Question

The people in a room are lined up from youngest to oldest. The age of the first person is 15, and the each person thereafter is two years older than the previous person. What is the sum of ages?
(1) The number of people in the room is 37.
(2) The age of the last person is 87.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

For arithmetic progression problems in data sufficiency, remember that to find the sum, you generally need to know three out of the four variables: first term (a), common difference (d), number of terms (n), and last term (l). The problem often gives you two, and each statement provides a way to find a third, which is sufficient.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The ages of the people form an arithmetic progression (AP). An arithmetic progression is a sequence of numbers such that the difference between the consecutive terms is constant.
The first term is \(a = 15\).
The common difference is \(d = 2\).
The ages are 15, 17, 19, ...
The question asks for the sum of the ages, which is the sum of this arithmetic progression.
Step 2: Key Formula or Approach:
The sum of the first \(n\) terms of an AP is given by the formula:
\[ S_n = \frac{n}{2} [2a + (n-1)d] \]
where \(n\) is the number of terms, \(a\) is the first term, and \(d\) is the common difference.
To find the sum, we need to know the number of people, \(n\).
The \(n\)-th term of an AP is given by:
\[ a_n = a + (n-1)d \]
Step 3: Detailed Explanation:
Analyzing Statement (1):
This statement directly gives the number of people, \(n = 37\).
With \(n=37\), \(a=15\), and \(d=2\), we can calculate the sum \(S_{37}\).
\[ S_{37} = \frac{37}{2} [2(15) + (37-1)2] = \frac{37}{2} [30 + (36)2] = \frac{37}{2} [30 + 72] = \frac{37}{2} [102] = 37 \times 51 \]
Since we can calculate a unique value for the sum, statement (1) alone is sufficient.
Analyzing Statement (2):
This statement gives the age of the last person, which is the last term of the AP, \(a_n = 87\).
We can use the formula for the \(n\)-th term to find the number of people, \(n\).
\[ 87 = 15 + (n-1)2 \]
\[ 87 - 15 = (n-1)2 \]
\[ 72 = (n-1)2 \]
\[ 36 = n-1 \]
\[ n = 37 \]
Since we found the value of \(n\), we can proceed to calculate the sum just as we did with statement (1). Therefore, statement (2) alone is also sufficient.
Step 4: Final Answer:
Both statement (1) and statement (2) independently provide enough information to calculate the sum of the ages. Therefore, each statement alone is sufficient.