1. **Problem a)** Complete the fraction addition pyramid: Bottom row: $\frac{1}{5}, \frac{1}{4}, \frac{3}{10}$ Middle row fractions are sums of adjacent bottom fractions: $$\text{Middle left} = \frac{1}{5} + \frac{1}{4} = \frac{4}{20} + \frac{5}{20} = \frac{9}{20}$$ $$\text{Middle right} = \frac{1}{4} + \frac{3}{10} = \frac{5}{20} + \frac{6}{20} = \frac{11}{20}$$ Top fraction is sum of middle fractions: $$\text{Top} = \frac{9}{20} + \frac{11}{20} = \frac{20}{20} = 1$$ 2. **Problem b)** Given bottom row: $\frac{1}{?}, \frac{1}{4}, \frac{1}{6}$ and middle row: $\frac{?}{12}, \frac{?}{12}$, top is 1. Middle left = $\frac{1}{?} + \frac{1}{4} = \frac{a}{12}$ Middle right = $\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$ Since middle right is $\frac{5}{12}$, middle left + middle right = top = 1: $$\frac{a}{12} + \frac{5}{12} = 1 \Rightarrow \frac{a}{12} = \frac{7}{12} \Rightarrow a=7$$ So middle left = $\frac{7}{12}$ Now solve for $\frac{1}{?} + \frac{1}{4} = \frac{7}{12}$: $$\frac{1}{?} = \frac{7}{12} - \frac{3}{12} = \frac{4}{12} = \frac{1}{3}$$ Therefore, bottom left fraction is $\frac{1}{3}$. 3. **Problem c)** Bottom row: $\frac{5}{9}, \frac{1}{?}, \frac{1}{9}$, middle row: $\frac{?}{18}, \frac{?}{18}$, top = 1. Middle left = $\frac{5}{9} + \frac{1}{?} = \frac{a}{18}$ Middle right = $\frac{1}{?} + \frac{1}{9} = \frac{b}{18}$ Sum of middle fractions = top = 1: $$\frac{a}{18} + \frac{b}{18} = 1 \Rightarrow a + b = 18$$ Express $\frac{5}{9} = \frac{10}{18}$ and $\frac{1}{9} = \frac{2}{18}$. Let $\frac{1}{?} = x$. Then: $$a = 10 + 18x$$ $$b = 18x + 2$$ Sum: $$a + b = 10 + 18x + 18x + 2 = 12 + 36x = 18$$ Solve for $x$: $$36x = 6 \Rightarrow x = \frac{1}{6}$$ So $\frac{1}{?} = \frac{1}{6}$. Middle left: $$a = 10 + 18 \times \frac{1}{6} = 10 + 3 = 13$$ Middle right: $$b = 18 \times \frac{1}{6} + 2 = 3 + 2 = 5$$ Middle fractions are $\frac{13}{18}$ and $\frac{5}{18}$. **Final answers:** - a) Top: $1$, Middle: $\frac{9}{20}, \frac{11}{20}$ - b) Bottom left: $\frac{1}{3}$, Middle: $\frac{7}{12}, \frac{5}{12}$ - c) Bottom middle: $\frac{1}{6}$, Middle: $\frac{13}{18}, \frac{5}{18}$