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

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Math Expert
Joined: 02 Sep 2009
Posts: 59727
The average of seven numbers is 20. The average of the first four numb  [#permalink]

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25 Sep 2018, 04:53
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25% (medium)

Question Stats:

77% (01:42) correct 23% (02:02) wrong based on 71 sessions

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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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Posts: 4155
Re: The average of seven numbers is 20. The average of the first four numb  [#permalink]

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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

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

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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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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.

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

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28 May 2019, 20:47
Lets consider 7 no.s to be a b c d e f g

Now a b c d should have a value of 19 each, and d e f g should have a value of 24 each

Point 1 --Now, we can see that there is a loss of 3 because of a b c. (19 - 20) * 3 = -3
Also we can see the surplus of 12 because of e f g . (24 - 20) * 3 = 12

Point 2 -- Finally, D has both a loss of 1 and surplus of 4.

Hence the value of D = Avg Value of D - (loss of -1(from point 2)) + (surplus of 4 (from point 2)) - 3 (point 1)+ 12 (point 1)

therefore d = 20 - 1 + 4 -3 + 12 = 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
CrackVerbal Representative
Joined: 13 May 2019
Posts: 69
Re: The average of seven numbers is 20. The average of the first four numb  [#permalink]

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28 May 2019, 22:55
Average of the 7 numbers = 20

Sum of the 7 numbers = 20*7 = 140

Average of 1st four numbers = 19

Sum of first four numbers = 19*4 = 76

Average of last four numbers = 24

Sum of last 4 numbers = 24*4 = 96

If we sum up the first four numbers and last four numbers, then the sum will be equal to the sum of 7 numbers + the 4th number, as the 4th number will be added twice.

76 + 96 = 140 + 4th number

4th number = 172 - 140 = 32

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Re: The average of seven numbers is 20. The average of the first four numb   [#permalink] 28 May 2019, 22:55
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