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Shortcut in Average Problems

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Intern
Joined: 24 Jul 2016
Posts: 1
Shortcut in Average Problems  [#permalink]

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06 Aug 2017, 21:05
Hey guys,

I have been trying to come up with shortcuts in Average Problems and would like to share two things that I came up with. Because I figured this out by trial, hopefully, someone can confirm that they are correct and applicable.

1) Consider this problem:
A certain bakery has 6 employees. It pays annual salaries of \$14,000 to each of 2 employees, \$16,000 to 1 employee, and \$17,000 to each of the remaining 3 employees. The average (arithmetic mean) annual salary of these employees is closest to which of the following?
A. \$15,200
B. \$15,500
C. \$15,800
D. \$16,000
E. \$16,400
OA: C

Suppose 3 employees earn 14(simplified) and 3 employees earn 17. We will have the average = 15.5

Next step we know that one employee actually gets 16, an increase of 2 from the hypothesized salary (14).

This employee has 1/6 of the impact on the 15.5 mean => it will increase the mean by 2*1/6 = 2/6 = 1/3 = 0.333 (NOTE: 2 is actually 2000 in this function, so 0.333 is actually 333.33)

0.33+15.5 = 15.83. Hence, closest to C: 15,800.

2) Consider the second problem:
The 15 homes in a new development are each to be sold for one of three different prices so that the developer receives an average (arithmetic mean) of \$200,000 per home. If 4 of the homes are to be sold for \$170,000 each and 5 are to be sold for \$200,000 each, what will be the selling price of each of the remaining 6 homes?
A\$200,000
B\$210,000
C \$215,000
D\$220,000
E\$230,000
OA: Below

In this problem, we have 4 houses sold for 170 and 5 houses sold for 200; and the mean is 200. If we only sell 4 more houses, we will sell them for 230, since the distance from 170 to 200 is balanced to the distance from 200 to 230. Easy to see right?

However, we actually sell 6 more houses. Therefore, we the prices of those 6 houses must be lower than 230 to maintain the mean of 200. But by how much?

A fast way to do it is that we sell 6 houses, which is 2 more than the balance of 4 houses => we need to weight back 2 houses of the 6 houses = 2/6 = 1/3.

The distance from 200 to 230 is 30 => Weight back = 1/3 * 30 = 10. After weight back, the price of 6 houses will be 230 - 10 =220. Hence D.

I apologize for posting 2 problems. I really need someone to confirm that those shortcuts are correct.

Thanks,
Intern
Joined: 29 May 2017
Posts: 16
Re: Shortcut in Average Problems  [#permalink]

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06 Aug 2017, 22:15
I still did not get the answer. Need more clear explanation. Can any one do it?
Math Expert
Joined: 02 Sep 2009
Posts: 55271
Re: Shortcut in Average Problems  [#permalink]

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06 Aug 2017, 22:18
protimakudnu@yahoo.com wrote:
I still did not get the answer. Need more clear explanation. Can any one do it?

Question 1: https://gmatclub.com/forum/a-certain-ba ... 48748.html
Question 2: https://gmatclub.com/forum/the-15-homes ... 42108.html

Hope it helps.
_________________
Re: Shortcut in Average Problems   [#permalink] 06 Aug 2017, 22:18
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Shortcut in Average Problems

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