Fraction Pyramids Ebb21C

1. **Problem Statement:** We have three pyramids of boxes with fractions in the bottom and middle rows, and empty boxes in the middle and top rows. The goal is to find the missing fractions in the empty boxes for each pyramid. 2. **Key Concept:** In a pyramid of fractions, each box in the upper row is typically the sum of the two boxes directly below it. 3. **Pyramid a) (top-left):** - Bottom row: $\frac{1}{10}$, $\frac{1}{5}$, $\frac{3}{5}$ - Middle row: two empty boxes - Top box: empty Calculate middle row: - First middle box = $\frac{1}{10} + \frac{1}{5} = \frac{1}{10} + \frac{2}{10} = \frac{3}{10}$ - Second middle box = $\frac{1}{5} + \frac{3}{5} = \frac{2}{10} + \frac{6}{10} = \frac{8}{10} = \frac{4}{5}$ Calculate top box: - Top box = $\frac{3}{10} + \frac{4}{5} = \frac{3}{10} + \frac{8}{10} = \frac{11}{10}$ 4. **Pyramid b) (center):** - Bottom row: $\frac{1}{9}$, $\frac{1}{3}$, empty - Middle row: empty box, $\frac{8}{9}$ - Top box: empty Calculate missing bottom box: - Let bottom right box be $x$ Calculate middle left box: - Middle right box = $\frac{8}{9}$ - Middle left box = $\frac{1}{9} + \frac{1}{3} = \frac{1}{9} + \frac{3}{9} = \frac{4}{9}$ Calculate bottom right box: - Since middle right box = $\frac{1}{3} + x = \frac{8}{9}$ - Solve for $x$: $x = \frac{8}{9} - \frac{1}{3} = \frac{8}{9} - \frac{3}{9} = \frac{5}{9}$ Calculate top box: - Top box = $\frac{4}{9} + \frac{8}{9} = \frac{12}{9} = \frac{4}{3}$ 5. **Pyramid c) (top-right):** - Bottom row: empty, $\frac{1}{6}$, $\frac{1}{3}$ - Middle row: $\frac{5}{6}$, empty - Top box: empty Calculate bottom left box: - Let bottom left box be $y$ Calculate middle right box: - Middle left box = $\frac{5}{6}$ - Middle right box = $\frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$ Calculate bottom left box: - Since middle left box = $y + \frac{1}{6} = \frac{5}{6}$ - Solve for $y$: $y = \frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$ Calculate top box: - Top box = $\frac{5}{6} + \frac{1}{2} = \frac{5}{6} + \frac{3}{6} = \frac{8}{6} = \frac{4}{3}$ **Final answers:** - Pyramid a): Middle row: $\frac{3}{10}$, $\frac{4}{5}$; Top box: $\frac{11}{10}$ - Pyramid b): Bottom right box: $\frac{5}{9}$; Middle left box: $\frac{4}{9}$; Top box: $\frac{4}{3}$ - Pyramid c): Bottom left box: $\frac{2}{3}$; Middle right box: $\frac{1}{2}$; Top box: $\frac{4}{3}$