1. **State the problem:** We need to find how many kids (k), parent (p), and sponsor (s) tickets were sold given the total tickets sold is 400 and total revenue is 6800. 2. **Define variables:** - $k$ = number of kids tickets - $p$ = number of parent tickets - $s$ = number of sponsor tickets 3. **Write the equations based on the problem:** - Total tickets: $$k + p + s = 400$$ - Total revenue: $$12k + 18p + 22s = 6800$$ - Relationship between parent and sponsor tickets: $$p = 2s$$ 4. **Substitute $p = 2s$ into the first equation:** $$k + 2s + s = 400 \implies k + 3s = 400$$ 5. **Substitute $p = 2s$ into the revenue equation:** $$12k + 18(2s) + 22s = 6800 \implies 12k + 36s + 22s = 6800 \implies 12k + 58s = 6800$$ 6. **From step 4, express $k$ in terms of $s$:** $$k = 400 - 3s$$ 7. **Substitute $k$ into the revenue equation:** $$12(400 - 3s) + 58s = 6800$$ $$4800 - 36s + 58s = 6800$$ $$4800 + 22s = 6800$$ 8. **Solve for $s$:** $$22s = 6800 - 4800 = 2000$$ $$s = \frac{2000}{22} = \frac{1000}{11} \approx 90.91$$ Since the number of tickets must be whole numbers, check for integer solutions near this value. 9. **Try $s = 40$ (from given options):** - $p = 2 \times 40 = 80$ - $k = 400 - 3 \times 40 = 400 - 120 = 280$ Check revenue: $$12 \times 280 + 18 \times 80 + 22 \times 40 = 3360 + 1440 + 880 = 5680$$ (less than 6800) 10. **Try $s = 20$:** - $p = 40$ - $k = 400 - 60 = 340$ Check revenue: $$12 \times 340 + 18 \times 40 + 22 \times 20 = 4080 + 720 + 440 = 5240$$ (less than 6800) 11. **Try $s = 50$:** - $p = 100$ - $k = 400 - 150 = 250$ Check revenue: $$12 \times 250 + 18 \times 100 + 22 \times 50 = 3000 + 1800 + 1100 = 5900$$ (less than 6800) 12. **Try $s = 60$:** - $p = 120$ - $k = 400 - 180 = 220$ Check revenue: $$12 \times 220 + 18 \times 120 + 22 \times 60 = 2640 + 2160 + 1320 = 6120$$ (less than 6800) 13. **Try $s = 70$:** - $p = 140$ - $k = 400 - 210 = 190$ Check revenue: $$12 \times 190 + 18 \times 140 + 22 \times 70 = 2280 + 2520 + 1540 = 6340$$ (less than 6800) 14. **Try $s = 80$:** - $p = 160$ - $k = 400 - 240 = 160$ Check revenue: $$12 \times 160 + 18 \times 160 + 22 \times 80 = 1920 + 2880 + 1760 = 6560$$ (less than 6800) 15. **Try $s = 90$:** - $p = 180$ - $k = 400 - 270 = 130$ Check revenue: $$12 \times 130 + 18 \times 180 + 22 \times 90 = 1560 + 3240 + 1980 = 6780$$ (close to 6800) 16. **Try $s = 91$:** - $p = 182$ - $k = 400 - 273 = 127$ Check revenue: $$12 \times 127 + 18 \times 182 + 22 \times 91 = 1524 + 3276 + 2002 = 6802$$ (very close to 6800) 17. **Conclusion:** The closest integer solution is approximately: $$k = 127, p = 182, s = 91$$ This satisfies the conditions closely. **Final answer:** $$\boxed{(k, p, s) = (127, 182, 91)}$$