} * 100\) = 50%

“A solid metallic cube is melted to form five solid cubes whose volumes”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. The GMAT Problem Solving questions test the abilities of candidates in interpreting quantitative problems. The candidates can practice varieties types of questions from GMAT Quant practice papers to improve their calculative skills.

Suggested GMAT Problem Solving Samples

• ABC is an equilateral triangle, and point D is the midpoint of side BC GMAT Problem Solving
• A Batsman Makes a Score of 87 Runs in the 17th Match and Thus Increases GMAT Problem Solving
• If M= √4+3√4+4√4, Then the Value of M is GMAT Problem Solving
• An Octagon Is Inscribed In A Circle As Shown Above. What Of The Area GMAT Problem Solving
• In a Company of Only 20 Employees, 10 Employees make $80,000/yr GMAT Problem Solving
• A bag contains blue and red balls only GMAT Problem Solving
• (4.8*10^9)^(1/2) is closest in value to GMAT Problem Solving
• What Is The Units Digit Of 2222^333 ∗ 3333^222? GMAT Problem Solving
• What Is The Tens Digit Of 6^17? GMAT Problem Solving
• If m=−2, What Is −m^(−m)? GMAT Problem Solving
• An Automated Manufacturing Plant Uses Robots To Manufacture Products GMAT Problem Solving
• The Surface Distance Between 2 Points on the Surface of a Cube is the GMAT Problem Solving
• The Average Monthly Expenditure of a Family for the First Four Months GMAT Problem Solving
• When a Certain Perfect Square is Increased by 148, the Result is GMAT Problem Solving
• If p#q Denotes the Least Common Multiple of p and q, Then ((12#16) GMAT Problem Solving
• In How Many Ways Can One Divide 12 Different Chocolate Bars Into Four GMAT Problem Solving
• Employee X's Annual Salary Is $12,000 More Than Half Of Employee Y's GMAT Problem Solving
• What is the Area of Quadrilateral ABCD Shown? GMAT Problem Solving
• Eleven Chairs are Numbered 1 Through 11. Four Girls and Seven Boys GMAT Problem Solving
• Two Friends, Tanaya and Stephen were Standing Together GMAT Problem Solving
• If x is a number such that x^2 - 3x + 2 = 0 and x^2 - x - 2 = 0 GMAT Problem Solving
• Barney is forming two large cubes, A and B, with small identical cubes GMAT Problem Solving
• It Takes Printer A 4 Minute More than Printer B to Print 40 GMAT Problem Solving
• A synchronized diving competition will feature 25 synchronized GMAT Problem Solving
• The Earth Travels Around The Sun At A Speed Of Approximately 18.5 miles GMAT Problem Solving
• There are 8 ounces in a 1/2 pound. How many ounces are in 7 3/4 lbs? GMAT Problem Solving
• In isosceles triangle EFG above, angle FEG measures 60 degrees GMAT Problem Solving
• In the Figure, O is the Center of the Circle. Which One of the Following GMAT Problem Solving
• If a and b are Positive Integers and (2a)^b= 2^3, What is the Value GMAT Problem Solving
• What is the Remainder When 3^243 is Divided by 5? GMAT Problem Solving
• A Fashion Designer Sold a Pair of Jeans to a Retail Store for 40 Percent GMAT Problem Solving
• A Rectangular-Shaped Carpet Remnant That Measures x Feet GMAT Problem Solving
• In The Figure Above, If The Square Inscribed In The Circle Has An Area GMAT Problem Solving
• A Right Triangle Is Inscribed In A Circle. The Legs Of The Triangle GMAT Problem Solving
• Tom and Jerry are running on the same road towards each other. If Tom GMAT Problem Solving
• A number is said to be prime saturated if the product of all the different GMAT Problem Solving
• What is the sum of first 10 non-negative even integers GMAT Problem Solving
• A certain computer program randomly generates equation of line is form GMAT Problem Solving
• A perfect number is one which is equal to the sum of all its positive GMAT Problem Solving
• At A Prestigious Dog Show, Six Dogs Of Different Breeds Are To Be GMAT Problem Solving
• The Average Wages of a Worker During a Fortnight Comprising 15 GMAT Problem Solving
• Which of the Following Fractions is the Largest? GMAT Problem Solving
• Alice and Bob Traveled in the Same Direction Along the Same Route at GMAT Problem Solving
• Paul, Quallis and Robert Divide a Sum of Money Among Themselves in the GMAT Problem Solving
• If A Sphere With Radius r Is Inscribed In A Cube With Edges Of Length GMAT Problem Solving
• A Globe Of Radius 8 Inches Is To Be Placed Into A Square Box For Shipment GMAT Problem Solving
• What is the sum of digits in decimal notation of number (10^20) - 16? GMAT Problem Solving
• Which of the following is equivalent to the pair of GMAT Problem Solving
• A password to a certain database consists of digits that can GMAT Problem Solving
• A Can do 1/3 of the Work in 5 Days and B Can do 2/5 of the Work in 10 Days GMAT Problem Solving