value of x is closest to which integer?

[Vemuri]

The numbers x and y are not integers. The value of x is closest to which integer?

1. 4 is the integer that is closest to x+y
2. 1 is the integer that is closest to x-y

[bluementor]

The numbers x and y are not integers. The value of x is closest to which integer?

1. 4 is the integer that is closest to x+y
2. 1 is the integer that is closest to x-y

Statement 1: 4 is the integer that is closest to x+y

this essentially means (x+y) can be anywhere between 3.5 and 4.5. We can represent this as an inequality:

3.5 < x + y < 4.5

we have no info on y, so insufficient.

Statement 2: 1 is the integer that is closest to x-y

Again, this can be represented in the form of an inequality:

0.5 < x - y < 1.5

here too we do not know anything about y, hence insufficient.

Both statements together:

3.5 < x + y < 4.5
0.5 < x - y < 1.5

add both inequalities:

3.5 + 0.5 < x + y + x - y < 4.5 + 1.5
4 < 2x < 6
2 < x < 3

x can be anywhere between 2 and 3. If x is 2.1, the nearest integer would be 2 and if x is 2.7, the nearest integer would be 3. Insufficient.

Choose E.

-BM-

[Vemuri]

The numbers x and y are not integers. The value of x is closest to which integer?

1. 4 is the integer that is closest to x+y
2. 1 is the integer that is closest to x-y

Statement 1: 4 is the integer that is closest to x+y

this essentially means (x+y) can be anywhere between 3.5 and 4.5. We can represent this as an inequality:

3.5 < x + y < 4.5

we have no info on y, so insufficient.

Statement 2: 1 is the integer that is closest to x-y

Again, this can be represented in the form of an inequality:

0.5 < x - y < 1.5

here too we do not know anything about y, hence insufficient.

Both statements together:

3.5 < x + y < 4.5
0.5 < x - y < 1.5

add both inequalities:

3.5 + 0.5 < x + y + x - y < 4.5 + 1.5
4 < 2x < 6
2 < x < 3

x can be anywhere between 2 and 3. If x is 2.1, the nearest integer would be 2 and if x is 2.7, the nearest integer would be 3. Insufficient.

Choose E.

-BM-

Cool explanation

[lunarpower]

it shouldn't take too long for you to conclude that the individual statements are insufficient.

statement 1 means that 3.5 < x + y < 4.5
this of course doesn't tell us anything about the sizes of x and y.
for instance, x and y could be 1.5 and 2.5. or, they could be -999.5 and 1003.5.
insufficient by itself.

--

statement 2 means that 0.5 < x - y < 1.5
this likewise tells us nothing about the individual values of x and y.
for instance, x and y could be 2.5 and 1.5. or, they could be 1001.5 and 1000.5.
insufficient by itself.

--

together, you can ADD THE INEQUALITIES, so that 'y' cancels out.
(TAKEAWAY: you can add inequalities whenever the 'alligators' - i.e., the "<" or ">" - face the SAME WAY.)
this gives
3.5 < x + y < 4.5
0.5 < x - y < 1.5
add
4 < 2x < 6
therefore
2 < x < 3

this is still insufficient, because x could be closer to 2, closer to 3, or neither (if it's exactly in the middle, at 2.5).

ans (e).

--

you could also just LOOK FOR SPECIFIC NUMBERS that satisfy the criteria in the problem. for instance, x = 2.7 and y = 1.5 satisfy both statements, making x closest to 3. also, x = 2.3 and y = 1.5 satisfy both statements, making x closest to 2.
so, both statements together are still insufficient, so, (e).