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The average of the numbers 1, 2, 3,..., 98, 99, and x is 100x. What is

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Joined: 02 Sep 2009
Posts: 58982
The average of the numbers 1, 2, 3,..., 98, 99, and x is 100x. What is  [#permalink]

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31 Mar 2019, 21:01
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Question Stats:

45% (02:35) correct 55% (02:36) wrong based on 60 sessions

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The average of the numbers 1, 2, 3,..., 98, 99, and x is 100x. What is x?

A. 49/101
B. 50/101
C. 1/2
D. 51/101
E. 50/99

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Re: The average of the numbers 1, 2, 3,..., 98, 99, and x is 100x. What is  [#permalink]

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31 Mar 2019, 22:25
Bunuel wrote:
The average of the numbers 1, 2, 3,..., 98, 99, and x is 100x. What is x?

A. 49/101
B. 50/101
C. 1/2
D. 51/101
E. 50/99

Sum of first n natural numbers = (n*(n+1))/2

Hence for first 99 numbers = (99 * 100)/2

From the question stem, we know:

(1+2+3+4....98+99+x)/100 = 100x
=> 1+2+3....99+x = 100*100x
=> (99*100)/2 + x = 100*100x
=> (99*100)/2 = 100*100x - x = 10000x - x = 9999x = 9(1111)x
=> 99*100 = 2*9*1111x
=> x = (99*100)/(2*9*1111) = 50/101

Hence B is the correct answer.
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Re: The average of the numbers 1, 2, 3,..., 98, 99, and x is 100x. What is  [#permalink]

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01 Apr 2019, 03:09
Bunuel wrote:
The average of the numbers 1, 2, 3,..., 98, 99, and x is 100x. What is x?

A. 49/101
B. 50/101
C. 1/2
D. 51/101
E. 50/99

sum of integers 1 to 99 ; 99 *100/2 = 4950
4950+x/100 = 100x
so
solve for x we get 50/101
IMO B
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Joined: 25 Feb 2019
Posts: 336
Re: The average of the numbers 1, 2, 3,..., 98, 99, and x is 100x. What is  [#permalink]

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01 Apr 2019, 03:32
Sum of numbers 1 to 99 = 4950 ( find the sum of AP 1, 2 ,3 ... 99 where a = 1 and d =1 , n = 99)

So sum of all numbers including x = 4950+x

and total numbers = 99+1 = 100 ( 99 are the original numbers and one extra is x ,so total numbers = 100)

average = total sum /total numbers = (4950+x)/100 = 100x ( given in question)

So we get 4950+x = 10000x
=> 4950 = 10000x-x
=> 4950 = 9999x

so x = 4950/9999 = 50/101

IMO B

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Re: The average of the numbers 1, 2, 3,..., 98, 99, and x is 100x. What is  [#permalink]

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01 Apr 2019, 12:16
Bunuel wrote:
The average of the numbers 1, 2, 3,..., 98, 99, and x is 100x. What is x?

A. 49/101
B. 50/101
C. 1/2
D. 51/101
E. 50/99

There are a total of 100 terms i.e. 1 to 99 and x and given average of these 100 terms is 100x

$$\frac{4950 +x}{100}={100x}$$ where 4950 sum of terms from 1 to 99 using $$\frac{n(n+1)}{2}$$ $$\Rightarrow$$$$\frac{99(99+1)}{2}$$ $$\Rightarrow$$ $$\frac{99(100)}{2}$$$$\Rightarrow$$ 4950

on solving we get x=$$\frac{4950}{9999}$$ on further reduction we get x =$$\frac{50}{101}$$
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Re: The average of the numbers 1, 2, 3,..., 98, 99, and x is 100x. What is   [#permalink] 01 Apr 2019, 12:16
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