GRE Math : How to find the number of integers between two other integers
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Example Questions
Example Question #1 : How To Find The Number Of Integers Between Two Other Integers
x = the total number of positive, odd 2 digit numbers less than 100
Column A
x
Column B
45
The two quantities are equal
The quantity in column B is greater
The quantity in column A is greater
The relationship cannot be determined from the given information
The two quantities are equal
There are 50 positive, odd numbers less than 100, and 45 of them are 2 digit numbers.
Example Question #1 : Counting
0 < x < y < z < 10
x, y, and z are integers.
Quantity A: –7
Quantity B: x + y – z
The relationship cannot be determined from the information given.
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
Quantity B is greater.
Since:
- There are only 9 integers between 0 and 10
- x, y, and z must all be unique
- They must be specifically ordered such that x < y < z
There are actually not too many ways in which these numbers can be chosen. So what we can do is find a range of answers for Quantity B, and see if 6 falls a) below b) above or c) in between the range.
For the maximum:
Note that the term (x + y) is maximized when x and y are maximum. The (–z) term is maximized when z is minimized. However, there are 2 terms in (x + y) and one term in (–z); thus intuitively it seems we should prioritize (x+y). To make x and y maximum:
0 < 7 < 8 < 9 < 10 since x, y, and z must be unique.
Thus maximum: (x + y – z) = 7 + 8 – 9 = 6
For the minimum:
Note that (x + y) is minimum when (x) and (y) are minimum, and (–z) is minimum when (z) itself is maximimized. However since there are 2 terms in (x+y) and1 of (–z) , again intuititively you should prioritize (x+y) over (-z). Then in order to make this the least number possible, x and y would be:
min(x + y – z) = 1 + 2 – 9 = –6
Thus, the range of possible answers is:
(x + y – z): [–6, 6]
and –7 is always less than this amount.
Example Question #2 : Counting
Quantity A: The number of positive even integers less than 1000
Quantity B: The number of positive odd integers less than 1000
Quantity B is greater.
Quantity A is greater.
The relationship cannot be determined from the information given.
The two quantities are equal.
Quantity B is greater.
The question asks for the number of positive even and odd integers less than 1000. Because 1000 is not included, the numbers to consider are 1 through 999. Every positive odd integer will have a corresponding even integer (1 and 2, 3 and 4, 5 and 6, etc.) until you get to 999. This gives the positive odd integers one more number than the number of positive even integers.
Example Question #2 : How To Find The Number Of Integers Between Two Other Integers
Miles is 3 years older than Ashley. Ashley is 5 years younger than Bill. How old is Ashley if together the three of their ages sum to 44?
8
10
14
16
12
12
Miles is 3 years older than Ashley, so M = A + 3. Also, Bill is 5 years older than Ashley, so B = A + 5. Together the three of their ages sum to 44, thus:
A + A + 3 + A + 5 = 44
3A + 8 = 44
3A = 36
A = 12
Example Question #205 : Arithmetic
Quantity A: The number of possible values of 
Quantity B: 31
Quantity B is greater
Quantity A is greater
The information cannot be determined from the information given.
The two quantities are equal
Quantity B is greater
An integer with a units digit of 5 occurs once every 10 consecutive integers. There are 300 integers between 200 and 500, so there must be 30 values with a units digit of 5.
Example Question #2 : How To Find The Number Of Integers Between Two Other Integers
In a certain game, integers are called magic numbers if they are multiples of either 
How many magic numbers are there in the game between 
There are 13 "magic" numbers: 3,4,6,8,9,12, their negative counterparts, and 0.
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