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Arithmetic Statistics Quant Review 2nd Ed #66

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Arithmetic Statistics Quant Review 2nd Ed #66

by runningguy » Wed Sep 18, 2013 8:30 am
If the average (arithmetic mean) of n consecutive odd integers is 10, what is the least of the integers?

1) The range of the n integers is 14.
2) The greatest of the n integers is 17.

Please help explain what concept this is testing and how to tackle a problem like this on the GMAT. Where can I find more helpful explanations on consecutive integer problems?
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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Wed Sep 18, 2013 9:51 am
runningguy wrote:If the average (arithmetic mean) of n consecutive odd integers is 10, what is the least of the integers?

1) The range of the n integers is 14.
2) The greatest of the n integers is 17.
Target question: What is the least of the integers?

Statement 1: The range of the n integers is 14.
There's a nice rule that says, "In a set where the numbers are equally spaced, the mean will equal the median."
Since consecutive odd integers are equally spaced, the mean of the integers will equal the median.
So, we know that the "middlemost" value is 10.
Since 10 is not odd, we now know that there must be an even number of odd integers in the set, and the average of the TWO MIDDLEMOST integers is 10.
If the median is 10, then half of the odd integer are greater than 10, and half are less than 10.
Since the range is 14, we know that the greatest value is 10+7 and the least value is 10-7
So, the integers are 3,5,7,9,11,13,15,17, which means the smallest value is 3
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The greatest of the n integers is 17
There's another nice rule that says, In a set where the numbers are equally spaced, the mean = (largest value - smallest value)/2
So, we know that 10 = (17 - smallest value)/2
From this, we can conclude that the smallest value must be 3
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

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Brent
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by theCodeToGMAT » Wed Sep 18, 2013 10:02 am
Let the sequence be p+2, p+4, p+6 --> p + 2n

Now
(p+2) + (p+4) + (p+6) .... + (p+2n) = 10
______________________________
n

np + 2 (1+2+3....+n) = 10n
np + 2 (n)(n+1)/2 = 10n
p + n + 1 = 10
p = 9 - n

Statement 1:
(p + 2n) - (p + 2) = 14
2n = 16 --> n = 8
p = 9 - 8 = 1
SUFFICIENT

Statement 2:
p + 2n=17
p + 2(9 - p) = 17
p + 18 - 2p = 17
p = 1

SUFFICIENT

hence, Answer [D]

Is the Answer [D] only???
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by [email protected] » Wed Sep 18, 2013 5:30 pm
Hi runningguy,

This DS question hides an interesting Number Property that you'll find useful.

If you're dealing with consecutive odd integers with an average of 10, then the possibilities will follow a pattern...

The numbers COULD be:

9, 11
7, 9, 11, 13
5, 7, 9, 11, 13, 15
3, 5, 7, 9, 11, 13, 15, 17
etc

The pattern is that there will be an EQUAL number of terms above 10 and below 10. You can use THAT pattern along with the other info to CRUSH this question relatively quickly.

Fact 1: The range is 14.

Look at the examples above. There's only one group with a range of 14. The other groups would either have a smaller range or a larger range.
Fact 1 is SUFFICIENT.

Fact 2: The greatest value is 17

Again, look at the examples above. There's only one group with 17 as the largest number.
Fact 2 is SUFFICIENT.

Final Answer: D

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