If m is an even integer, v is an odd integer, and m v 0, which of

Question

If  m  is an even integer,  v  is an odd integer, and  m > v > 0 , which of the following represents the number of even integers less than  m  and greater than  v  ?

A.  \frac{m-v}{2} -1

B.  \frac{m-v-1}{2}

C.  \frac{m-v}{2}

D.  m-v-1

E.  m-v

Mo2men · Accepted Answer

Let m = 10 & v = 1...........There are 4 Even number less than 10 and greater than 1

A.  \frac{10-1}{2} -1 = Fraction number.... Eliminate

B.  \frac{10-1-1}{2}  = 4......................... Keep

C.  \frac{10-1}{2}  = Fraction................... Eliminate

D.  10-1-1  = 8........................................ Eliminate

E.  10-1 =9............................................. Eliminate

Answer: B

Dkingdom · Answer

Hi Devbek, 
I'll try to explain.

Number of even numbers between 2 different even numbers (eg, a,b) can be given by ((a-b)/2)+1 (1 is added to include the first even number in the series)
In the current question we need to find number of even numbers "greater than v" and "less than m".
The even number after v(which is odd) is v+1 and the even number before m(which is even) is m-2. (This step is done to exclude both v and m from the total number of even integers)
Hence the number of even numbers between v and m (excluding v and m) should be ((m-2-v-1)/2) + 1) (1 added to include the number v+1)
Upon solving you will get (m-v-1)/2

Hope you got it.

generis · Answer

Greyfield , it looks as if you are interpreting one little part incorrectly. The wording is terse. And to use x can be confusing here. We are looking for a specially defined kind of number, defined by the prompt. Not a value. The answers are supposed to tell us how many of those specially defined numbers there are -- not the actual values of those specially defined numbers. We need the number of numbers between yet two other numbers on a number line. We choose a value for m . We choose a value for v . We count how many even integers exist between our m and our v . Then we plug our m value and v values into the answer choices. We are looking for the answer choice that "spits out" the number that matches the quantity of even integers that we counted. Try rewriting this: ...which of the following represents the number of even integers less than m and greater than v to THIS: ...which of the following represents how many even integers exist that are less than m and greater than v You wrote Mmm... no. x is how many even integers there are. If we pick values, in one case, the number of integers will be even. If we increase m, the number of integers will be odd. Count how many integers there are between your chosen values. When you plug in your m and v Option B will always give you that quantity, that number of integers. • Let's use the numbers in your quote, m = 8, v = 3 <----(v = 3 )----- 4 -----5----- 6 -----7-----(m= 8 )-----> Between the integer 3 and the integer 8, there are two even integers: 4 and 6. How many even integers? Two : (1) the number four; and (2) the number six. The number of even numbers is 2. Answer B) \frac{(m-v-1)}{2} Plug in your numbers: \frac{(8-3-1)}{2}=\frac{4}{2}=2 Perfect. That is, on the number line, between 8 and 3, there are two even numbers: 4 and 6. Both are less than m=8 and greater than v=3 . Answer B does not tell you "4 and 6." You have to see that fact yourself. • Case #2: Your numbers: m = 10, v = 3 <----(v = 3 )---- 4 ----5---- 6 ----7---- 8 -----9-----(m= 10 > Between 10 and 3, there are, just as you calculated from B, three even integers: 8, 6, and 4. All are less than 10 and greater than 3. Once you have figured out how many of those special integers there are, you cannot "go again" and plug your "new" choices into (B) to find out which even integers they are. If you choose different values for the variables, although B's answer or output will change, it will still match your answer -- your new answer, which has also changed. v is any odd number. m is any even number. m > v > 0 I could use m = 100 and v = 7. The answer will be B. The number of even numbers will change depending on what you plug in for m and v , but option B will "spit out" the number of numbers that matches the values you choose. Hope that helps.

Devbek · Answer

The fastest way is plugging in some numbers.

Can someone show how to solve the problem algebraically?

ScottTargetTestPrep · Answer

We can let m = 6 and v = 5. Since there are no (or zero) even integers less than m but greater than v, we see that the answer can be either B, (m - v - 1)/2, or D, m - v - 1, since either choice will produce 0 when we substitute m = 6 and v = 5.

Now, let’s let m = 8 and v = 5. Since there is 1 even integer (namely, 6) less than m but greater than v, we see that the answer must be choice B, since (8 - 5 - 1)/2 = 1 whereas choice D will yield 8 - 5 - 1 = 2. Thus, the correct answer choice must be B.

Answer: B

Dkingdom · Answer

Formula to find count of even numbers in a consecutive series between two numbers is ((a-b)/2) + 1. In this formula both a and b are included in the count.
let m = 10 (even), v = 3(odd)
What the question is asking : find the number of even numbers greater than v (3) and less than m (10).
Breaking the question down :1) We need to exclude m and v from the total count of even numbers. v obviously is out because it is odd. To exclude m we need to use an even number less than m.
Hence in order to apply the above stated formula we will use v+1 (4) and m-2 (8). (Since we need to find only the count of even numbers that's why it is easier to keep the starting and ending numbers as even digits.
Applying the formula : ((8-4)/2)+1) = 3 (i.e. 4,6,8) Remember :numbers are more than 3 and less than 10 and even.
The above equation can be written as ((m-2-v-1)/2)+1 = (m-v-1)/2

nganpham · Answer

m is even and v is odd -> (m-v) is odd -> (m-v)/2 is not an interger => elimate A, C imidiately

SOLVE: Between m and v, we have   = (m-v-1) numbers, include both odd and even -> the number of even is half of (m-v-1)
=> The answer is B

Greyfield · Answer

Hi all,
 I still don't understand. 
M>v>0 and we have to find a number x which is a even integer and m>x>v>0.
Because x is a even integer, x can only be (B) or (D).
If m=8, v=3:
(m-v-1)/2=2
m-v-1= 4
2<v=3; 4>v=3
So (B) is wrong.
If m=10, v =3
(m-v-1)/2=3 is an odd integer
m-v-1=6
So (B) is wrong
Please help me to explain these situations. Thank you

xlgoh1992 · Answer

Hi, I am not really sure why are you canceling out Choice D. I thought the question require us to find an EVEN integer that is LESS than m but GREATER than v? If you use m =10, v=1, isn't 8 fulfill this condition as well? since 8 is an even integer <10 but >1?

chetan2u · Answer

HI..

we rae looking for NUMBERS of even numbers between v and m..
if v=1 and m=10, even integers are 2,4,6,8.. HOW many 4

so our answer should be 4

8 is  ONE of the even integer  BUT we are looking for  NUMBERS  of even integers between v and m

adkikani · Answer

Bunuel   niks18   pushpitkc   chetan2u   VeritasPrepKarishma

Usually in a PS if I take numbers satisfying the property given in question stem
(here even/odd) I am usually left with only one answer choice. Here as suggested by 
 ScottTargetTestPrep  I had to further narrow down my options from 2 to 1 by taking more
set of numbers. Is there any other time efficient way to come to OA?

chetan2u · Answer

to find the number of items between x and y inclusive, it is x-y+1..
say between 2 and 7... 7-2+1=6..... 2,3,4,5,6,7
so if we have consecutive numbers starting with O/E and finishing with E/O... half of them will be odd and half even
so here m is even so m-1 is odd and v is odd so v+1 will be even..
numbers between m-1 and v+1 = m-1-(v+1)+1 = m-1-v-1+1=m-v-1..
half will be even and half odd
so  \frac{m-v-1}{2}

Sri07 · Answer

Hi can you please help me understand the question? I am not clear with what is asked? Thanks in advance

ScottTargetTestPrep · Answer

Solution:

When it is not clear what is being asked in a question like this, you can always assign values to the variables and think in terms of those numbers, which will help you understand what is being asked.

For instance, we can let m = 4 and v = 1. Notice that 4 is an even number, 1 is an odd number and 4 > 1 > 0 is satisfied. For these values, the question now becomes “What is the number of even integers less than 4 and greater than 1?” We see that 2 is the only even integer less than 4 and greater than 1, so the answer to the question for m = 4 and v = 1 is “there is only one even integer between 4 and 1.”

As another example, let’s let m = 10 and v = 5. Again, 10 is even, 5 is odd and we have 10 > 5 > 0. In this case, the question is “What is the number of even integers less than 10 and greater than 5?” Well, there are two such integers: 8 and 6. The answer when m = 10 and v = 5 is “there are two such integers.”

Using the above examples (in addition to the ones I provided in my solution), we understand that we are being asked for the number of even integers between m and v, not inclusive.

To determine the answer, we can also use the new numbers I provided. For instance, if we choose to use m = 4 and v = 1, we need to determine which answer choice gives us 1 when we substitute m = 4 and v = 1. Substituting these values for m and v in the expressions given in the answer choices, we obtain 1/2, 1, 3/2, 2, and 3. As you can see, only the expression in answer choice B results in 1 when we substitute m = 4 and v = 1. That’s why B is the answer.

Answer: B

kittle · Answer

Can someone explain why the answer is not A?

chetan2u · Answer

m is even and v is odd, so m-v = even-odd=odd. 
Thus (m-v)/2=odd/2=fraction

But we cannot have number of even integers as a fraction

CrackverbalGMAT · Answer

There are 2 ways to approach this question.

Approach 1: Plugging values
Let’s say m= 8, n = 3
There are only 2 even numbers between 3 and 8 i.e., 4 and 6
So, when we substitute 8 and 3 as m and n in the answer options, you should get 2 as the answer.

A. (m−v)/2−1 ==> (8-3)/2 – 1 is a decimal number. So, its eliminated
B. (m−v−1)/2 ==> (8-3-1)/2 = 2
C. (m−v)/2 ==> (8-3)/2 is a decimal number. Eliminated
D. m−v−1 ==> 8-3-1 = 4. Eliminated
E. m−v ==> 8-3 = 5. Eliminated

Option B is the answer.

Approach 2:

If a and b are even numbers and a > b.
No of even numbers between a and b (including a and b) = (a-b)/2 + 1

Here m is an even integer, v is an odd integer, and m > v > 0.
V+1 is an even integer.
No of even number between m and v+1 (including m and v+1) =(m-(v+1))/2 +1

No of even numbers between m and v = (m-(v+1))/2 +1 -1 = (m – v – 1)/2
We subtract 1 from the total because m is not included the range 
 
Option B is the answer.

Thanks,
 Clifin J Francis,
GMAT SME

woohoo921 · Answer

chetan2u
Hi friend! I thought for choosing smart numbers, in problems like this where no real numbers are given, you only NEED to choose one set of smart numbers (e.g., just use m = 6 and v = 1 to solve). However, I see an expert on this form choosing two different cases for smart numbers to find the answer. Please correct me if I am wrong, and if you are REQUIRED to try another set of smart numbers. Thank you

chetan2u · Answer

Hi

No, you don’t require to check for the second pair here.

You require to go for second value/values only when the values you have taken give you two options as answer. Then, to eliminate one option, you choose another value. 
But here, any set of values will get you to the answer.

If m is an even integer, v is an odd integer, and m > v> 0, which of

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GMAT Problem Solving (PS) Questions

If m is an even integer, v is an odd integer, and m > v> 0, which of

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