1. **Stating the problem:** We have a 4x4 grid with shapes in each cell and totals for each row and column. We need to find the missing column total marked with a question mark. 2. **Assign variables to each shape:** - Let $G$ = value of green triangle - Let $O$ = value of orange square - Let $Y$ = value of yellow hexagon - Let $R$ = value of red circle 3. **Write equations from row totals:** - Row 1: $G + O + O + G = 28$ which simplifies to $2G + 2O = 28$ - Row 2: $Y + O + Y + O = 30$ which simplifies to $2Y + 2O = 30$ - Row 3: $R + G + R + R = 18$ which simplifies to $3R + G = 18$ - Row 4: $R + O + R + R = 20$ which simplifies to $3R + O = 20$ 4. **Simplify row equations:** - From Row 1: $2G + 2O = 28$ divide both sides by 2: $$\cancel{2}G + \cancel{2}O = \frac{28}{2}$$ $$G + O = 14$$ - From Row 2: $2Y + 2O = 30$ divide both sides by 2: $$\cancel{2}Y + \cancel{2}O = \frac{30}{2}$$ $$Y + O = 15$$ 5. **Express $G$ from Row 1:** $$G = 14 - O$$ 6. **From Row 3:** $$3R + G = 18$$ substitute $G$: $$3R + (14 - O) = 18$$ $$3R = 18 - 14 + O$$ $$3R = 4 + O$$ 7. **From Row 4:** $$3R + O = 20$$ 8. **Use equations from steps 6 and 7:** From step 6: $3R = 4 + O$ From step 7: $3R + O = 20$ Substitute $3R$ from step 6 into step 7: $$(4 + O) + O = 20$$ $$4 + 2O = 20$$ $$2O = 16$$ $$O = 8$$ 9. **Find $G$ and $R$:** - $G = 14 - O = 14 - 8 = 6$ - From step 6: $3R = 4 + O = 4 + 8 = 12$ $$R = \frac{12}{3} = 4$$ 10. **Find $Y$ from step 4:** $$Y + O = 15$$ $$Y = 15 - O = 15 - 8 = 7$$ 11. **Calculate the missing column total (first column):** Column 1 shapes: Row 1: $G = 6$ Row 2: $Y = 7$ Row 3: $R = 4$ Row 4: $R = 4$ Sum: $$6 + 7 + 4 + 4 = 21$$ **Final answer:** The missing total is $21$.