The Smallest of Six Consecutive Odd Integers Whose Average (arithmetic mean) is x + 2 GMAT Problem Solving

Question: What is the smallest of six consecutive odd integers whose average (arithmetic mean) is x + 2?

1. x - 5
2. x - 3
3. x - 1
4. x
5. x + 1

“What is the smallest of six consecutive odd integers whose average (arithmetic mean) is x + 2?” - 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:

We can let the first (or smallest) odd integer = n, and thus we have:

n, n + 2, n + 4, n + 6, n + 8, and n + 10 as the six consecutive odd integers.

Since the average of terms in an evenly spaced set is (first number + last number)/2:

=>(n + n + 10)/2 = x + 2
=>2n + 10 = 2x + 4
=>2n = 2x - 6
=>n = x - 3

Correct Answer: B

Approach Solution 2:

Let us consider that k = the smallest odd integer
So, k + 2 = the next odd integer (since each consecutive odd integer is always 2 greater than the odd integer before it)
And k + 4 = the next odd integer
And k + 6 = the next odd integer
And k + 8 = the next odd integer
And k + 10 = the last (6th) odd integer

Since the average of the six integers is x+2, we can write:

\(\frac{k+(k+2)+(k+4)+(k+6)+(k+8)+(k+10)}{6} = x+2\)

If we simplify, we get :
(6k+30)/ 6 = x+2

Now, we will multiply both sides of the equation by 6 to get:
6k+30 = 6x+12

Coming to the question: What is the smallest of six consecutive odd integers?

Since k is the smallest odd integer, we need to solve this equation for k.
6k+30 = 6x+12
We can subtract 30 from both sides:
6k = 6x-18
If we divide both sides by 6 to get:
K = x-3

Correct Answer: B

Approach Solution 3:

Let us consider that the integers to be (n - 5), (n - 3), (n- 1), (n+1), (n + 3), (n + 5)
The arithmetic mean= 6n/6 = n
It is given in the problem, that
n = x + 2
Hence, smallest term = n - 5 = x + 2 - 5 = x - 3

Correct Answer: B

Suggested GMAT Problem Solving Questions

• Two Consultants, Mary and Jim, Can Type up a Report in 12.5 Hours and Edit it in 7.5 Hours GMAT Problem Solving 
• The Value of k if the Sum of Consecutive Odd Integers From 1 to k Equals 441 GMAT Problem Solving 
• A Circle with a Radius R is Inscribed into a Square with a Side K. GMAT Problem Solving 
• Machine A can do a Certain Job in 12 Days Working 2 Full Shifts GMAT Problem Solving 
• The square of 5√252 =? GMAT Problem Solving 
• After 6 Games, Team B Had an Average of 61.5 Points Per Game. GMAT Problem Solving 
• If 12 Ounces of a Strong Vinegar Solution are Diluted with 50 Ounces of Water to Form a Three-Percent Vinegar Solution GMAT Problem Solving 
• A Contractor Estimated that his 10-Man Crew Could Complete the Construction in 110 Days if There Was no Rain. GMAT Problem Solving 
• A Circle is Inscribed in a Square with the Diagonal of 4 Centimeters.  GMAT Problem Solving 
• At a Dog Competition, a Dog is Awarded 10 Points if it Runs Through 4 Pipes, Makes 10 Jumps, and Walks on 2 Beams. GMAT Problem Solving 
• The Population of the Bacteria Colony Doubles Every Day GMAT Problem Solving
• If tu=xytu=xyand ty=uxty=ux Where t, u, x, and y are Non-Zero Integers GMAT Problem Solving 
• A Farm has Chickens, Cows and Sheep. The Number of Chickens and Cows Combined is 3 Times the Number of Sheep. GMAT Problem Solving 
• In how Many Different Ways Can a Group of 8 People be Divided into 4 Teams of 2 People Each? GMAT Problem Solving 
• If 4 People are Selected from a Group of 6 Married Couples, What is the Probability That none of Them would be Married to Each Other? GMAT Problem Solving 
• If m is Three Times n, and if 2n + 3 is 20% of 25, What is the value of m? GMAT Problem Solving
• If Ben Were to Lose the Championship, Mike would be the Winner GMAT Problem Solving
• A Train Travelling at a Certain Constant Speed takes 30 seconds GMAT Problem Solving
• A Conference Room is Equipped with a Total of 45 Metal or Wooden Chairs.GMAT Problem Solving 
• A Welder Received an Order to Make a 1 Million Litre Cube-Shaped Tank. GMAT Problem Solving