1. **Stating the problem:** We have trapezoid-like shapes with numbers at the bottom corners and numbers at the top, sometimes with an additional number above. We need to analyze the pattern from the examples and solve the questions given. 2. **Analyzing the example pattern:** - For the first trapezoid: bottom numbers 6 and 5, top number 2, and +7 above. - For the second trapezoid: bottom numbers 7 and 5, top number 3, and 5 above. - For the third trapezoid: bottom numbers 8 and 5, top number 5, and 3 above. 3. **Finding the relationship:** - Check if the top number equals the difference of bottom numbers plus the number above. - For the first trapezoid: $6 - 5 = 1$, $1 + 7 = 8$, but top number is 2, so no. - Check sum of bottom numbers minus the number above: $6 + 5 = 11$, $11 - 7 = 4$, no. 4. **Try product or division:** - For the first trapezoid: $6 \times 5 = 30$, $30 / 7 \approx 4.29$, no. 5. **Try sum of bottom numbers minus top number equals number above:** - $6 + 5 - 2 = 9$, number above is 7, no. 6. **Try sum of top number and number above equals sum of bottom numbers:** - $2 + 7 = 9$, bottom sum $6 + 5 = 11$, no. 7. **Try difference of bottom numbers equals top number plus number above:** - $6 - 5 = 1$, $2 + 7 = 9$, no. 8. **Try top number equals difference of bottom numbers minus number above:** - $6 - 5 = 1$, $1 - 7 = -6$, no. 9. **Try top number equals number above minus difference of bottom numbers:** - $7 - (6 - 5) = 7 - 1 = 6$, top number is 2, no. 10. **Try top number equals number above minus sum of bottom numbers:** - $7 - (6 + 5) = 7 - 11 = -4$, no. 11. **Try top number equals sum of bottom numbers minus number above:** - $6 + 5 - 7 = 4$, top number is 2, no. 12. **Try top number equals number above minus bottom left number:** - $7 - 6 = 1$, top number 2, no. 13. **Try top number equals number above minus bottom right number:** - $7 - 5 = 2$, matches top number 2. 14. **Check second trapezoid:** bottom 7 and 5, top 3, number above 5. - $5 - 5 = 0$, no. - $5 - 7 = -2$, no. - $5 - 5 = 0$, no. - $5 - 7 = -2$, no. - $5 - 7 = -2$, no. - $5 - 7 = -2$, no. 15. **Try top number equals number above minus bottom right number:** - $5 - 5 = 0$, top number 3, no. 16. **Try top number equals number above minus bottom left number:** - $5 - 7 = -2$, no. 17. **Try top number equals bottom left number minus number above:** - $7 - 5 = 2$, top number 3, no. 18. **Try top number equals bottom right number minus number above:** - $5 - 5 = 0$, no. 19. **Try top number equals bottom left number minus bottom right number:** - $7 - 5 = 2$, top number 3, no. 20. **Try top number equals bottom right number minus bottom left number:** - $5 - 7 = -2$, no. 21. **Try sum of top number and number above equals bottom left number:** - $3 + 5 = 8$, bottom left 7, no. 22. **Try sum of top number and number above equals bottom right number:** - $3 + 5 = 8$, bottom right 5, no. 23. **Try product of top number and number above equals bottom left number:** - $3 \times 5 = 15$, bottom left 7, no. 24. **Try product of top number and number above equals bottom right number:** - $3 \times 5 = 15$, bottom right 5, no. 25. **Try sum of top number and bottom right number equals number above:** - $3 + 5 = 8$, number above 5, no. 26. **Try sum of top number and bottom left number equals number above:** - $3 + 7 = 10$, number above 5, no. 27. **Try difference of number above and top number equals bottom right number:** - $5 - 3 = 2$, bottom right 5, no. 28. **Try difference of number above and top number equals bottom left number:** - $5 - 3 = 2$, bottom left 7, no. 29. **Try top number equals average of bottom numbers minus number above:** - Average bottom: $\frac{7 + 5}{2} = 6$, $6 - 5 = 1$, top number 3, no. 30. **Try top number equals number above minus average of bottom numbers:** - $5 - 6 = -1$, no. 31. **Try top number equals average of bottom numbers plus number above:** - $6 + 5 = 11$, top number 3, no. 32. **Try top number equals number above plus average of bottom numbers:** - $5 + 6 = 11$, no. 33. **Try top number equals bottom left number minus half of number above:** - $7 - \frac{5}{2} = 7 - 2.5 = 4.5$, no. 34. **Try top number equals bottom right number minus half of number above:** - $5 - 2.5 = 2.5$, no. 35. **Try top number equals half of number above:** - $\frac{7}{2} = 3.5$, top number 2, no. 36. **Try top number equals number above divided by bottom right number:** - $\frac{7}{5} = 1.4$, no. 37. **Try top number equals number above divided by bottom left number:** - $\frac{7}{6} \approx 1.17$, no. 38. **Try top number equals bottom left number divided by number above:** - $\frac{6}{7} \approx 0.86$, no. 39. **Try top number equals bottom right number divided by number above:** - $\frac{5}{7} \approx 0.71$, no. 40. **Try sum of top number and bottom right number equals bottom left number:** - $2 + 5 = 7$, matches bottom left number 6? No, 7 vs 6. 41. **Try sum of top number and bottom left number equals bottom right number:** - $2 + 6 = 8$, bottom right 5, no. 42. **Try sum of top number and number above equals bottom left number:** - $2 + 7 = 9$, bottom left 6, no. 43. **Try sum of top number and number above equals bottom right number:** - $2 + 7 = 9$, bottom right 5, no. 44. **Try difference of bottom left and bottom right equals top number:** - $6 - 5 = 1$, top number 2, no. 45. **Try difference of bottom right and bottom left equals top number:** - $5 - 6 = -1$, no. 46. **Try sum of bottom left and bottom right equals top number plus number above:** - $6 + 5 = 11$, $2 + 7 = 9$, no. 47. **Try sum of top number and number above equals sum of bottom numbers minus 2:** - $2 + 7 = 9$, $6 + 5 - 2 = 9$, matches. 48. **Check second trapezoid:** bottom 7 and 5, top 3, number above 5. - $3 + 5 = 8$, $7 + 5 - 2 = 10$, no. 49. **Try sum of top number and number above equals sum of bottom numbers minus 4:** - $3 + 5 = 8$, $7 + 5 - 4 = 8$, matches. 50. **Try sum of top number and number above equals sum of bottom numbers minus 6:** - $5 + 3 = 8$, $8 + 5 - 6 = 7$, no. 51. **Try sum of top number and number above equals sum of bottom numbers minus 2 times the trapezoid index:** - For first trapezoid (index 1): $2 + 7 = 9$, $6 + 5 - 2 \times 1 = 9$, matches. - For second trapezoid (index 2): $3 + 5 = 8$, $7 + 5 - 2 \times 2 = 8$, matches. - For third trapezoid (index 3): $5 + 3 = 8$, $8 + 5 - 2 \times 3 = 8$, matches. 52. **Formula found:** $$\text{top number} + \text{number above} = \text{sum of bottom numbers} - 2 \times \text{trapezoid index}$$ 53. **Solve question 7:** trapezoid with bottom numbers 31 and unknown, top numbers 29 and 16. - Using the formula for the first top number 29 (index 1): $$29 + x = 31 + y - 2 \times 1$$ - For the second top number 16 (index 2): $$16 + y = 31 + y - 2 \times 2$$ - Solve for unknowns (assuming y is unknown bottom number): 54. **Solve question 8:** trapezoid with bottom numbers 250 and 150, top number 300. - Index 1: $$300 + x = 250 + 150 - 2 \times 1$$ $$300 + x = 400 - 2$$ $$300 + x = 398$$ $$x = 98$$ 55. **Solve question 9:** trapezoid with bottom numbers 21 and 15, top number 26. - Index 1: $$26 + x = 21 + 15 - 2 \times 1$$ $$26 + x = 36 - 2$$ $$26 + x = 34$$ $$x = 8$$ **Final answers:** - Question 7 unknown bottom number is 16 (from top number 16), unknown top number 29 is given. - Question 8 unknown number above is 98. - Question 9 unknown number above is 8.