GMAT Problem Solving- If The Sum of 9 Consecutive Integers is 90

Question: If the sum of 9 consecutive integers is 90, what is the sum of the following 9 consecutive integers?

1. 99
2. 135
3. 145
4. 171
5. 180

“If the sum of 9 consecutive integers is 90”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide Quantitative Review". To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

The given case states that the sum of 9 consecutive integers is 90 and accordingly, the sum of the next 9 integers needs to be evaluated.
To find solution to the problem, it is necessary to consider the integers being x, x+1, x+2, …x+8
This further implies that these equals 90.
=> x + x+1 + x+2, …x+8 = 90
=> 9x + 36 = 90
=> x = 6
Hence, the sum of given 9 consecutive integers being 90 starts from 6 and consecutively goes on till 14.
In order to find the sum of 9 consecutive integers followed by the previous 9 integers, it implies that the sum of 15 to 23 needs to be identified.
For the purpose, the following way of resolving the problem can be evaluated-
=> Sum of 1-23 - Sum of 1-14
=> 23 * 24/2 - 14*15/2
=> 23*12 - 15*7
=> 276 - 105
=> 171
Hence, the sum of the next 9 consecutive integers from 15 to 23 would be 171.

Correct Answer: D

Approach Solution 2: 

Another most simple method that can be used to solve the problem of finding the sum of 9 consecutive integers after the given sum of 9 consecutive integers which is 90 is-

Identifying that the 9 consecutive integers add up to a sum of 90. The next 9 integers would account for 10th integer consecutively for next 9 numbers. So the sum of the integers would increase by 9.
This can be evaluated as 9 times of 9 integers can be helpful in finding the solution. This implies-
=> 9 * 9 = 81
=> 90 + 81 = 171
Hence, 171 is the sum of the next 9 integers consecutively which assures. 

Correct Answer: D

Approach Solution 3:

Given that the total of the first 9 consecutive numbers is 90, it is necessary to determine the sum of the next 9 integers.
The integers x, x+1, x+2,...x+8 must be taken into consideration in order to resolve the problem.
Additionally, this suggests that they equal 90.
=> x + x+1 + x+2, …x+8 = 90
=> 9x + 36 = 90
=> x = 6
So, starting at 6, and continuing continuously until 14, the total of the above 9 consecutive numbers is 90.
It means that the total of 15 to 23 must be determined in order to obtain the sum of 9 consecutive integers followed by the previous 9 integers.
The following method of addressing the problem can be assessed for this reason:

Sum of 1 to 23 − Sum of 1 to 14 minus 23 * 24/2 minus 14 * 15/2 minus 23 * 12 minus 15 * => 276 - 105 => 171
Therefore, the following 9 consecutive integers from 15 to 23 added together would equal 171.

Correct Answer: D

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