Which of the Following is the Largest? GMAT Problem Solving

Question: Which of the following is the largest?

(A) 2^27.3
(B) 3^18.2
(C) 5^11.1
(D) 7^9.1
(E)11^5.1

Correct Answer: B
Solution and Explanation:
Approach Solution 1:

The problem statement asks to find the largest integers from the following options.
To solve the question, we need to look for some commonality that is either power or base.
We can say 27.3 is 3*9.1
We can also deduce 18.2 is 2*9.1. Let’s analyse options A, B and D:
(A) 2^27.3 = 8^9.1
(B) 3^18.2 = 9^9.1
(D) 7^9.1

Therefore, we can say out of (A), (B) and (D), option (B) is the largest.

Now, let us compare (C) 5^11 with (E) 11^5.
Out of a^b and b^a, the greater is usually the one with the smaller base but higher power.
Only in early powers of 2 is this pattern not followed. 2^3 < 3^2, 2^4 = 4^2 but 2^5 > 5^2 and the gap keeps widening.
Similarly, 3^4 > 4^3 and the gap keeps widening.
Therefore, 5^11 will be greater than 11^5. Hence option (C) is greater.
Now we just need to compare option (B) with (C)
(B) 3^18 = 27^6
(C) 5^11=25^5.5

Therefore option (B) has a greater base and a greater power. Hence, it is obviously greater.

Approach Solution 2:

The problem statement asks to find the largest integers from the following options.

As per the rules of mathematics, we know that 2^3 < 3^2.
By raising both sides to the power of 9.1, we get 3^18.2 > 2^27.3
Therefore, we can say, option B is greater than option A.

Again, according to the rules of mathematics, we know that 3^2 >7.
By raising both sides to the power of 9.1, we get 3^18.2 > 7^9.1
Therefore, we can say, option B is greater than option D.

Furthermore, as per the rules of mathematics, we know that 5^3 > 11^2.
By raising both sides to the power of 3.7, we have 5^3 ∗ 3.7 > 11^2 ∗ 3.7
Therefore, it can be deduced as 5^11.1 > 11^7.4.
Thus, 5^11.1 has to be greater than 11^5.1.
Hence, option C is greater than option E

Now it is required to determine whether option B is greater than option C to find the answer to this problem.
Finally, according to the rules of mathematics, we know that 3^3 > 5^2
By raising both sides to the power of 6, we have 3^18 > 5^12.
Thus, 3^18.1 will always be greater than 5^11.1.

Hence, we can conclude that option B is greater than option C.

Approach Solution 3:

The problem statement asks to find the largest integers from the following options.
Let’s analyse each of the options to solve the problem.
Each of these options can be defined as below:

Option (A) 2^27.3 can be derived as (2)^3∗9.1. This can be deduced as (2^3)^9.1 i.e equals to 8^9.1.
Option (B) 3^18.2 can be derived as (3)^2∗9.1. This can be deduced as (3^2)^9.1 i.e equals to 9^9.1
Option (C) 5^11.1 can be derived as (10/2)^11.1. This can be deduced as 10^11.1 / 2^11.1
Option (D)7^9.1 can be derived as 7^9.1
Option (E) 11^5.1 can be derived as approximately 10^5

Among A, B & D we can infer that option B is the clear winner.

Now choices are down to B, C & E. We can approximate further the options to get the actual answer.
Option (B) 9^9.1 can be further approximated to 10^9
Option C) 10^11 / 2^11 can be further deduced as 10^11/1024 which equals 10^11/10^3. Therefore option C can be approximated as 10^8.
Option (E) is approximated earlier as 10^5.

Among these approximations of the given options, it can be inferred that option B is the largest.

“Which of the following is the largest?”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. GMAT Problem Solving questions help the candidates to upgrade their mathematical skills by solving quantitative problems. It tests candidates’ efficiency in numerical calculation and abilities in solving quantitative problems. GMAT Quant practice papers enable the candidates to learn different sorts of questions. This helps the candidates to be familiar with lots of topics that will enhance their mathematical knowledge and understanding.

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