If `p,q,r,s,t` are numbers such that `p+q lt r+s` `q+r lt s+t` `r+s lt t+p` `s+t lt p+q` thn the largest and smallest numbers are

Options

A. p and q respectively
B. r and t respectively
C. r and q respectively
D. q and p respectively

Answer

To solve the problem, we need to analyze the given inequalities step by step. We have the following inequalities:

1. \( p + q < r + s \) (Equation 1)
2. \( q + r < s + t \) (Equation 2)
3. \( r + s < t + p \) (Equation 3)
4. \( s + t < p + q \) (Equation 4)

We will use these inequalities to find the largest and smallest numbers among \( p, q, r, s, t \).

### Step 1: Analyze Equations 1 and 3
From Equation 1:
\[ p + q < r + s \]
From Equation 3:
\[ r + s < t + p \]

Combining these two inequalities, we can write:
\[ p + q < t + p \]

Subtracting \( p \) from both sides gives:
\[ q < t \] (Relation 1)

### Step 2: Analyze Equations 2 and 4
From Equation 2:
\[ q + r < s + t \]
From Equation 4:
\[ s + t < p + q \]

Combining these two inequalities, we can write:
\[ q + r < p + q \]

Subtracting \( q \) from both sides gives:
\[ r < p \] (Relation 2)

### Step 3: Analyze Equations 1 and 4
From Equation 1:
\[ p + q < r + s \]
From Equation 4:
\[ s + t < p + q \]

Combining these two inequalities, we can write:
\[ s + t < r + s \]

Subtracting \( s \) from both sides gives:
\[ t < r \] (Relation 3)

### Step 4: Compile the Relations
Now we have the following relations:
1. \( q < t \)
2. \( r < p \)
3. \( t < r \)

From \( t < r \) and \( r < p \), we can conclude:
\[ t < r < p \]

Also, from \( q < t \), we can conclude:
\[ q < t < r < p \]

### Conclusion
From the relations we derived, we can see that:
- The smallest number is \( q \).
- The largest number is \( p \).

Thus, the largest and smallest numbers are:
- Largest: \( p \)
- Smallest: \( q \)