1. Let's define variables based on the problem: - Let $F$ be the number of students who learn the flute. - Let $V$ be the number of students who learn the violin. - Let $x$ be the number of students who learn both the flute and the violin. 2. From the problem, we have two key pieces of information: - Half of the students who learn the flute also learn the violin, so $$x = \frac{1}{2}F$$. - Three times as many students learn the violin as learn the flute, so $$V = 3F$$. 3. We want to find $x$ in terms of $F$ and $V$ and check consistency. 4. Since $x = \frac{1}{2}F$ and $V = 3F$, substitute $F$ from the second equation into the first: $$x = \frac{1}{2}F = \frac{1}{2} \times \frac{V}{3} = \frac{V}{6}$$ 5. Interpretation: - The number of students who learn both instruments, $x$, is half the number of flute learners. - It is also one-sixth the number of violin learners. Final answer: $$x = \frac{1}{2}F = \frac{V}{6}$$