The highest score in an inning was $\frac{3}{11}$ of the total and the next highest was $\frac{3}{11}$ of the remainder. If the scores differed by 9, the total score was

### Concept & Strategy This problem requires formulating an equation for successive remainders. By defining the total score as an algebraic variable or choosing a perfect denominator square value, we can model the relationship between the highest and second-highest scores. ### Step-by-Step Solution * **Given:** * Highest score = $\frac{3}{11}$ of Total Score * Next highest score = $\frac{3}{11}$ of Remainder Score * Difference between these two scores = 9 * **Calculation:** * Let the total score be $S$. * $\text{Highest Score} = \frac{3}{11}S$ * $\text{Remainder} = S - \frac{3}{11}S = \frac{8}{11}S$ * $\text{Next Highest Score} = \frac{3}{11} \times \left(\frac{8}{11}S\right) = \frac{24}{121}S$ * Setup the difference equation: $$ \frac{3}{11}S - \frac{24}{121}S = 9 $$ * Equate the denominators by scaling up $\frac{3}{11}$ to $\frac{33}{121}$: $$ \frac{33}{121}S - \frac{24}{121}S = 9 $$ $$ \frac{9}{121}S = 9 \implies S = 121 $$ ### Exam Strategy & Shortcut **Intelligent Choice of Variable Multiplier:** Since the fraction $\frac{3}{11}$ is applied twice sequentially, let the total score be $11 \times 11 = 121$ units. * Highest score = $\frac{3}{11} \times 121 = 33$ units. * Remainder = $121 - 33 = 88$ units. * Next highest score = $\frac{3}{11} \times 88 = 24$ units. * Difference between the two scores = $33 - 24 = 9$ units. * Since the actual difference given is exactly 9, our assumed base scale value matches perfectly. Therefore, total score = 121. ### Common Pitfall A standard error is taking the second highest score as $\frac{3}{11}$ of the total score instead of the remainder, resulting in an incorrect difference step. Always isolate the phrase "of the remainder". ### Final Answer **Therefore, the correct answer is 121.**