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11 consecutive integers

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11 consecutive integers

by rahul.s » Sun Jan 24, 2010 3:59 am
If eleven consecutive integers are listed from least to greatest, what is the average (arithmetic mean) of the eleven integers?

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

OA: D
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by tanviet » Sun Jan 24, 2010 5:55 am
the first number is x, second x+a, ....x+6a, so we have total first 9 is 9x+36a=7*9

we can infer total of 11x+(a+10a)*1/2*10

2, is similar
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by papgust » Sun Jan 24, 2010 7:14 am
11 consecutive numbers are listed from smallest to greatest. Mean of 11 numbers?

1. Avg of first nine integers is 7.

This means that 5th integer is 7. So 1st integer is 3. From here it is enough to know that mean could be calculated.
Sufficient.

2. Avg of last nine integers is 9.

This means that 7th integer is 9. So 1st integer is 3. Again sufficient.
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by rahul.s » Sun Jan 24, 2010 7:19 am
papgust wrote:11 consecutive numbers are listed from smallest to greatest. Mean of 11 numbers?

1. Avg of first nine integers is 7.

This means that 5th integer is 7. So 1st integer is 3. From here it is enough to know that mean could be calculated.
Sufficient.

2. Avg of last nine integers is 9.

This means that 7th integer is 9. So 1st integer is 3. Again sufficient.
thanks pagust :D
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by VikingWarrior » Sun Jan 24, 2010 1:02 pm
11 consecutive numbers are listed from smallest to greatest. Mean of 11 numbers?

1. Avg of first nine integers is 7.

This means that 5th integer is 7. So 1st integer is 3. From here it is enough to know that mean could be calculated.
Sufficient.

2. Avg of last nine integers is 9.

This means that 7th integer is 9. So 1st integer is 3. Again sufficient.
I got the correct answer by the earlier method but I think I am missing a concept here. How do you know that the 5th integer is the mean of 1st five and 7th integer is mean of last nine. Please explain, Thanks :)
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by ajith » Sun Jan 24, 2010 1:20 pm
VikingWarrior wrote:

I got the correct answer by the earlier method but I think I am missing a concept here. How do you know that the 5th integer is the mean of 1st five and 7th integer is mean of last nine. Please explain, Thanks :)
say the 10 integers are a, a+1, a+2, .... ,a+10

The mean of first 9 integers is(a+ a+1 + a+2.........+a+8)/9 = (9a+1+2+3+...+8)/9 = 9a+36/9 = a+4 (which indeed is the fifth integer)

The mean of last 9 integers is (a+2+ a+3 + a+4 + .....+ a+10) = (9a + 2+3+4....+10)/9= 9a+54/9 =a+6 (which is seventh integer in the list)
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by VikingWarrior » Sun Jan 24, 2010 1:26 pm
Ajith, I know the method you have used. I used the same method and arrived at the answer but I was wondering if pagust used some theory to directly find out which no. was at which position by knowing the mean of consecutive nos.
Thanks for your explanation, appreciate the effort. :)
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by Stuart@KaplanGMAT » Sun Jan 24, 2010 7:43 pm
VikingWarrior wrote:Ajith, I know the method you have used. I used the same method and arrived at the answer but I was wondering if pagust used some theory to directly find out which no. was at which position by knowing the mean of consecutive nos.
Thanks for your explanation, appreciate the effort. :)
When we have an odd number of terms in a set of consecutive numbers, the middle term is both the average (mean) and the median.

So, the average of the first 9 numbers (i.e. terms 1 through 9) will be the middle term in that subset, or the 5th term.

Similarly, the average of the last 9 numbers (i.e. terms 3 through 11) will be the middle term in that subset, or the 7th term.
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by VikingWarrior » Sun Jan 24, 2010 11:47 pm
Thanks Stuart! Could anyone please give me a link to a good resource for this kind of theory (math) that is required/specific/important for GMAT? thanks :)
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