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The average of three equally spaced numbers is 12. If the largest

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MHIKER
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The average of three equally spaced numbers is 12. If the largest [#permalink] New post   11 Dec 2020, 03:56
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The average of three equally spaced numbers is 12. If the largest number equals twice the smallest, what is the value of the smallest number?

(A) 4

(B) 6

(C) 8

(D) 10

(E) 12
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C
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The average of three equally spaced numbers is 12. If the largest [#permalink] New post   11 Dec 2020, 04:07
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SOLUTION:

Let the three numbers be (x-d),x,( x +d)
"d" is the space.

Average = (x-d)+x+(x+ d)/3 = 12
=> x=36/3=12

According to the question:
(x+d) = 2(x-d)
=> x=3d
Using x=12 from (i) we have d=4

Substituting the values-
(12+4) , 12 , (12-4)
16 , 12 , 8
Hence the smallest number is 8. OPTION (C)

Hope this helps :thumbsup:
Devmitra Sen (Math)
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Re: The average of three equally spaced numbers is 12. If the largest [#permalink] New post   13 Dec 2020, 22:08
Since the 3 numbers are evenly spaced,
I used: x, (3/2)x, 2x

The middle number is (3/2)x because it must be the mean of the smallest and largest numbers.

Add them all together and equal them to 36.
Answer is 8
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Re: The average of three equally spaced numbers is 12. If the largest [#permalink] New post   14 Dec 2020, 06:47
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jerryjz2690 wrote:
Since the 3 numbers are evenly spaced,
I used: x, (3/2)x, 2x

The middle number is (3/2)x because it must be the mean of the smallest and largest numbers.

Add them all together and equal them to 36.
Answer is 8


For your case, the difference among each number is x/2; and the answer is correct

What if I take the first number is x, the second number 2x and the third number is 3x; for each number the difference among them is x.

\(But \ I \ am \ coming \ up \ with \ the \ wrong \ answer \ of \ x=6, \ can \ you \ tell \ me \ why?\)
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Re: The average of three equally spaced numbers is 12. If the largest [#permalink] New post   14 Dec 2020, 07:09
three equally spaced numbers mean number are in AP
average of AP is always middle number, therefor middle number is 12
this means sum of other 2 numbers = 24
a + 2a = 3a = 24 .. a= 8

Answer - c
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The average of three equally spaced numbers is 12. If the largest [#permalink] New post   14 Dec 2020, 09:09
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MHIKER wrote:
The average of three equally spaced numbers is 12. If the largest number equals twice the smallest, what is the value of the smallest number?

(A) 4

(B) 6

(C) 8

(D) 10

(E) 12



Now, What I did is:

The three numbers are equally spaced; So this is an arithmetic progression.

In an arithmetic progression, the average is equal to the median (or middle number) of the series.

The first number is \(= a\)

The second number is \(=12\) [as there are three numbers]

The third number is \(= 2a\) [As the largest number equals twice the smallest]

The sum of the three numbers is=

\(=a+12+2a=36\)

\(=3a=36-12=24\)

\(So, \ a=8\)


The smallest number is 8

\(The \ numbers \ are: \ 8,\ 12, \ & \ 16\)

The answer is C
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Re: The average of three equally spaced numbers is 12. If the largest [#permalink] New post   14 Dec 2020, 12:03
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MHIKER wrote:
jerryjz2690 wrote:
Since the 3 numbers are evenly spaced,
I used: x, (3/2)x, 2x

The middle number is (3/2)x because it must be the mean of the smallest and largest numbers.

Add them all together and equal them to 36.
Answer is 8


For your case, the difference among each number is x/2; and the answer is correct

What if I take the first number is x, the second number 2x and the third number is 3x; for each number the difference among them is x.

\(But \ I \ am \ coming \ up \ with \ the \ wrong \ answer \ of \ x=6, \ can \ you \ tell \ me \ why?\)


Thank you for pointing this out. I'm not quite sure if this response pertains to this exact question. The question states that the largest number is twice the smallest, so the third number has to be 2x.
If the set is x, 2x, 3x, then the largest number would be 3 times the smallest number. That's why the answer is different.
Am I right in this?
Thank you
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Re: The average of three equally spaced numbers is 12. If the largest [#permalink] New post   14 Dec 2020, 12:53
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jerryjz2690 wrote:
MHIKER wrote:
jerryjz2690 wrote:
Since the 3 numbers are evenly spaced,
I used: x, (3/2)x, 2x

The middle number is (3/2)x because it must be the mean of the smallest and largest numbers.

Add them all together and equal them to 36.
Answer is 8


For your case, the difference among each number is x/2; and the answer is correct

What if I take the first number is x, the second number 2x and the third number is 3x; for each number the difference among them is x.

\(But \ I \ am \ coming \ up \ with \ the \ wrong \ answer \ of \ x=6, \ can \ you \ tell \ me \ why?\)


Thank you for pointing this out. I'm not quite sure if this response pertains to this exact question. The question states that the largest number is twice the smallest, so the third number has to be 2x.
If the set is x, 2x, 3x, then the largest number would be 3 times the smallest number. That's why the answer is different.
Am I right in this?
Thank you


Yes, you are right.

At first I took a long approach, as the Crackverbal did and the official solution also in that way. Now I learned a shorter way that I posted now. Your right too.
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Re: The average of three equally spaced numbers is 12. If the largest [#permalink]
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