Are the integers z and f to the right of 0 on the number

Are the integers z and f to the right of 0 on the number line?

Question basically asks whether z and f are both positive.

(1) The product of z and f is positive --> zf>0 --> z and f have the same sign, they are either both positive or both negative. Not sufficient.

(2) The sum of z and f is positive --> z+f>0 --> it's certainly possible that both are positive but it's also possible one to be positive and another negative (for example z=2 and f=-1). Not sufficient.

(1)+(2) Since from (2) z+f>0 then from (1) it's not possible that z and f to be both negative. Hence, z and f are both positive. Sufficient.

Answer: C.
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Given :
Q- Y-N;
is Z,F < 0;

1. Z * F = +Ve ==
(Z,F) = ( +v,+v) : No
(Z,F) = (-v , -v ) : Yes
Insufficient:

2.Z+F = +v
Z,F = (+v,+v) : No
Z > F ( +v,-v ) : F <0, Z<0 :
Insufficient:

1+2:
from 1 ; It (Z,F) has to (-ve,-ve) or (+v,v)
from 2 : (Z,F) (+v,+v)
Hence : (Z,F) = (+v,+v) : Z,F >0 : No

C: Sufficient.

Target question: Are the integers z and f to the right of 0 on the number line?
This is a good candidate for rephrasing the target question.

If z and f to the right of 0 on the number line, then that means z and f are both POSITIVE
Let's now REPHRASE the target question....
REPHRASED target question: Are integers z and f both POSITIVE?

Aside: the video below has tips on rephrasing the target question

Statement 1: The product of z and f is positive.
There are two possible scenarios that satisfy statement 1:
Case a: z and f are both positive. In this case, the answer to the REPHRASED target question is YES, z and f are both POSITIVE
Case b: z and f are both negative. In this case, the answer to the REPHRASED target question is NO, z and f are not both POSITIVE
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The sum of z and f is positive.
There are infinitely-many values of z and f that satisfy statement 2. Here are two:
Case a: z = 1 and f = 1. In this case, the answer to the REPHRASED target question is YES, z and f are both POSITIVE
Case b: z = -1 and f = 2. In this case, the answer to the REPHRASED target question is NO, z and f are not both POSITIVE
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
From statement 1, we concluded that EITHER z and f are both positive OR z and f are both negative
If z and f are both positive, then the sum of z and f will be POSITIVE
If z and f are both negative, then the sum of z and f will be NEGATIVE

Statement 2 tells us that The sum of z and f is positive.
AHA!
This means it must be the case that z and f are both positive
So, the answer to the REPHRASED target question is YES, z and f are both POSITIVE
Since we can answer the REPHRASED target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

RELATED VIDEO

In other words, we need to determine: are both z and f positive?

Statement One Alone:

The product of z and f is positive.

Since the product of z and f is positive, either z and f are both positive or both negative. Thus, we cannot determine on which side of the number line they fall. Statement one is not sufficient to answer the question.

Statement Two Alone:

The sum of z and f is positive.

Although the sum is positive, either both z and f are positive or one is positive and one is negative. Thus, we cannot determine on which side of the number line they fall. Statement two is not sufficient to answer the question.

Statements One and Two Together:

Using the statements together, we see that the only possible scenario that works is when both z and f are positive.

Answer: C
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Step 1: Analyse Question Stem

z and f are integers.
We have to find out if z and f are both positive.

Step 2: Analyse Statements Independently (And eliminate options) – AD / BCE

Statement 1: The product of z and f is positive.

The product of two numbers is positive if both have the same signs.
Therefore, z and f can either both be positive or both be negative.

The data in statement 1 is insufficient to answer the question with a definite Yes or No.
Statement 1 alone is insufficient. Answer options A and D can be eliminated.

Statement 2: The sum of z and f is positive.

When the sum of two numbers is positive,
Both numbers are positive
OR
The bigger number (in terms of absolute value) is positive and the smaller number (again, in terms of absolute value) would be negative

The data in statement 2 is insufficient to answer the question with a definite Yes or No.
Statement 2 alone is insufficient. Answer option B can be eliminated.

Step 3: Analyse Statements by combining

From statement 1: The product of z and f is positive.
From statement 2: The sum of z and f is positive.

For satisfying both constraints, both numbers have to be positive.

The combination of statements is sufficient to answer the question with a definite YES.
Statements 1 and 2 together are sufficient. Answer option E can be eliminated.

The correct answer option is C.
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