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

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M7MBA: Moderator; Posts: 2058; Joined: Sun Oct 29, 2017 4:24 am; Thanked: 1 times; Followed by:5 members

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

by M7MBA » Sat Nov 25, 2017 8:15 am

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

$$\left(A\right)\ x+\frac{5}{y}+5$$ $$\left(B\right)\frac{xy+5y}{y-1}$$ $$\left(C\right)\frac{xy-5}{y+1}$$ $$\left(D\right)\frac{x}{y+1}+5y$$ $$\left(E\right)x-\frac{5}{y}$$ I'd really like some help here. Experts, may you explain this PS question to me? Thanks.

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Source: Beat The GMAT — Problem Solving | Sat Nov 25, 2017 8:15 am

EconomistGMATTutor: GMAT Instructor; Posts: 555; Joined: Wed Oct 04, 2017 4:18 pm; Thanked: 180 times; Followed by:12 members

by EconomistGMATTutor » Mon Nov 27, 2017 9:02 am

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

$$\left(A\right)\ x+\frac{5}{y}+5$$ $$\left(B\right)\frac{xy+5y}{y-1}$$ $$\left(C\right)\frac{xy-5}{y+1}$$ $$\left(D\right)\frac{x}{y+1}+5y$$ $$\left(E\right)x-\frac{5}{y}$$ I'd really like some help here. Experts, may you explain this PS question to me? Thanks.

Hi M7MBA,
Let's take a look at your question.

The average (arithmetic mean) of y numbers is x, therefore, if the sum of y numbers is Y then it can be represented as:
$$\frac{Y}{y}=x$$
$$Y=xy ... (i)$$

If z is added to the numbers, the new average (arithmetic mean) will be z-5,
$$\frac{Y+z}{y+1}=z-5$$

Plugin Y=xy from eq(i),
$$\frac{xy+z}{y+1}=z-5$$
$$xy+z=\left(z-5\right)\left(y+1\right)$$
$$xy+z=yz-5y+z-5$$
$$xy=yz-5y-5$$
$$xy+5y+5=yz$$
$$z=\frac{1}{y}\left(xy+5y+5\right)$$
$$z=\frac{xy}{y}+\frac{5y}{y}+\frac{5}{y}$$
$$z=x+5+\frac{5}{y}$$

Therefore, Option A is correct.

Hope it helps.
I am available if you'd like any follow up.

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by Brent@GMATPrepNow » Mon Nov 27, 2017 9:21 am

M7MBA wrote:The average (arithmetic mean) of y numbers is x. If z is added to the numbers, the new average (arithmetic mean) will be z-5. What is the value of z in terms of x and y?

$$\left(A\right)\ x+\frac{5}{y}+5$$ $$\left(B\right)\frac{xy+5y}{y-1}$$ $$\left(C\right)\frac{xy-5}{y+1}$$ $$\left(D\right)\frac{x}{y+1}+5y$$ $$\left(E\right)x-\frac{5}{y}$$ I'd really like some help here. Experts, may you explain this PS question to me? Thanks.

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 + y + 5/y

Answer: A

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8086; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

Re: The average (arithmetic mean) of y numbers is x.

by Scott@TargetTestPrep » Tue Jan 21, 2020 8:05 am

M7MBA wrote: ↑
Sat Nov 25, 2017 8:15 am
The average (arithmetic mean) of y numbers is x. If z is added to the numbers, the new average (arithmetic mean) will be z-5. What is the value of z in terms of x and y?

$$\left(A\right)\ x+\frac{5}{y}+5$$ $$\left(B\right)\frac{xy+5y}{y-1}$$ $$\left(C\right)\frac{xy-5}{y+1}$$ $$\left(D\right)\frac{x}{y+1}+5y$$ $$\left(E\right)x-\frac{5}{y}$$ I'd really like some help here. Experts, may you explain this PS question to me? Thanks.

We use this formula for the average: average = sum/number, or equivalently, sum/number = average. We know that the average of y numbers is x, which is expressed as:

x = sum/y

xy = sum

Thus, we see that the sum of y numbers is xy. When z is added to the numbers, we have:

(xy + z)/(y + 1) = z - 5

xy + z = (y + 1)(z - 5)

xy + z = zy + z - 5y - 5

zy = xy + 5y + 5

z = (xy + 5y + 5)/y = x + 5 + 5/y

Answer: A

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