In a cricket innings, the highest individual score is equal to 3/11 of the total team score. The next highest individual score is equal to 3/11 of the remaining runs after subtracting the highest score from the total. If the difference between the highest and the next highest scores is 9 runs, what was the total team score?

Introduction / Context:
This question comes from sports based arithmetic, often used to make ratio and fraction topics more interesting. It involves two players scores that are given as fractions of the total team score and of the remaining runs. By converting these statements into algebra, we can find the total score. This type of problem tests your skill in writing expressions with fractions and solving a linear equation based on a given difference between two quantities.

Given Data / Assumptions:

• Total team score is T runs.
• Highest individual score = 3/11 of T.
• Remaining runs after removing the highest score = T - 3/11 T.
• Next highest score = 3/11 of the remaining runs.
• Difference between highest and next highest scores is 9 runs.
• All scores are assumed to be whole numbers of runs.

Concept / Approach:
We introduce a variable T for the total team score and express both the highest and next highest scores in terms of T. Since the next highest is taken as a fraction of the remaining runs, we must carefully subtract the highest score before applying the fraction. The relationship that their difference is 9 runs gives us a straightforward equation in T. Solving that equation provides the total team score. The fractions and arithmetic must be handled with care to avoid mistakes.

Step-by-Step Solution:
Let the total team score be T runs. Highest score H = (3/11) * T. Remaining runs after removing H = T - (3/11) * T. Compute remaining runs: T - 3T/11 = (11T/11 - 3T/11) = 8T/11. Next highest score S = (3/11) of remaining runs = (3/11) * (8T/11) = 24T/121. Given that H - S = 9 runs. So, (3/11) * T - 24T/121 = 9. Convert 3/11 to a fraction with denominator 121: 3/11 = 33/121. Thus, H - S = (33T/121) - (24T/121) = 9T/121. So, 9T/121 = 9. Multiply both sides by 121/9 to get T = 9 * 121 / 9 = 121. Therefore, the total team score is 121 runs.

Verification / Alternative check:
Check the internal values to be sure. If T = 121, then the highest score H = 3/11 * 121 = 33 runs. Remaining runs = 121 - 33 = 88. Next highest score S = 3/11 of 88 = 3 * 8 = 24 runs. Difference between highest and next highest = 33 - 24 = 9 runs, which matches the given information exactly. This confirms that the total team score of 121 runs is correct.

Why Other Options Are Wrong:
110 runs: If T = 110, then H = 30 and the remaining runs are 80. S would be 3/11 of 80, which is not an integer equal to 21, and the difference is not 9.
132 runs: For T = 132, H = 36 and remaining runs are 96, so S = 3/11 of 96 which is not 27, and the difference does not become 9.
143 and 154 runs: Similar substitution shows that the difference between highest and next highest does not equal 9 for these totals either.

Common Pitfalls:
Many learners mistakenly take both scores directly as fractions of the total score, ignoring that the second one is a fraction of the remaining runs. This changes the equation and leads to a wrong total value. Another pitfall is mismanaging the fraction arithmetic, especially the conversion of 3/11 to 33/121. Some students also forget to apply the correct order of operations when subtracting. Writing out each step clearly with a common denominator helps avoid these mistakes.

Final Answer:
The total team score in the innings was 121 runs.