In a One Day cricket match of 50 overs, Team A scores at an average rate of 4.5 runs per over for the first 30 overs and 5.5 runs per over for the remaining 20 overs. Team B bats the full 50 overs while chasing this target but finally loses the match by 10 runs. What is the average number of runs per over scored by Team B over its 50 overs?

Introduction / Context:
This problem is based on the concept of average run rate in a limited overs cricket match. We are given the over wise scoring pattern of Team A and the margin of defeat for Team B. Using this information we have to determine the average runs per over scored by Team B over the full 50 overs of its innings.

Given Data / Assumptions:

• Total overs in the match for each team = 50 overs.
• For Team A, average in the first 30 overs = 4.5 runs per over.
• For Team A, average in the last 20 overs = 5.5 runs per over.
• Team B bats all 50 overs and loses by 10 runs.
• We assume standard one day rules where every over has 6 legal balls and totals are simple sums of runs.

Concept / Approach:
The key ideas are:

• Runs scored = average run rate * number of overs.
• Team A total gives the target for Team B.
• If Team B loses by 10 runs, its total is 10 runs less than Team A.
• Average run rate of Team B = (total runs of Team B) / 50.

We will first compute Team A total, then Team B total, and finally Team B average run rate.

Step-by-Step Solution:
Step 1: Compute runs scored by Team A in the first 30 overs. Runs in first 30 overs = 30 * 4.5 = 135 runs. Step 2: Compute runs scored in the last 20 overs. Runs in last 20 overs = 20 * 5.5 = 110 runs. Step 3: Compute Team A total score. Team A total = 135 + 110 = 245 runs. Step 4: Compute Team B total, given that it loses by 10 runs. Team B total = 245 - 10 = 235 runs. Step 5: Compute Team B average run rate. Average of Team B = 235 / 50 = 4.7 runs per over.

Verification / Alternative check:
We can check reasonableness. Team A scores 245 in 50 overs, which is an overall run rate of 245 / 50 = 4.9 runs per over. Team B scores slightly lower, 235 runs, so its average must be slightly less than 4.9 and more than 4.5. The value 4.7 fits well between these, confirming our calculation is consistent.

Why Other Options Are Wrong:
The option 4.4 is too low and would give a total of only 4.4 * 50 = 220 runs, which would mean a defeat margin of 25 runs, not 10. The option 5.2 is higher than Team A run rate and would give 260 runs, implying Team B wins, which contradicts the question. The option 4.6 gives 230 runs, corresponding to a 15 run defeat, again not matching the given 10 run margin. Only 4.7 produces exactly 235 runs and a 10 run defeat.

Common Pitfalls:
Students often make the mistake of taking the simple average of run rates, that is (4.5 + 5.5) / 2 = 5, which is wrong because the number of overs in each segment is different. Another common error is to forget that Team B total is 10 runs less than Team A, or to divide the total runs by 20 or 30 instead of the full 50 overs. Always relate runs, overs and run rate using total runs = overs * run rate.

Final Answer:
The average number of runs per over scored by Team B is 4.7 runs per over.