M03-33

Official Solution:


Are integers \(x\), \(y\), and \(z\) consecutive?

(1) The average (arithmetic mean) of \(x\) and \(z\) is equal to \(y\).

This statement implies that \(y=\frac{x+z}{2}\). If \(x=y=z=0\), then the answer is NO, but if \(x=-1\), \(y=0\), and \(z=1\), then the answer is YES. This statement alone is not sufficient to determine if the integers are consecutive.

(2) \(x = -z\).

This statement alone is not sufficient, as it provides no information about \(y\).

(1)+(2) The examples discussed for statement (1) remain valid, so we can have both NO (\(x=y=z=0\)) and YES (\(x=-1\), \(y=0\), and \(z=1\)) answers. Combining both statements is still not sufficient to determine if the integers are consecutive.


Answer: E
_________________

I think a set of 3 equally spaced numbers would also show that statement 1 is insufficient because of the property that mean = median.

I think this is a high-quality question and I agree with explanation. i think it s 700 lv q

I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.
_________________

Hi Bunuel,

For a problem like this, when it asks are x,y,z consecutive, does that mean that they have to be consecutive in that order?

For example, I tested x=3,z-1,y=2. It satisfies S1, but they are not consecutive in the order that is listed on the question. So I am not sure if my test case is allowed or not. Wouldn't the question need to use the word "respectively" to convey that the order matters?

It's not necessary x, y, and z to be consecutive in that order. They would be considered consecutive if say z=1, x=2 and y=3.
_________________

Statement 2 here says x=-z < the only number equal to it's opposite is 0. Therefore x and z are zero. Hence x,y,z can never be consecutive integer . Bunuel can you please explain what am i missing here?

The above is not correct. x = -z can be true for infinitely many numbers:

x = 1 and z = -1 --> x = -z.
x = 2 and z = -2 --> x = -z.
x = 3 and z = -3 --> x = -z.
...
x = -1 and z = 1 --> x = -z.
...
_________________

Key points: We need to see whether the variables in the question are consecutive, as in 1, 2, 3; do not be afraid to calculate an average.
Statements: The first statement tells us that \(\frac{(x + z)}{2} = y\); the second is a basic equation relating x to z.
Breakdown: I typically like to start with what appears to be the easier of the two statements in order to help with timing. I would say statement (2) seems like an easier point of entry. If x = -z, then we can use some easy numbers to test for sufficiency. How about -1 and 1 for x and z, respectively? That would give us -1 and 1, but does y have to equal 0 in that case? Of course not! It could be, but it could just as easily be anything else. With a quick substitution, then, we have ruled out (B) and (D). Onto statement (1).

If the average of x and z is y, then they could be consecutive, as in 1, 2, 3: \(\frac{(1 + 3)}{2} = 2\). Does that have to be the case, though? Since we have already dealt with statement (2), we could test values for x and z that are equidistant from y, which we can set to 0, and see what happens. For instance, let x = -4 and z = 4. Now, we have

\(\frac{(-4 + 4)}{2} = 0\) True.

Since we have tested two valid combinations of integers x and z that average to y, one in which the three are consecutive, and the other in which they are not, we know that statement (1) is NOT sufficient. Together, the two statements achieve nothing more than what we just tested above for the first statement. (We could test -1, 0, and 1 to see that the integers could be consecutive, but there is no need.) Choice (E) must be the answer.

Guessing: Because statement (2) takes little in the way of interpretation to figure out, it should be pretty simple to arrive at an (A), (C), or (E) split. Statement (1), for all its words, deals with nothing more than an average, and with three variables that represent integers, intuitive numbers can crack the problem fairly quickly, as demonstrated above. With a 50/50 between (C) or (E), I would like to think that anyone who had spent a little time figuring out statement (1) would immediately see that putting (1) and (2) together will shed no new light on the problem. Choice (C) is just a trap, NOT to be used as a crutch because you do not want to do a little work. Remember, this is a test of reasoning ability more than one that measures mathematical prowess.

As always, good luck with your studies.

- Andrew
_________________

(1) Yes if 1,2,3
No if 1,3,5
A and D are out

(2) Yes if -1,0,1
No if -2,0,2
B is out

(1) + (2) Yes if -1,0,1
No if -2,0,2
C is out

Therefore, we cannot determine whether x,y,z are consecutive.
Answer: E

Bunuel when it asks for consecutive does it necessarily imply spaced by 1 or could ti also mean spaced by 2 pr spaced by 3?

When we see "consecutive integers" it ALWAYS means integers that follow each other in order with common difference of 1: ... x-3, x-2, x-1, x, x+1, x+2, ...

-7, -6, -5 are consecutive integers.

2, 4, 6 ARE NOT consecutive integers, they are consecutive even integers.

3, 5, 7 ARE NOT consecutive integers, they are consecutive odd integers.

2, 5, 8, 11 ARE NOT consecutive integers, they are terms of arithmetic progression with common difference of 3.

All sets of consecutive integers represent arithmetic progression but not vise-versa.

Hope it's clear.
_________________

I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. (x) 0 + (z) 0 = 0; 0/2=0; if y = 0 why it is not sufficient?

Though your question is not very clear I assume, you think x = y = z = 0 is the only solution for (1)+(2), and thus conclude that x, y, and z, are not consecutive integers. This is not true as the examples in the solution clearly indicate. Infinitely many sets satisfy y=(x+z)/2 and x=-z, for example:

x = 1 and y = 0 and z = -1
x = 0 and y = 0 and z = 0
x = 2 and y = 0 and z = -2
x = 3 and y = 0 and z = -3
...

As you can see, only one set gives consecutive integers (x = 1 and y = 0 and z = -1) while all others do not.
_________________

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
_________________

I think this is a high-quality question and I agree with explanation.
_________________