Five friends

Five friends ate a 12-slice pizza. Samir ate three times as many slices as Joe. Joe ate half as many slices as Emily. Kendra and Fong together ate half a pizza. Kendra ate one-third as many slices as Samir. Fong ate the most slices. What fraction of a pizza did each person eat?

Correct answer:

S =  1/4
J =  1/12
M =  1/6
K =  1/12
F =  5/12

Step-by-step explanation:

$$\begin{gathered} S=3\cdot J \\ J=M/2 \\ K+F=1/2 \\ K=S/3 \\ K+J+S+F+M=1 \\ 3J-S=0 \\ 2J-M=0 \\ 2F+2K=1 \\ 3K-S=0 \\ F+J+K+M+S=1 \\ Pivot:Row1\leftrightarrow Row3 \\ 2F+2K=1 \\ 2J-M=0 \\ 3J-S=0 \\ 3K-S=0 \\ F+J+K+M+S=1 \\ Row5-\frac{1}{2}\cdot Row1\to Row5 \\ 2F+2K=1 \\ 2J-M=0 \\ 3J-S=0 \\ 3K-S=0 \\ J+M+S=0.5 \\ Pivot:Row2\leftrightarrow Row3 \\ 2F+2K=1 \\ 3J-S=0 \\ 2J-M=0 \\ 3K-S=0 \\ J+M+S=0.5 \\ Row3-\frac{2}{3}\cdot Row2\to Row3 \\ 2F+2K=1 \\ 3J-S=0 \\ -M+0.66667S=0 \\ 3K-S=0 \\ J+M+S=0.5 \\ Row5-\frac{1}{3}\cdot Row2\to Row5 \\ 2F+2K=1 \\ 3J-S=0 \\ -M+0.66667S=0 \\ 3K-S=0 \\ M+1.33333S=0.5 \\ Pivot:Row3\leftrightarrow Row4 \\ 2F+2K=1 \\ 3J-S=0 \\ 3K-S=0 \\ -M+0.66667S=0 \\ M+1.33333S=0.5 \\ Row5+Row4\to Row5 \\ 2F+2K=1 \\ 3J-S=0 \\ 3K-S=0 \\ -M+0.66667S=0 \\ 2S=0.5 \\ S=\frac{0.5}{2}=0.25 \\ M=\frac{0-0.66666666666667S}{-1}=\frac{0-0.66666667\cdot 0.25}{-1}=0.16666667 \\ K=\frac{0+S}{3}=\frac{0+0.25}{3}=0.08333333 \\ J=\frac{0+S}{3}=\frac{0+0.25}{3}=0.08333333 \\ F=\frac{1-2K}{2}=\frac{1-2\cdot 0.08333333}{2}=0.41666667 \\ F=\frac{5}{12}\doteq 0.416667 \\ J=\frac{1}{12}\doteq 0.083333 \\ K=\frac{1}{12}\doteq 0.083333 \\ M=\frac{1}{6}\doteq 0.166667 \\ S=\frac{1}{4}=0.25 \end{gathered}$$

$J=0.0833=\frac{1}{12}$

$M=0.1667=\frac{1}{6}$

$K=0.0833=\frac{1}{12}$

$F=0.4167=\frac{5}{12}$

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