1. **Problem:** Given $A \times B = 24$, $C \times D = 32$, $B \times D = 48$, and $B \times C = 24$, find $A \times B \times C \times D$. 2. **Formula and approach:** We want to find the product $A \times B \times C \times D$. Notice that this can be grouped as $(A \times B) \times (C \times D)$ or $(B \times C) \times (A \times D)$, but we don't have $A \times D$ directly. We can use the given products to find $A \times D$. 3. **Step-by-step solution:** - From the given, $A \times B = 24$ and $B \times C = 24$. - Also, $B \times D = 48$ and $C \times D = 32$. 4. **Find $A \times C$:** - Divide $B \times C$ by $B$ to get $C$, but we don't know $B$ yet. Instead, use ratios. 5. **Express variables in terms of $B$:** - From $A \times B = 24$, $A = \frac{24}{B}$. - From $B \times C = 24$, $C = \frac{24}{B}$. - From $B \times D = 48$, $D = \frac{48}{B}$. 6. **Calculate $A \times B \times C \times D$:** $$ A \times B \times C \times D = (A \times B) \times (C \times D) = 24 \times 32 = 768 $$ 7. **Answer:** $$ \boxed{768} $$