The average (arithmetic mean) of y numbers is x. If z is added

Question

The average (arithmetic mean) of a set of y numbers is x. If z is added to the set, the new average (arithmetic mean) will be z-5. What is the value of z in terms of x and y?

A) x +  \frac{5}{y}  + 5

B)  \frac{xy + 5y}{y - 1}

C)  \frac{xy - 5}{y+1}

D)  \frac{x}{y+1}  - 5y

E) x -  \frac{5}{y}

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nailin16 · Answer

A
Sum = yx
yx + z = (z-5)(y+1)
=> yx + z = zy + z -5y -5
=> zy = yx+ 5y + 5
So A

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jpeeples85 · Answer

I plugged in numbers for a pretty quick solution. I said:

\frac{sum}{y}=x  or  sum=x*y

Using y=2 and x=3, I found the sum to equal 6.

I then looked at the situation when z is added:

\frac{(sum+z)}{(y+1)}=z-5

I plugged in my values from the first equation to get:

\frac{(6+z)}{(2+1)}=z-5 
 6+z=3(z-5) 
 6+z=3z-15 
 2z=21 
 z=\frac{21}{2}

I then looked at the answer choices and plugged in y=2 and x=3, expecting an answer of  \frac{21}{2} . Answer A was  \frac{21}{2}  but I wanted to check the other answer choices as well.

I eliminated choices B, C, and D right away after plugging in y=2 because their denominators wouldn't support a final answer of  \frac{21}{2} . Answer E was also clearly not correct as it returned  3-\frac{5}{2}  or  \frac{1}{2} . Therefore the answer is A.

Edit: Nailin's solution is better.

BrentGMATPrepNow · Answer

The average (arithmetic mean) of y numbers is x.  
In other words: (sum of all y numbers)/y = x
We can also write:  sum of all y numbers = xy

If z is added to the numbers, the new average (arithmetic mean) will be z-5  
With the addition of this number (z), our NEW SUM =  xy  + z
Also, with the addition of this number (z), we now have y+1 numbers

If the new average is z-5, we can write: ( xy  + z)/(y+1) = z - 5
Multiply both sides by (y+1) to get: xy + z = (z - 5)(y + 1)
Expand: xy + z = yz + z - 5y - 5
Subtract z from both sides: xy = yz - 5y - 5
Rearrange to get yz on one side: yz = xy + 5y + 5
Divide both sides by y to get: z = (xy + 5y + 5)/y
Simplify: z = x + 5 + 5/y

Answer: A

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Abhishek009 · Answer

Total of y numbers is  yx

When z is added to the number it becomes =  ( yx + z )

Average of  ( y + 1 )  numbers is  \frac{( yx + z )}{( y + 1 )} = ( z - 5 )

Or,  ( yx + z ) = ( yz - 5y + z - 5 )

Or,  yx = yz - 5y - 5

Or, yz = yx + 5y + 5

Or,  z = \frac{yx + 5y + 5}{y}

Or,  z = x + 5 + \frac{5}{y}

Hence, answer will be (A)  z = x + 5 + \frac{5}{y}

IanStewart · Answer

I wasn't sure how to interpret the problem. We have y numbers with an average of x. So the sum of the numbers is xy. Then "z is added to the numbers". That could mean one of two things:

• we are inserting into the set a new value, equal to "z". Then our new set will sum to xy + z, and we will have y+1 values in the set. So the new average will be (xy + z)/(y + 1), and if that equals z-5, we get an equation

(xy + z) /(y + 1) = z - 5
xy + z = yz - 5y + z - 5
xy +5y + 5 = yz
z = (xy + 5y + 5)/y = x + 5 + (5/y), which is answer A.

• but the wording could also mean that we're adding the value z to each number in our set; if I say "I have some numbers, and if 3 is added to the numbers, the new average is 20", that could mean I'm just increasing each number in the set by 3, and not that I'm inserting a new value of "3" into the set. If we interpret the question this way, then by adding z to each value, we increase the mean by z. So the new average of the set is x+z, and if that equals z-5, all we can learn is that x = -5. We can't get an answer this way, so this clearly is not the intended meaning.

BrentGMATPrepNow · Answer

Good point, Ian. Thanks!
I've edited the question to prevent such ambiguity. 
Kudos for you!!

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The average (arithmetic mean) of y numbers is x. If z is added

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