1. **State the problem:** We need to find the values of $A$, $B$, and $C$ in the diagram so that the sum of the three numbers on each line equals 9. 2. **Identify the lines and write equations:** - Top vertical line: $4 + A + C = 9$ - Horizontal line: $B + C + 3\frac{1}{2} = 9$ - Bottom vertical line: $1 + C + 1\frac{1}{2} = 9$ 3. **Convert mixed numbers to improper fractions or decimals:** - $3\frac{1}{2} = 3.5$ - $1\frac{1}{2} = 1.5$ 4. **Write equations with decimals:** - $4 + A + C = 9$ - $B + C + 3.5 = 9$ - $1 + C + 1.5 = 9$ 5. **Simplify the third equation to find $C$:** $$1 + C + 1.5 = 9$$ $$C + 2.5 = 9$$ $$C = 9 - 2.5 = 6.5$$ 6. **Substitute $C=6.5$ into the first equation to find $A$:** $$4 + A + 6.5 = 9$$ $$A + 10.5 = 9$$ $$A = 9 - 10.5 = -1.5$$ 7. **Substitute $C=6.5$ into the second equation to find $B$:** $$B + 6.5 + 3.5 = 9$$ $$B + 10 = 9$$ $$B = 9 - 10 = -1$$ 8. **Final answers:** - $A = -1.5$ - $B = -1$ - $C = 6.5$