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Table with Missing Values
A coaching institute conducted a test with three sections: Quantitative Ability (QA), Verbal Ability (VA), and Data Interpretation & Logical Reasoning (DILR). The maximum score for each section is $100$. The overall score is calculated as a weighted average of the three section scores. The weights for QA, VA, and DILR are $w_Q$, $w_V$, and $w_D$ respectively, such that $w_Q = 0.2$, $w_V = 0.3$, and $w_D = 0.5$. The following table provides some of the scores for five students: $$\begin{array}{|c|c|c|c|c|} \hline \text{Student} & \text{QA Score} & \text{VA Score} & \text{DILR Score} & \text{Overall Score} \\ \hline A & 80 & X & 70 & 79.5 \\ \hline B & Y & 90 & 60 & 77 \\ \hline C & 60 & 70 & Z & OS_C \\ \hline D & 75 & 85 & 65 & OS_D \\ \hline E & P & Q & R & OS_E \\ \hline \end{array}$$ **Additional Information:** 1. No student scored less than $50$ in any section. 2. The score $X$ is the highest VA score among all students. 3. The score $Y$ is the highest QA score among all students. 4. $Z$ is the average of $X$ and $Y$. What is the overall score of Student C? A. $81.75$ B. $79.50$ C. $80.25$ D. $78.00$
Correct Answer: A **Step-by-step derivation:** 1. **Understand the Overall Score Calculation:** The overall score for any student is given by the weighted average of their section scores. The weights are $w_Q = 0.2$, $w_V = 0.3$, and $w_D = 0.5$. Overall Score $= (0.2 \times \text{QA Score}) + (0.3 \times \text{VA Score}) + (0.5 \times \text{DILR Score})$. 2. **Determine the value of X (Student A's VA Score):** For Student A, we have: QA Score = $80$ VA Score = $X$ DILR Score = $70$ Overall Score = $79.5$ Using the overall score formula: $0.2 \times 80 + 0.3 \times X + 0.5 \times 70 = 79.5$ $16 + 0.3X + 35 = 79.5$ $51 + 0.3X = 79.5$ $0.3X = 79.5 - 51$ $0.3X = 28.5$ $X = \frac{28.5}{0.3} = 95$ Given that $X$ is the highest VA score among all students, and the other known VA scores are $90$ (Student B), $70$ (Student C), and $85$ (Student D), $X=95$ satisfies this condition. 3. **Determine the value of Y (Student B's QA Score):** For Student B, we have: QA Score = $Y$ VA Score = $90$ DILR Score = $60$ Overall Score = $77$ Using the overall score formula: $0.2 \times Y + 0.3 \times 90 + 0.5 \times 60 = 77$ $0.2Y + 27 + 30 = 77$ $0.2Y + 57 = 77$ $0.2Y = 77 - 57$ $0.2Y = 20$ $Y = \frac{20}{0.2} = 100$ Given that $Y$ is the highest QA score among all students, and the other known QA scores are $80$ (Student A), $60$ (Student C), and $75$ (Student D), $Y=100$ satisfies this condition. 4. **Determine the value of Z (Student C's DILR Score):** It is given that $Z$ is the average of $X$ and $Y$. $Z = \frac{X + Y}{2}$ $Z = \frac{95 + 100}{2} = \frac{195}{2} = 97.5$ 5. **Calculate the Overall Score of Student C ($OS_C$):** For Student C, we have: QA Score = $60$ VA Score = $70$ DILR Score = $Z = 97.5$ Using the overall score formula: $OS_C = (0.2 \times 60) + (0.3 \times 70) + (0.5 \times 97.5)$ $OS_C = 12 + 21 + 48.75$ $OS_C = 81.75$ **Final Answer:** The overall score of Student C is $81.75$.
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