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Intern  Joined: 18 Apr 2010
Posts: 4
The average of 4 consecutive odd numbers is half that of the  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 70% (03:08) correct 30% (03:03) wrong based on 228 sessions

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The average of 4 consecutive odd numbers is half that of the average of 5 consecutive even numbers. If the sum of these two average is 18, then the difference between the largest and smallest of these numbers is

A. 10
B. 21
C. 7
D. 13
E. 5
##### Most Helpful Expert Reply
Math Expert V
Joined: 02 Sep 2009
Posts: 57022
Re: average problem  [#permalink]

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ENAFEX wrote:
Is there a different approach to this problem? I find the explanations above tough!! The average of 4 consecutive odd numbers is half that of the average of 5 consecutive even numbers. If the sum of these two average is 18, then the difference between the largest and smallest of these numbers is
A. 10
B. 21
C. 7
D. 13
E. 5

Some notes:
The average of evenly spaced set with even number of terms (4 in our case) is the average of two middle terms.
The average of evenly spaced set with odd number of terms (5 i our case) is the middle term.

Say the average of 4 consecutive odd numbers is $$x$$ and the average of 5 consecutive even numbers is $$y$$.

Given: $$x=\frac{y}{2}$$ and $$x+y=18$$ --> solve for $$x$$ and $$y$$: $$x=6$$and $$y=12$$.

So, we have that the average of 4 consecutive odd numbers is 6, which means that those numbers are: {3, 5, 7, 9} (6 is the average of two middle terms);

Similarly we have that the average of 5 consecutive even numbers is 12, which means that those numbers are: {8, 10, 12, 14, 16} (12 is the middle term);

The difference between the largest and smallest of these numbers is 16-3=13.

Answer: D.

Hope it's clear.
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Manager  Joined: 13 Dec 2009
Posts: 108
Re: average problem  [#permalink]

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gmat2012 wrote:
The average of 4 consecutive odd numbers is half that of the average of 5 consecutive even numbers. If the sum of these two average is 18, then the difference between the largest and smallest of these numbers is
a.10
b.21
c.7
d.13
e.5
OA d
please explain

Let odd numbers be 2n-3, 2n-1, 2n + 1, 2n + 3. Average = 2n.
Let even numbers be 2m- 4, 2m - 2, 2m, 2m + 2, 2m + 4. Average = 2m
it is given that 2m = 4n
Also 2n + 2m = 18 => 2n + 4n = 18.
6n = 18, 2n = 6 & 2m = 12. Largest = 16, smallest = 3.
Difference = 16 - 3 = 13.
hope this will help
Intern  Joined: 15 Mar 2010
Posts: 7
Re: average problem  [#permalink]

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2
There's a simple solution to this.

To find the average for a set of consecutive numbers, you add the first and last terms and divide by 2. In other words, the average is essentially center/pivot point of the series, whether or not it is a number in the series. (e.g. 1, 3, 5, 7 - the average is 4)

Now we look at the other information given. the average of the odd series is half the average of the even series and they sum up to 18. So let e be the average of the even series. We get 1.5e = 18
=> e = 12

12 will be the middle term of the series and since there are 5, we now know the series look like this: (8, 10, 12, 14, 16)
12/2 = 6, the pivot point of the odd series, since there are 4, we know the series look like this: (3, 5, 7, 9)

16 - 3 = 3.

QED.
Manager  Joined: 13 Dec 2009
Posts: 108
Re: average problem  [#permalink]

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1
thanatoz wrote:
There's a simple solution to this.

To find the average for a set of consecutive numbers, you add the first and last terms and divide by 2. In other words, the average is essentially center/pivot point of the series, whether or not it is a number in the series. (e.g. 1, 3, 5, 7 - the average is 4)

Now we look at the other information given. the average of the odd series is half the average of the even series and they sum up to 18. So let e be the average of the even series. We get 1.5e = 18
=> e = 12

12 will be the middle term of the series and since there are 5, we now know the series look like this: (8, 10, 12, 14, 16)
12/2 = 6, the pivot point of the odd series, since there are 4, we know the series look like this: (3, 5, 7, 9)

16 - 3 = 3.

QED.

good thought, i essentially solved using conventional method like assuming even and odd series numbers.. thanks for giving different prospective to the solution.
Manager  Status: And the Prep starts again...
Joined: 03 Aug 2010
Posts: 100
Re: average problem  [#permalink]

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Is there a different approach to this problem? I find the explanations above tough!! _________________
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Joined: 16 Oct 2010
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Re: The average of 4 consecutive odd numbers is half that of the  [#permalink]

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2
3
gmat2012 wrote:
The average of 4 consecutive odd numbers is half that of the average of 5 consecutive even numbers. If the sum of these two average is 18, then the difference between the largest and smallest of these numbers is

A. 10
B. 21
C. 7
D. 13
E. 5

Start with what you have been given so that you don't need to take variables. One average is half of the other and the sum of both is 18.
So a + 2a = 18
a = 6

Avg of 4 consecutive odd numbers is 6. The consecutive odd numbers will be 3, 5, 7 and 9. (avg lies in between the middle two numbers)
Avg of 5 consecutive even numbers is 12. The consecutive even numbers will be 8, 10, 12, 14, 16 (avg is the middle number).

Largest - smallest number = 16 - 3 = 13

Answer (D)
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Joined: 07 Dec 2014
Posts: 1225
The average of 4 consecutive odd numbers is half that of the  [#permalink]

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1
The average of 4 consecutive odd numbers is half that of the average of 5 consecutive even numbers. If the sum of these two average is 18, then the difference between the largest and smallest of these numbers is

A. 10
B. 21
C. 7
D. 13
E. 5

odd average=(4x+12)/4=x+3
even average=(5y+20)/5=y+4
y+4=2(x+3)➡2x-y=-2
(x+3)+(y+4)=18➡x+y=11
adding, 3x=9
x=3
y=11-3=8
8+4*2=16
16-3=13
D.
Intern  B
Joined: 13 Mar 2019
Posts: 27
Re: The average of 4 consecutive odd numbers is half that of the  [#permalink]

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1
Clearly there are two equations possible.
And by the averages' property -
1.x=y/2 and x+y=18 Solving- x=6x=6and y=12.
Now the highest and lowest - 16 and 3.
So 16-3=13 Re: The average of 4 consecutive odd numbers is half that of the   [#permalink] 08 Aug 2019, 10:56
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