Multiple choice

In a Mathematics test, the average marks of boys is $x\%$ and the average marks of girls is $y\%$ with $x\neq y$. If te average marks of all students is $z\%$ the ratio of the number of girls to the total number of students is?

  1. $\displaystyle\frac{z-x}{y-x}$
  2. $\displaystyle\frac{z-y}{y-x}$
  3. $\displaystyle\frac{z+y}{y-x}$
  4. $\displaystyle\frac{z+x}{y-x}$
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A Correct answer
Explanation

Let B be the number of boys and G be the number of girls. Total students = B + G. The weighted average is (Bx + Gy) / (B + G) = z. Rearranging: Bx + Gy = zB + zG, so B(x - z) = G(z - y). Thus, G/B = (x - z) / (z - y) = (z - x) / (y - z). The ratio of girls to total students is G / (B + G) = (z - x) / (y - x).

AI explanation

Let the number of boys be B and the number of girls be G, making the total number of students B + G. The overall total marks are found by adding the boys' total marks to the girls' total marks, giving B times x + G times y. We set this equal to the overall average multiplied by the total number of students, giving B times x + G times y = (B + G) times z. Expanding the right side and rearranging the terms to group the boys and girls together yields B times (x - z) = G times (z - y). Solving for the ratio of the number of girls to the total number of students, we divide G by B + G, which results in the fraction (z - x) / (y - x).

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