A student represents his scores in Mathematics, Statistics and Economics examinations in a pie chart. The central angle for Mathematics is $180^{\circ}$. He scored $80$ in Statistics and $70$ in Economics. His score in Mathematics is:

Question

Prepp Team · Accepted Answer

The problem involves interpreting a pie chart representing examination scores. We are given the central angle for Mathematics and the scores for Statistics and Economics. We need to find the score for Mathematics.

Pie Chart Score Calculation

In a pie chart, the central angle is proportional to the value it represents. A full circle represents a total angle of $360^{\circ}$.

We are given:

Central angle for Mathematics = $180^{\circ}$
Score in Statistics = $80$
Score in Economics = $70$

The central angle for Mathematics ($180^{\circ}$) is exactly half of the total angle ($360^{\circ}$). This implies that the score for Mathematics must be half of the total score obtained across all three subjects.

The combined score for Statistics and Economics is $80 + 70 = 150$.

Since the Mathematics angle ($180^{\circ}$) represents half the pie chart, its score must account for the remaining portion not covered by Statistics and Economics *if* those two represented the other half. However, the calculation works differently: the angle $180^{\circ}$ is half the total $360^{\circ}$. Therefore, the Mathematics score ($M_{score}$) must be half of the total score ($Total_{score}$).

The scores for Statistics and Economics together amount to $150$. Let the Mathematics score be $M_{score}$. The total score is $Total_{score} = M_{score} + 80 + 70 = M_{score} + 150$.

Using the proportionality:

$ \frac{\text{Mathematics Angle}}{\text{Total Angle}} = \frac{\text{Mathematics Score}}{\text{Total Score}} $ $ \frac{180^{\circ}}{360^{\circ}} = \frac{M_{score}}{M_{score} + 150} $ $ \frac{1}{2} = \frac{M_{score}}{M_{score} + 150} $

Cross-multiplying gives:

$ 1 \times (M_{score} + 150) = 2 \times M_{score} $ $ M_{score} + 150 = 2 M_{score} $

Subtracting $M_{score}$ from both sides:

$ 150 = M_{score} $

Alternatively, since the Mathematics angle is $180^{\circ}$, it represents $50\%$ of the total. The remaining subjects (Statistics and Economics) account for the other $180^{\circ}$ (the other $50\%$) of the pie chart. Their combined score is $80 + 70 = 150$. Because this combined score represents $50\%$ of the total score, the Mathematics score (representing the other $50\%$) must also be $150$.