1. **State the problem:** We want to find the weight of the elephant based on the given scales and relationships between animals. 2. **Define variables:** Let the weights be: - $F$ = weight of one frog - $L$ = weight of one lion - $K$ = weight of one koala - $T$ = weight of one tiger - $E$ = weight of one elephant 3. **Write equations from the scales:** - Top-left scale: $F = 2 + 3 = 5$ - Top-center-left scale: $L = F \times 1 = F = 5$ - Top-center-right scale: $2F + K = F + L \Rightarrow 2F + K = F + L$ - Top-right scale: $F + 2L = 3F + L$ - Bottom-left scale: $T = 2K$ - Bottom-center scale: $3K = 3T$ - Bottom-right scale: $3T = E$ 4. **Simplify and solve step-by-step:** - From top-left: $F = 5$ - From top-center-left: $L = 5$ - From top-center-right: $2F + K = F + L \Rightarrow 2(5) + K = 5 + 5 \Rightarrow 10 + K = 10 \Rightarrow K = 0$ - From top-right: $F + 2L = 3F + L \Rightarrow 5 + 2(5) = 3(5) + 5 \Rightarrow 5 + 10 = 15 + 5 \Rightarrow 15 = 20$ which is false, so check carefully: Rearranged: $F + 2L = 3F + L \Rightarrow F + 2L - 3F - L = 0 \Rightarrow -2F + L = 0 \Rightarrow L = 2F$ But from above $L=5$ and $F=5$, so $L=5$ and $2F=10$, contradiction. So the correct equation is $F + 2L = 3F + L$. Substitute $F=5$, $L=5$: $5 + 2(5) = 3(5) + 5 \Rightarrow 5 + 10 = 15 + 5 \Rightarrow 15 = 20$ no. So the assumption $L=5$ must be reconsidered. 5. **Re-examine top-center-left scale:** It says 1 lion = 1 frog × 1 (implied balance), so $L = F$. 6. **Use $L=F$ and $F=5$ from top-left:** So $L=5$. 7. **Use top-center-right scale:** $2F + K = F + L \Rightarrow 2(5) + K = 5 + 5 \Rightarrow 10 + K = 10 \Rightarrow K = 0$ 8. **Use bottom-left scale:** $T = 2K = 2(0) = 0$ 9. **Use bottom-center scale:** $3K = 3T \Rightarrow 3(0) = 3(0) \Rightarrow 0=0$ (valid) 10. **Use bottom-right scale:** $3T = E \Rightarrow 3(0) = E \Rightarrow E = 0$ 11. **Interpretation:** The koala weight $K=0$ leads to tiger $T=0$ and elephant $E=0$, which is not realistic. This suggests the problem's scales imply the koala has zero weight or the problem is inconsistent. 12. **Alternative approach:** Since $K=0$ is unrealistic, check if the top-left scale is interpreted correctly. 13. **Top-left scale:** 1 frog = 2 + 3 weight units means $F=5$. 14. **Top-center-left scale:** 1 lion = 1 frog × 1 means $L=F=5$. 15. **Top-center-right scale:** $2F + K = F + L \Rightarrow 2(5) + K = 5 + 5 \Rightarrow 10 + K = 10 \Rightarrow K=0$ 16. **Top-right scale:** $F + 2L = 3F + L \Rightarrow 5 + 2(5) = 3(5) + 5 \Rightarrow 15 = 20$ contradiction. 17. **Resolve contradiction:** Rearranged top-right scale: $F + 2L = 3F + L \Rightarrow F + 2L - 3F - L = 0 \Rightarrow -2F + L = 0 \Rightarrow L = 2F$ 18. **But from top-center-left scale, $L=F$, so $F = 2F \Rightarrow F=0$ which contradicts $F=5$. 19. **Conclusion:** The scales are inconsistent as given. 20. **If we ignore the top-center-left scale and use $L=2F$ from top-right scale, then:** - $F=5$ - $L=2(5)=10$ - From top-center-right scale: $2F + K = F + L \Rightarrow 10 + K = 5 + 10 \Rightarrow K = 5$ - Bottom-left scale: $T = 2K = 2(5) = 10$ - Bottom-center scale: $3K = 3T \Rightarrow 3(5) = 3(10) \Rightarrow 15 = 30$ contradiction again. 21. **Try bottom-center scale as $3K = 3T$ implies $K = T$** - From bottom-left scale: $T = 2K$ and from bottom-center: $K = T$ implies $K = 2K \Rightarrow K=0$ 22. **Again $K=0$ leads to $T=0$ and $E=0$** 23. **Final step:** From bottom-right scale: $3T = E$ and if $T=0$, then $E=0$. **Answer:** The elephant weighs $0$ weight units based on the given scales and their relationships, indicating an inconsistency or that the koala, tiger, and elephant weights are zero in this puzzle. **Summary:** $$F=5, L=5, K=0, T=0, E=0$$