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Arithmetic Mean - Confusing

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by kop » Wed Nov 13, 2013 6:37 am
what is the average(Arithmetic mean) of eleven consecutive integers?

1. The average of first nine integers is 7
2. The average of last nine integers is 9

Please explain how to solve problems involved with A.M and consecutive integers?
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by Brent@GMATPrepNow » Wed Nov 13, 2013 7:23 am
kop wrote:What is the average (arithmetic mean) of 11 consecutive integers?

1. The average of first nine integers is 7
2. The average of last nine integers is 9
Target question: What is the average (arithmetic mean) of 11 consecutive integers?

Statement 1: The average of first nine integers is 7
IMPORTANT: There's a nice rule that says, "In a set where the numbers are equally spaced, the mean will equal the median."
For more on this concept, see https://www.beatthegmat.com/mba/2012/05/ ... d-the-mean

We can apply this rule to this question, since consecutive integers are equally spaced.
So, if the mean of the first nine integers is 7, then the median of the first nine integers is 7.
So, the first nine integers must be 3, 4, 5, 6, 7, 8, 9, 10, 11
This means the 11 numbers are 3,4,5,6,7,8,9,10,11,12,13
Since the median of this set is 8, the mean must be 8
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The average of last nine integers is 9
If the mean of the last nine integers is 9, then the median of the last nine integers is 9.
So, the last nine integers must be 5, 6, 7, 8, 9, 10, 11, 12, 13
This means the 11 numbers are 3,4,5,6,7,8,9,10,11,12,13
Since the median of this set is 8, the mean must be 8
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent
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by theCodeToGMAT » Wed Nov 13, 2013 8:53 am
Let numbers be n, n+1,... n+10

TO find: mean = (n) + (n+1) + .. (n+10)/11 =11n/11 + some integer.

Rephrase: What is the value of "n"

Statement 1:
(n) + (n+1) ... (n+8) / 9 = 7
9n/9 + someInteger = 63
We can solve to get "n"
SUFFICIENT

Statement 2:
(n+2)+(n+3)...(n+11)/9 = 9
9n/9 + someInteger = 81
We can solve to get "n"
SUFFICIENT

Answer [spoiler]{D}[/spoiler]
R A H U L
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by gmattesttaker2 » Tue Jan 14, 2014 10:54 pm
Brent@GMATPrepNow wrote:
kop wrote:What is the average (arithmetic mean) of 11 consecutive integers?

1. The average of first nine integers is 7
2. The average of last nine integers is 9
Target question: What is the average (arithmetic mean) of 11 consecutive integers?

Statement 1: The average of first nine integers is 7
IMPORTANT: There's a nice rule that says, "In a set where the numbers are equally spaced, the mean will equal the median."
For more on this concept, see https://www.beatthegmat.com/mba/2012/05/ ... d-the-mean

We can apply this rule to this question, since consecutive integers are equally spaced.
So, if the mean of the first nine integers is 7, then the median of the first nine integers is 7.
So, the first nine integers must be 3, 4, 5, 6, 7, 8, 9, 10, 11
This means the 11 numbers are 3,4,5,6,7,8,9,10,11,12,13
Since the median of this set is 8, the mean must be 8
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The average of last nine integers is 9
If the mean of the last nine integers is 9, then the median of the last nine integers is 9.
So, the last nine integers must be 5, 6, 7, 8, 9, 10, 11, 12, 13
This means the 11 numbers are 3,4,5,6,7,8,9,10,11,12,13
Since the median of this set is 8, the mean must be 8
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent

Hello Brent,

I was just wondering if the following approach is OK?

We need the average of 11 consecutive integers.

Let the integers be: x, x+1, x+2 ,..., x+10

Since the integers are evenly spaced, Average = x + (x+10)/2

Hence, x = ?

1) Given, x + (x+8) / 2 = 7
=> x = 3
Hence, Sufficient

2) Given, (x+2)+(x+10)/2 = 9
=> x = 3
Hence, Sufficient

Hence D

Thanks a lot for your help.

Best Regards,
Sri
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by gmattesttaker2 » Tue Jan 14, 2014 10:55 pm
Brent@GMATPrepNow wrote:
kop wrote:What is the average (arithmetic mean) of 11 consecutive integers?

1. The average of first nine integers is 7
2. The average of last nine integers is 9
Target question: What is the average (arithmetic mean) of 11 consecutive integers?

Statement 1: The average of first nine integers is 7
IMPORTANT: There's a nice rule that says, "In a set where the numbers are equally spaced, the mean will equal the median."
For more on this concept, see https://www.beatthegmat.com/mba/2012/05/ ... d-the-mean

We can apply this rule to this question, since consecutive integers are equally spaced.
So, if the mean of the first nine integers is 7, then the median of the first nine integers is 7.
So, the first nine integers must be 3, 4, 5, 6, 7, 8, 9, 10, 11
This means the 11 numbers are 3,4,5,6,7,8,9,10,11,12,13
Since the median of this set is 8, the mean must be 8
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The average of last nine integers is 9
If the mean of the last nine integers is 9, then the median of the last nine integers is 9.
So, the last nine integers must be 5, 6, 7, 8, 9, 10, 11, 12, 13
This means the 11 numbers are 3,4,5,6,7,8,9,10,11,12,13
Since the median of this set is 8, the mean must be 8
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent

Hello Brent,

I was just wondering if the following approach is OK?

We need the average of 11 consecutive integers.

Let the integers be: x, x+1, x+2 ,..., x+10

Since the integers are evenly spaced, Average = x + (x+10)/2

Hence, x = ?

1) Given, x + (x+8) / 2 = 7
=> x = 3
Hence, Sufficient

2) Given, (x+2)+(x+10)/2 = 9
=> x = 3
Hence, Sufficient

Hence D

Thanks a lot for your help.

Best Regards,
Sri
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by Brent@GMATPrepNow » Tue Jan 14, 2014 11:22 pm
gmattesttaker2 wrote:
Brent@GMATPrepNow wrote:
kop wrote:What is the average (arithmetic mean) of 11 consecutive integers?

1. The average of first nine integers is 7
2. The average of last nine integers is 9
Target question: What is the average (arithmetic mean) of 11 consecutive integers?

Statement 1: The average of first nine integers is 7
IMPORTANT: There's a nice rule that says, "In a set where the numbers are equally spaced, the mean will equal the median."
For more on this concept, see https://www.beatthegmat.com/mba/2012/05/ ... d-the-mean

We can apply this rule to this question, since consecutive integers are equally spaced.
So, if the mean of the first nine integers is 7, then the median of the first nine integers is 7.
So, the first nine integers must be 3, 4, 5, 6, 7, 8, 9, 10, 11
This means the 11 numbers are 3,4,5,6,7,8,9,10,11,12,13
Since the median of this set is 8, the mean must be 8
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The average of last nine integers is 9
If the mean of the last nine integers is 9, then the median of the last nine integers is 9.
So, the last nine integers must be 5, 6, 7, 8, 9, 10, 11, 12, 13
This means the 11 numbers are 3,4,5,6,7,8,9,10,11,12,13
Since the median of this set is 8, the mean must be 8
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent

Hello Brent,

I was just wondering if the following approach is OK?

We need the average of 11 consecutive integers.

Let the integers be: x, x+1, x+2 ,..., x+10

Since the integers are evenly spaced, Average = x + (x+10)/2

Hence, x = ?

1) Given, x + (x+8) / 2 = 7
=> x = 3
Hence, Sufficient

2) Given, (x+2)+(x+10)/2 = 9
=> x = 3
Hence, Sufficient

Hence D

Thanks a lot for your help.

Best Regards,
Sri
Perfect approach, Sri - great work!

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by gmattesttaker2 » Wed Jan 15, 2014 6:42 pm
Brent@GMATPrepNow wrote:
gmattesttaker2 wrote:
Brent@GMATPrepNow wrote:
kop wrote:What is the average (arithmetic mean) of 11 consecutive integers?

1. The average of first nine integers is 7
2. The average of last nine integers is 9
Target question: What is the average (arithmetic mean) of 11 consecutive integers?

Statement 1: The average of first nine integers is 7
IMPORTANT: There's a nice rule that says, "In a set where the numbers are equally spaced, the mean will equal the median."
For more on this concept, see https://www.beatthegmat.com/mba/2012/05/ ... d-the-mean

We can apply this rule to this question, since consecutive integers are equally spaced.
So, if the mean of the first nine integers is 7, then the median of the first nine integers is 7.
So, the first nine integers must be 3, 4, 5, 6, 7, 8, 9, 10, 11
This means the 11 numbers are 3,4,5,6,7,8,9,10,11,12,13
Since the median of this set is 8, the mean must be 8
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The average of last nine integers is 9
If the mean of the last nine integers is 9, then the median of the last nine integers is 9.
So, the last nine integers must be 5, 6, 7, 8, 9, 10, 11, 12, 13
This means the 11 numbers are 3,4,5,6,7,8,9,10,11,12,13
Since the median of this set is 8, the mean must be 8
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent

Hello Brent,

I was just wondering if the following approach is OK?

We need the average of 11 consecutive integers.

Let the integers be: x, x+1, x+2 ,..., x+10

Since the integers are evenly spaced, Average = x + (x+10)/2

Hence, x = ?

1) Given, x + (x+8) / 2 = 7
=> x = 3
Hence, Sufficient

2) Given, (x+2)+(x+10)/2 = 9
=> x = 3
Hence, Sufficient

Hence D

Thanks a lot for your help.

Best Regards,
Sri
Perfect approach, Sri - great work!

Cheers,
Brent
Hello Brent,

Thanks a lot.

Best Regards,
Sri
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