OG The people in a line waiting to buy tickets

AbeNeedsAnswers wrote:The people in a line waiting to buy tickets to a show are standing one behind the other. Adam and Beth are among the people in the line, and Beth is standing behind Adam with a number of people between them. If the number of people in front of Adam plus the number of people behind Beth is 18, how many people in the line are behind Beth?

(1) There are a total of 32 people in the line.
(2) 23 people in the line are behind Adam.

C

Target question: How many people in the line are behind Beth?

Given: Beth is standing behind Adam with a number of people between them. The number of people in front of Adam plus the number of people behind Beth is 18
So, we have: FRONT.....x people...ADAM......y people.....BETH......z people......BACK
We can write: x + z = 18
NOTE: Our goal is to determine the value of z

Statement 1: There are a total of 32 people in the line
We can write: x + y + z + 2 = 32 (the 2 represents Adam and Beth)
Simplify: x + y + z = 30
Is this information, along with x + z = 18, enough to determine the value of z?
No. There are several values of x, y and z that satisfy statement 1. Here are two:
Case a: x = 1, y = 12 and z = 17. In this case, z = 17
Case b: x = 2, y = 12 and z = 16. In this case, z = 16
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: 23 people in the line are behind Adam
We can write: y + z + 1 = 23 (the 1 represents Beth)
Simplify: y + z = 22
Is this information, along with x + z = 18, enough to determine the value of z?
No. There are several values of x, y and z that satisfy statement 1. Here are two:
Case a: x = 2, y = 6 and z = 16. In this case, z = 16
Case b: x = 3, y = 7 and z = 15. In this case, z = 15
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x + y + z = 30
Statement 2 tells us that y + z = 22
Plus, it's given that x + z = 18
Since we have three different equations with 3 variables, we COULD solve this system for x, y and z, which means we COULD determine the value of z (the number of people behind Beth) . Of course, we're not going to waste valuable time solving the system, since our sole goal is to determine the sufficiency of the statements.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

AbeNeedsAnswers wrote:The people in a line waiting to buy tickets to a show are standing one behind the other. Adam and Beth are among the people in the line, and Beth is standing behind Adam with a number of people between them. If the number of people in front of Adam plus the number of people behind Beth is 18, how many people in the line are behind Beth?

(1) There are a total of 32 people in the line.
(2) 23 people in the line are behind Adam.

C

We need to determine the number of people in the line who are behind Beth, given that Beth is standing behind Adam and the number of people in front of Adam plus the number of people behind Beth is 18. If we know the number of people who are in front of Adam and the number of people who are strictly between Adam and Beth, then we can determine the number of people who are behind Beth.

Statement One Alone:

There are a total of 32 people in the line.

Since there are a total of 32 people in the line, subtracting Adam, Beth, and the 18 people who are either in front of Adam or behind Beth, we know that there are 12 people between Adam and Beth. However, since we don’t know the exact number of people who are in front of Adam, we can’t determine the number of people who are behind Beth. Statement one alone is not sufficient.

Statement Two Alone:

23 people in the line are behind Adam.

We are given that 23 people in the line are behind Adam. However, since we don’t know the number of people who are between Adam and Beth, we can’t determine the number of people who are behind Beth. Statement two alone is not sufficient.

Statements One and Two Together:

From the two statements, we know that there are a total of 32 people in the line and that 23 people in the line are behind Adam. Thus, there must be 32 - 23 - 1 = 8 people who are in front of Adam (note the number 1 is for Adam). From statement one, we also know that 12 people are between Adam and Beth. Thus, there are 32 - 8 - 12 - 2 = 10 people behind Beth (note the number 2 is for Adam and Beth).

Answer: C