# The average of 5 numbers is 64. When a sixth number is added, the average becomes 66. What is the sixth number? *?

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• Average of 5 numbers=64

Total of 5 numbers=64×5=320

Average of 6 numbers=66

Total of 6 numbers=66×6=396

Sixth number=396-320= 76

• There are two competition ways to do this, besides the regular algebra method.

One is to realize that when a sixth number is added to a set, it pulls the average with a weight of 1/6.  We need to pull the average up by 2, so with a weight of 1/6, we need to pull by 2*6=12.  So the last number should be 64+12=76.

Method 2 would be to realize that since they don't tell us what the first 5 numbers are, we are free to choose them.  My choice for 5 numbers that average to 64 is all 64's.  The sum of these is 320.  And actually, 320, no matter what the numbers are.  We don't know the 6 numbers that average to 66, so I can choose them all to be 66, which would be a total of 396.  And again, 396 no matter what the 6 numbers are.  396 - 320 = 76, which must be that 6th number.

• I get 70...  70 - 76

• Average = sum / count

64 = (sum of 5 numbers) / 5

66 = (sum of 6 numbers) / 6

sum of 5 numbers = 320

sum of 6 numbers = 396

6th number  = 76

• avg  is  that is the  number it would be if all five were the same

5 x 64 = 320

add a 6th call it x  then the avg is  66

(320 + x)/6 = 66

320 + x = 66 x 6 = 396

x is then 76

basicaally to  increase the avg  by 2  you have to add  2x 6 more percent to the  added grade over the avg previous

64 + 12 = 76

• Mean(5) ; (x(1) + x(2)+... +x(5)) / 5 = 64

Mean(6) ; (x(1) + x(2)+... +x(5) + x(6)) / 6 = 66

Multiply both sides by '5' & '6' respectively

Hence

(x(1) + x(2)+... +x(5)) = 320

(x(1) + x(2)+... +x(5) + x(6)) = 396

Subtract the two lines

x(6) = 396 - 320 = 76

x(6) = 76   ( the sixth number)

• The average of 5 numbers is 64

Σ = 5*64 = 320

When a sixth number is added, the average becomes 66.

Σ + n = 6*66 = 396

What is the sixth number?

n = 396-320 = 76

Ans: 76

• The average of 5 numbers is 64.

When a sixth number is added,

the average becomes 66.

(320 + 76)/6 = 66

The sixth number is 76.

• First imagine you added 64 as the sixth number. The average would stay the same.

To raise the average 2 points (for 6 numbers), you need to add 12 more points. So the sixth number should actually be 76.

64 + 12 = 76

• You are told that (a1+a2+a3+a4+a5)/5 = 64. Thus (a1 + ...+a5) = [left for you].

Then told that ( (a1+...+a5) + a6 )/6 = 66.

Since you found the (a1+...+a5), just isolate a6. Done!

Hopefully no one will spoil you the answer thereby depriving you from your personal enhancement; that would be very inconsiderate of them.