It costs Rs p to send each of the first 1000 messages and Rs q to send each message thereafter. If r is greater than 1000, what is the total cost in rupees to send r messages in this messaging plan?

Introduction:
This question tests the concept of total cost when there are two different rates. The first part of the service charges one rate up to a certain limit, and any usage beyond that limit is charged at a different rate. Such problems are common in mobile plans, internet packages and messaging services, and they can be solved using simple algebra and careful counting of how many units are billed at each rate.

Given Data / Assumptions:
- Cost for each of the first 1000 messages is Rs p. - Cost for each message after the first 1000 is Rs q. - The total number of messages sent is r, and r is greater than 1000. - We need the total cost in rupees expressed in terms of p, q and r.

Concept / Approach:
The key idea is to split the total messages into two groups. The first group contains the first 1000 messages, all of which are charged at the rate p. The second group contains the remaining messages, which are charged at the rate q. The total cost is the cost of the first group plus the cost of the second group. This is a direct application of the idea: total cost = number of units * rate per unit for each slab.

Step-by-Step Solution:
Step 1: Number of messages in the first slab = 1000. Step 2: Cost for the first 1000 messages = 1000 * p = 1000p. Step 3: Since r is greater than 1000, the number of additional messages beyond 1000 is r - 1000. Step 4: Each additional message is charged Rs q, so cost for the extra messages = q * (r - 1000) = q(r - 1000). Step 5: Total cost = cost for first 1000 messages + cost for additional messages. Step 6: Therefore, total cost = 1000p + q(r - 1000).

Verification / Alternative Check:
If r = 1000, the number of additional messages is r - 1000 = 0, so the cost becomes 1000p + q(0) = 1000p, which matches the idea that only the first slab is used. If r = 1001, the cost should be 1000p for the first 1000 messages plus q for one extra message. The formula gives 1000p + q(1001 - 1000) = 1000p + q, which is correct. This confirms that the expression behaves correctly at the boundary.

Why Other Options Are Wrong:
Option b: 1000p + qr ignores that only r - 1000 messages are billed at rate q and therefore overcounts the cost of messages at rate q. Option c: pr + qr uses p for all r messages, which does not match the slab structure given in the question. Option d: 1000q + p(r - 1000) swaps the roles of p and q and so assigns the wrong rate to each slab. Option e: 1000(q - p) + qr does not simplify to the correct expression and gives an incorrect dependence on p and q.

Common Pitfalls:
A common mistake is to multiply r directly by either p or q without splitting the total messages into the two slabs. Another frequent error is to use qr for the second part instead of q(r - 1000). It is important to clearly separate the first 1000 messages and the remaining r - 1000 messages before writing the cost expression.

Final Answer:
The total cost in rupees to send r messages is 1000p + q(r - 1000).