What is the average (arithmetic mean) of x, y, and z?

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DivyaD: Junior | Next Rank: 30 Posts; Posts: 10; Joined: Fri Jan 18, 2019 3:48 am

by DivyaD » Fri Jan 18, 2019 4:00 am

What is the average (arithmetic mean) of x, y, and z?

(1) -x âˆ’ 4y + 3z = 14

(2) 5x + 8y + z = 26

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Source: Beat The GMAT — Quantitative Reasoning | Fri Jan 18, 2019 4:00 am

deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pmFollowed by: 5 members

by deloitte247 » Sun Jan 20, 2019 3:37 am

Statement 1
$$-x-4y+3z=14$$ =
Values of x, y and z are unknown, so we can't arrive at a definite answer hence, Statement 1 is INSUFFICIENT.

Statement 2
$$5x+8y+z=26$$
Value of x, y and z are unknown, so we can't arrive at a definite answer. Hence, Statement 2 is INSUFFICIENT.

Combining statements 1 and 2 together
Adding Statement 1 and 2 together
$$-x-4y+3z=14$$ ..............................1
$$5x+8y+z=26$$ ...............................2

We obtain
$$4x+4y+4z=40$$
Dividing both sides of the equation by 4, we obtain
$$\frac{\left(4x+4y+4z\right)}{4}=\frac{40}{4}$$
$$\left(x+y+z\right)=10$$
$$Average\ arithemetic\ mean=\frac{EFX}{EF}=\frac{10}{3}$$
both statement together is SUFFICIENT.

$$answer\ is\ Option\ C$$

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ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pmFollowed by: 247 members; Thanked: 1443 times

by ceilidh.erickson » Mon Jan 21, 2019 12:56 pm

You should always start by REPHRASING any DS question to determine: what's the minimum information that would be sufficient to answer this question?

Our question is "what is the average of x, y, and z?" First, rephrase algebraically:
(x + y + z)/3 = ?

What would we need in order to find that? Just the sum (x + y + z). We don't necessarily need the values of the variables individually.

Target question: what is the value of x + y + z?

(1) -x - 4y + 3z = 14
We cannot rearrange or simplify this to get a value for x + y + z. Insufficient.

(2) 5x + 8y + z = 26
We cannot rearrange or simplify this to get a value for x + y + z. Insufficient.

(1) and (2) together:
If we combine the 2 equations, we get:
-x - 4y + 3z = 14
+ 5x + 8y + z = 26
____________________
4x + 4y + 4z = 40

Divide both sides by 4 and we get a value for x + y + z. Sufficient. (We don't actually need to do the work to solve, or to find the average. We know that our target question is answerable, so that's enough).

The answer is C.

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

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fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Mon Jan 21, 2019 3:51 pm

DivyaD wrote:What is the average (arithmetic mean) of x, y, and z?

(1) -x âˆ’ 4y + 3z = 14

(2) 5x + 8y + z = 26

$$? = \frac{{x + y + z}}{3}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\boxed{\,? = x + y + z\,}$$
$$\left( 1 \right)\,\, - x - 4y + 3z = 14\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y,z} \right) = \left( { - 14,0,0} \right)\,\,\,\, \Rightarrow \,\,\,? = - 14 \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y,z} \right) = \left( { - 10, - 1,0} \right)\,\,\,\, \Rightarrow \,\,\,? = - 11 \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,{\rm{INSUFF}}.$$
$$\left( 2 \right)\,\,5x + 8y + z = 26\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y,z} \right) = \left( {0,0,26} \right)\,\,\,\, \Rightarrow \,\,\,? = 26 \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y,z} \right) = \left( {4,0,6} \right)\,\,\,\, \Rightarrow \,\,\,? = 10 \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,{\rm{INSUFF}}{\rm{.}}$$
$$\left( {1 + 2} \right)\,\,\,\left\{ \matrix{
\, - x - 4y + 3z = 14 \hfill \cr
\,5x + 8y + z = 26 \hfill \cr} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,4\left( {x + y + z} \right) = 40\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,{\rm{SUFF}}{\rm{.}}$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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