In a certain year, the difference between Mary's and Jim's

[guest]

i cant understand how the answer is b? how can we answer the question with 3 variables and only one given??

In a certain year, the difference between Mary's and Jim's annual salaries was twice the difference between Mary's and Kate's annual salaries. If Mary's annual salary was the highest of the three people, what was the average annual salary of the three people that year??

A) Jim's annual salary was 30,000 that year
B) Kate's annual salary was 40,000 that year

[Guest]

In a certain year, the difference between Mary's and Jim's annual salaries was twice the difference between Mary's and Kate's annual salaries. If Mary's annual salary was the highest of the three people, what was the average annual salary of the three people that year??

A) Jim's annual salary was 30,000 that year
B) Kate's annual salary was 40,000 that year



M-J = 2(M-K)

M-J=2M-2K

-M-J = -2K

2K= M+J

2(40K) = M+J

M+J= 80K

[RonPurewal]

i cant understand how the answer is b? how can we answer the question with 3 variables and only one given??

In a certain year, the difference between Mary's and Jim's annual salaries was twice the difference between Mary's and Kate's annual salaries. If Mary's annual salary was the highest of the three people, what was the average annual salary of the three people that year??

A) Jim's annual salary was 30,000 that year
B) Kate's annual salary was 40,000 that year

this is another twist on an old classic: these three points are in ARITHMETIC PROGRESSION, meaning that they are EQUALLY SPACED on the number line. try drawing out a number line and see for yourself: if the distance between M and J is twice the distance between M and K, then this puts K at the exact midpoint between J (on the left) and M (on the right).

whenever you have points in arithmetic progression, no matter how many of them there are, symmetry dictates that the median and the mean are the same value. in the case of this problem, the median is kate's salary, so choice (b) is actually giving us both the median and the mean of this particular set.

(a) is insufficient because it's the leftmost point, giving us no information about the spacing of the points, and therefore no information about the location of point K (= kate's salary, the only one we actually care about).

the algebraic approach in the other post here also works.

[ryan.m.doyle]

Adding to the above:

Simplifying the question:
M - J = 2(M - K)
M - J = 2M - 2K
-J = M - 2K
2K = M + J

Question is asking for the average, so all you really need to solve for is:

M + J + K = ?

Substituting M + J with 2K
2K + 1= ?
3K = ?

So all you need before you even look at the statements is K

1. Insufficient - still two variables (M and K)
2. Sufficient

[RonPurewal]

Adding to the above:

Simplifying the question:
M - J = 2(M - K)
M - J = 2M - 2K
-J = M - 2K
2K = M + J

Question is asking for the average, so all you really need to solve for is:

M + J + K = ?

Substituting M + J with 2K
2K + 1= ?
3K = ?

So all you need before you even look at the statements is K

1. Insufficient - still two variables (M and K)
2. Sufficient

sure

[rkafc81]

this is a sneaky question!!! It's a typical "C-trap" question I reckon...

i translated it like this:

m - j = 2(m - k)
--> so m - j = 2m - 2k
--> so 2k - j = m
--> so m = 2k - j

...the question is asking what is the value of (m + j + k) / 3 = ?
...the value of m + j + k is missing here and is what we need to find...

(1)
j = 30,000
so, substituting in j...
m = 2k - 30,000
m + j + k = 2k - 30,000 + 30,000 + 2k
= 4k
NOT sufficient, as we don't know the value of k

(2) k = 40,000
so, substituting in k...
m = 2(40,000) - j
= 80,000 - j

m + j + k = 80,000 - j + j + 40,000
= 120,000

SUFFICIENT

this is a sneaky question because statements 1 and 2 both seem insufficient... but actually when you work statement 2 through all the way to the end, and don't just intuitively say 'that's not sufficient', then you see that in fact you CAN calculate m + j + k...

i guess the takeaways are as such (am I right?) -->

*whenever two statements seem to be clearly insufficient, and that both statements together are clearly going to be sufficient, LOOK FOR THE TRAP! - the question can't surely be that easy!!! *

and

*when you are dealing with algebraic tranlsations, always use all the given information in the question prompt and see if the statement that you are evaluating actually does contain enough info to solve the question...*

[jlucero]

I'll agree with everything you said and will add one more of my own:

Write down what the question is looking to solve from the very beginning. Even a simple "avg=?" helps remind me that I'm not looking for everything in this problem, but a very specific aspect of the information given.

[supratim7]

"In a certain year, the difference between Mary's and Jim's annual salaries was twice the difference between Mary's and Kate's annual salaries. Mary's annual salary was the highest of the three people"

I came up up 2 rephrases of this,

A) M = y + 2x, K = y + x, J = y
{M - J or 2x = 2(M - K) or 2x}

B) M = 3x, K = 2x, J = x
{M - J or 2x = 2(M - K) or 2x}

Are both these rephrases valid??
(A) vs. (B) What are the implications??
Struggling a bit with the concept/theory here..

Many thanks | Supratim

[RonPurewal]

A) M = y + 2x, K = y + x, J = y

yes.

because this is what they are telling you:
jim = jim
kate = jim + (whatever difference)
mary = jim + 2(whatever difference)
so, then, the difference between mary and kate is also "whatever difference".

B) M = 3x, K = 2x, J = x

no. this is one possibility, but most sets of salaries satisfying the statement won't happen to be in this exact ratio.

for instance, the following salaries work with the statement:
mary $100,000
kate $90,000
jim $80,000
these are definitely not 3x, 2x, and x. but they can be represented as you've done in your first rephrase.

[supratim7]

aah.. big conceptual flaw..
So, the takeaway would look like this.. right??

Case-1
"In a certain year, the difference between Mary's and Jim's annual salaries was twice the difference between Mary's and Kate's annual salaries. Mary's annual salary was the highest of the three people" --» add/subtract something to illustrate the difference between a & b.

Rephrase..
(A) M = y + 2x, K = y + x, J = y
or
(B) M = y, K = y - x, J = y - 2x

multiplication/division CAN ALSO illustrate the "difference" between a & b, but doing so is NOT WARRANTED.

Case-2
"In a certain year, the ratio of Mary's annual salary to Jim's annual salary was twice the ratio of Mary's annual salary to Kate's annual salary. Mary's annual salary was the highest of the three people" --» multiply/divide something to illustrate ratio between a & b

Rephrase..
(A) M = 4x, K = 2x, J = x
or
(B) M = x, K = x/2, J = x/4

Many thanks | Supratim

[jlucero]

aah.. big conceptual flaw..
So, the takeaway would look like this.. right??

Case-1
"In a certain year, the difference between Mary's and Jim's annual salaries was twice the difference between Mary's and Kate's annual salaries. Mary's annual salary was the highest of the three people" --» add/subtract something to illustrate the difference between a & b.

Rephrase..
(A) M = y + 2x, K = y + x, J = y
or
(B) M = y, K = y - x, J = y - 2x

multiplication/division CAN ALSO illustrate the "difference" between a & b, but doing so is NOT WARRANTED.

Correct. What we know is that there is a certain constant that goes up from Jim to Kate to Mary.

Case-2
"In a certain year, the ratio of Mary's annual salary to Jim's annual salary was twice the ratio of Mary's annual salary to Kate's annual salary. Mary's annual salary was the highest of the three people" --» multiply/divide something to illustrate ratio between a & b

Rephrase..
(A) M = 4x, K = 2x, J = x
or
(B) M = x, K = x/2, J = x/4

Many thanks | Supratim

Not quite. Again, this could be the case, but doesn't have to be. When you talk about ratios, you know that the thing that is changing (Kate/Mary's salary) must be changed in order for the ratio to change. But we know nothing about the constant's (Mary's) salary. Numbers:

Could be the case:
M= 8
K = 4
J = 2

M/J = 8/2; M/K = 8/4 --- 8/2 = 2*(8/4)

Doesn't have to be the case:
M= 40
K = 4
J = 2

M/J = 40/2; M/K = 40/4 --- 40/2 = 2*(40/4)

[supratim7]

Not quite. Again, this could be the case, but doesn't have to be. When you talk about ratios, you know that the thing that is changing (Kate/Mary's salary) must be changed in order for the ratio to change. But we know nothing about the constant's (Mary's) salary. Numbers:

Could be the case:
M= 8
K = 4
J = 2

M/J = 8/2; M/K = 8/4 --- 8/2 = 2*(8/4)

Doesn't have to be the case:
M= 40
K = 4
J = 2

M/J = 40/2; M/K = 40/4 --- 40/2 = 2*(40/4)

Thank you Joe for the correction...
Would following be OK?

Case-2
"In a certain year, the ratio of Mary's annual salary to Jim's annual salary was twice the ratio of Mary's annual salary to Kate's annual salary. Mary's annual salary was the highest of the three people"

Rephrase..
(M/K)*2 = M/J

M = x, K = 2y, J = y
or
M = x, K = y, J = y/2

--» multiplication/division by something CAN also illustrate the ratio, but doing so is NOT WARRANTED.
--» 2 variables are involved.

[RonPurewal]

Case-2
"In a certain year, the ratio of Mary's annual salary to Jim's annual salary was twice the ratio of Mary's annual salary to Kate's annual salary. Mary's annual salary was the highest of the three people"

Rephrase..
(M/K)*2 = M/J

M = x, K = 2y, J = y
or
M = x, K = y, J = y/2

--» multiplication/division by something CAN also illustrate the ratio, but doing so is NOT WARRANTED.
--» 2 variables are involved.

that's pretty much the idea.

at that point, though, you might want to just go ahead and test numbers"”that will make your task MUCH easier, conceptually at least.
Viz., look how much easier it is to understand Joe's explanation (in terms of hard numbers) than to understand a bunch of algebra.

[supratim7]

Thank you for the reply Ron.

at that point, though, you might want to just go ahead and test numbers"”that will make your task MUCH easier, conceptually at least.
Viz., look how much easier it is to understand Joe's explanation (in terms of hard numbers) than to understand a bunch of algebra.

Totally agree.. sometimes you can't brush off the algebra out of your head though.. ominous allure of innocent looking variables :D

Many thanks | Supratim

[RonPurewal]

Totally agree.. sometimes you can't brush off the algebra out of your head though.. ominous allure of innocent looking variables :D

Then you need to work on that.

This exam is specifically designed not to give the highest scores to people who automatically do algebra every single time they see variables.
If mental flexibility is a problem, then you need to work on developing mental flexibility.