Question

Four friends Amrita, Deepa, Smita and Rhea complete their PhD in different number of years.
(K) The one who took maximum time took eight years to complete her PhD, while the one who took the least time took only three years to complete it.
(L) Rhea took more time only than Amrita and completed her PhD in five years.
(M) Smita did not take longer time than Deepa to complete her PhD.
Arrange the names of the friends in order from the minimum to the maximum number of years taken to complete their PhD.

• A. Amrita \(\rightarrow\) Rhea \(\rightarrow\) Smita \(\rightarrow\) Deepa
• B. Amrita \(\rightarrow\) Rhea \(\rightarrow\) Deepa \(\rightarrow\) Smita
• C. Smita \(\rightarrow\) Amrita \(\rightarrow\) Rhea \(\rightarrow\) Deepa
• D. Smita \(\rightarrow\) Amrita \(\rightarrow\) Deepa \(\rightarrow\) Rhea

Hint:

Pay close attention to the word "only" in statement L. "Rhea took more time ONLY than Amrita" means that except for Amrita, everyone else took more time than Rhea. This immediately places Amrita at the absolute bottom of the list and Rhea in the second position from the bottom.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to determine the chronological order (from minimum to maximum years) of the time taken by four friends to complete their PhD based on the given inequality clues.

Step 2: Detailed Explanation:
Let the number of years taken by Amrita, Deepa, Smita, and Rhea be \( T_A, T_D, T_S, \) and \( T_R \).
- Clue (K):
- Maximum time = 8 years
- Minimum time = 3 years
- Clue (L): "Rhea took more time only than Amrita and completed her PhD in five years."
- This means Rhea took more time than only Amrita, and everyone else took more time than Rhea.
- This implies Amrita took the least time, which is 3 years.
- So we have: \( T_A = 3 \, \text{years} \) and \( T_R = 5 \, \text{years} \).
- Since \( T_R = 5 \) and both Deepa and Smita took more time than Rhea, their times must be greater than 5 years (and one of them must be the maximum of 8 years).
- Clue (M): "Smita did not take longer time than Deepa to complete her PhD."
- This translates to: \( T_S \le T_D \). Since they took different numbers of years, \( T_S < T_D \).
- Thus, Deepa must have taken the maximum time of 8 years.
Ordering the times from minimum to maximum:
\[ T_A < T_R < T_S < T_D \implies \text{Amrita} \rightarrow \text{Rhea} \rightarrow \text{Smita} \rightarrow \text{Deepa} \] This matches Option (A).

Step 3: Final Answer:
(A) Amrita \(\rightarrow\) Rhea \(\rightarrow\) Smita \(\rightarrow\) Deepa