1. Stating the problem: We are given ratios $A:B=2:3$, $B:C=4:5$, $C:D=7:10$ and asked to find the values of $A$, $B$, $C$, and $D$. 2. Since the problem involves direct and inverse variation but doesn't specify which pairs vary directly or inversely, we'll assume the ratios imply proportional relationships. 3. First, combine the ratios for $B$ from the two given ratios $A:B=2:3$ and $B:C=4:5$ by equating $B$: - From $A:B=2:3$, $B=3k$ - From $B:C=4:5$, $B=4m$ We want $3k=4m$. Let $m=3n$, then $k=4n$ for some $n$. 4. Substitute values back: - $A=2k=2\times4n=8n$ - $B=3k=3\times4n=12n$ - $C=5m=5\times3n=15n$ 5. Now include $C:D=7:10$ with $C=15n$: - $C=7p=15n$, so $p=\frac{15n}{7}$ - $D=10p=10\times\frac{15n}{7}=\frac{150n}{7}$ 6. Thus, the ratios are: $$A=8n,\quad B=12n,\quad C=15n,\quad D=\frac{150n}{7}$$ 7. For simplicity, choose $n=7$ to clear denominators: - $A=8\times7=56$ - $B=12\times7=84$ - $C=15\times7=105$ - $D=\frac{150\times7}{7}=150$ Hence, $$\boxed{A=56, B=84, C=105, D=150}$$