GMAT Practice Question: If m is an even integer, v is an odd integer, and m>v>0...
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 ?
- \frac{m-v}{2}-1
- \frac{m-v-1}{2}
- \frac{m-v}{2}
- m-v-1
- m-v
Topics: odds/evens, counting evenly spaced integers, translations
Solution
Step 1
We translate “m is an even integer” and “v is an odd integer” into expressions using integer parameters.
v = 2j + 1 m = 2k
Step 2
The even integers between v and m start at the first even after v and end at the last even before m.
\text{first even} = v + 1 = 2(j + 1) \text{last even} = m - 2 = 2(k - 1)
Step 3
We count the number of terms in this evenly spaced sequence with step 2.
\frac{2(k - 1) - 2(j + 1)}{2} + 1 = (k - 1 - (j + 1)) + 1 = k - j - 2 + 1 = k - j - 1
Step 4
We substitute back k = \frac{m}{2} and j = \frac{v - 1}{2} into k - j - 1.
k - j - 1 = \frac{m}{2} - \frac{v - 1}{2} - 1
Step 5
We simplify the expression in terms of m and v.
= \frac{m - (v - 1)}{2} - 1 = \frac{m - v + 1}{2} - 1 = \frac{m - v - 1}{2}
Answer
\frac{m - v - 1}{2}
