1. **Problem Statement:** Consider the following scenarios: - Direct variation: $y$ varies directly as $x$. - Inverse variation: $z$ varies inversely as $w$. - Joint variation: $m$ varies jointly as $p$ and $q$. - Combined variation: $r$ varies directly as $s$ and inversely as $t$. Let: - When $x=4$, $y=12$. - When $w=3$, $z=8$. - When $p=2$ and $q=5$, $m=20$. - When $s=6$ and $t=2$, $r=9$. Find $y$ when $x=10$, $z$ when $w=6$, $m$ when $p=4$ and $q=3$, and $r$ when $s=8$ and $t=4$. 2. **Direct Variation:** Since $y$ varies directly as $x$, $y=kx$ for some constant $k$. Given $y=12$ when $x=4$, solve for $k$: $$k=\frac{y}{x}=\frac{12}{4}=3$$ To find $y$ when $x=10$: $$y=3\times10=30$$ 3. **Inverse Variation:** Since $z$ varies inversely as $w$, $z=\frac{k}{w}$ for some constant $k$. Given $z=8$ when $w=3$, solve for $k$: $$k=zw=8\times3=24$$ To find $z$ when $w=6$: $$z=\frac{24}{6}=4$$ 4. **Joint Variation:** Since $m$ varies jointly as $p$ and $q$, $m=kpq$ for some constant $k$. Given $m=20$, $p=2$, $q=5$, solve for $k$: $$k=\frac{m}{pq}=\frac{20}{2\times5}=2$$ To find $m$ when $p=4$ and $q=3$: $$m=2\times4\times3=24$$ 5. **Combined Variation:** Since $r$ varies directly as $s$ and inversely as $t$, $r=\frac{ks}{t}$. Given $r=9$ when $s=6$ and $t=2$, solve for $k$: $$k=\frac{rt}{s}=\frac{9\times2}{6}=3$$ To find $r$ when $s=8$ and $t=4$: $$r=\frac{3\times8}{4}=6$$ **Final Answers:** - $y=30$ - $z=4$ - $m=24$ - $r=6$