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The average of seven numbers is 20. The average of the first four numb

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The average of seven numbers is 20. The average of the first four numb  [#permalink]

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New post 25 Sep 2018, 04:53
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The average of seven numbers is 20. The average of the first four numbers is 19 and that of the last four is 24. What is the value of the fourth number?

(A) 23
(B) 25
(C) 32
(D) 43
(E) 63

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Re: The average of seven numbers is 20. The average of the first four numb  [#permalink]

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New post 25 Sep 2018, 07:29
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Bunuel wrote:
The average of seven numbers is 20. The average of the first four numbers is 19 and that of the last four is 24. What is the value of the fourth number?

(A) 23
(B) 25
(C) 32
(D) 43
(E) 63


Let the 7 numbers be {a, b, c, d, e, f, g}

The average of seven numbers is 20
We can write: (a + b + c + d + e + f + g)/7 = 20
Multiply both sides by 7 to get: a + b + c + d + e + f + g = 140 [equation #1]

The average of the first four numbers is 19
So, (a + b + c + d)/4 = 19
Multiply both sides by 4 to get: a + b + c + d = 76 [equation #2]

The average of the last four numbers is 24
So, (e + f + g + d)/4 = 24
Multiply both sides by 4 to get: e + f + g + d = 96 [equation #3]

What is the value of the fourth number?
We want to find the value of d

Take the following two equations:
a + b + c + d = 76 [equation #2]
e + f + g + d = 96 [equation #3]

Add them to get: a + b + c + 2d + e + f + g = 172 [equation #4]
Notice that this equation looks A LOT like the first equation.

In fact, we have:
a + b + c + 2d + e + f + g = 172 [equation #4]
a + b + c + d + e + f + g = 140 [equation #1]
Subtract the bottom equation from the top equation to get: d = 32

Answer: C

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The average of seven numbers is 20. The average of the first four numb  [#permalink]

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New post 25 Sep 2018, 17:09
Bunuel wrote:
The average of seven numbers is 20. The average of the first four numbers is 19 and that of the last four is 24. What is the value of the fourth number?

(A) 23
(B) 25
(C) 32
(D) 43
(E) 63


sum of first four numbers=19*4=76
sum of last four numbers=24*4=96
sum of seven numbers=20*7=140
76+96-140=32
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Re: The average of seven numbers is 20. The average of the first four numb  [#permalink]

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New post 27 Sep 2018, 17:32
Bunuel wrote:
The average of seven numbers is 20. The average of the first four numbers is 19 and that of the last four is 24. What is the value of the fourth number?

(A) 23
(B) 25
(C) 32
(D) 43
(E) 63


The sum of all 7 numbers is 20 x 7 = 140.

The sum of the first 4 numbers is 4 x 19 = 76.

The sum of the last 4 numbers is 4 x 24 = 96.

Note that when the sum of the first 4 numbers and the sum of the last 4 numbers are added together, the result will be the sum of the 7 numbers plus the 4th number, which has been added twice since it belongs to both the sums.

So the 4th number is 96 + 76 - 140 = 32.

Answer: C
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Re: The average of seven numbers is 20. The average of the first four numb &nbs [#permalink] 27 Sep 2018, 17:32
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