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Four red candies and three green candies can be combined to make many different “flavors.” Flavors are different if the percent red is different, so “3 red / 0 green” is the same flavor as “2 red / 0 green,” and likewise “4 red / 2 green” is the same flavor as “2 red / 1 green.” If a flavor is to be made using some or all of the seven candies, how many different flavors are possible?

May 27, 2020

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Four red candies and three green candies can be combined to make many different “flavors.”
Flavors are different if the percent red is different,
so “3 red / 0 green” is the same flavor as “2 red / 0 green,”
and likewise “4 red / 2 green” is the same flavor as “2 red / 1 green.”
If a flavor is to be made using some or all of the seven candies,
how many different flavors are possible?

$$\begin{array}{|c|c|c|c|c|} \hline & {\color{green}0} & {\color{green}1} &{\color{green}2} &{\color{green}3} \\ \hline {\color{red}0} & - & 0+1=1 & 0+2=2 & 0+3=3 \\ & & \frac{{\color{red}0}}{1}: \frac{{\color{green}1}}{1} &\frac{{\color{red}0}}{2}: \frac{{\color{green}2}}{2}&\frac{{\color{red}0}}{3}: \frac{{\color{green}3}}{3}\\ & & {\color{red}0}: {\color{green}1} & {\color{red}0}: {\color{green}1}&{\color{red}0}: {\color{green}1}\\ \hline {\color{red}1} & 1+0=1 & 1+1=2 & 1+2=3 & 1+3=4 \\ & \frac{{\color{red}1}}{1}: \frac{{\color{green}0}}{1} & \frac{{\color{red}1}}{2}: \frac{{\color{green}1}}{2} &\frac{{\color{red}1}}{3}: \frac{{\color{green}2}}{3}&\frac{{\color{red}1}}{4}: \frac{{\color{green}3}}{4}\\ & {\color{red}1}: {\color{green}0} & \\ \hline {\color{red}2} & 2+0=2 & 2+1=3 & 2+2=4 & 2+3=5 \\ & \frac{{\color{red}2}}{2}: \frac{{\color{green}0}}{2} & \frac{{\color{red}2}}{3}: \frac{{\color{green}1}}{3} &\frac{{\color{red}2}}{4}: \frac{{\color{green}2}}{4}&\frac{{\color{red}2}}{5}: \frac{{\color{green}3}}{5}\\ & {\color{red}1}: {\color{green}0} & &\frac{{\color{red}1}}{2}: \frac{{\color{green}1}}{2} \\ \hline {\color{red}3} & 3+0=3 & 3+1=4 & 3+2=5 & 3+3=6 \\ & \frac{{\color{red}3}}{3}: \frac{{\color{green}0}}{3} & \frac{{\color{red}3}}{4}: \frac{{\color{green}1}}{4} &\frac{{\color{red}3}}{5}: \frac{{\color{green}2}}{5}&\frac{{\color{red}3}}{6}: \frac{{\color{green}3}}{6}\\ & {\color{red}1}: {\color{green}0} & & &\frac{{\color{red}1}}{2}: \frac{{\color{green}1}}{2}\\ \hline {\color{red}4} & 4+0=4 & 4+1=5 & 4+2=6 & 4+3=7 \\ & \frac{{\color{red}4}}{4}: \frac{{\color{green}0}}{4} & \frac{{\color{red}4}}{5}: \frac{{\color{green}1}}{5} &\frac{{\color{red}4}}{6}: \frac{{\color{green}2}}{6}&\frac{{\color{red}4}}{7}: \frac{{\color{green}3}}{7}\\ & {\color{red}1}: {\color{green}0} & &\frac{{\color{red}2}}{3}: \frac{{\color{green}1}}{3} \\ \hline \end{array}$$

The percent red

$$\begin{array}{|c|c|c|c|c|} \hline & {\color{green}0} & {\color{green}1} &{\color{green}2} &{\color{green}3} \\ \hline {\color{red}0} & & {\color{red}0} & {\color{red}0}&{\color{red}0}\\ \hline {\color{red}1} & {\color{red}1} & \frac{{\color{red}1}}{2} &\frac{{\color{red}1}}{3}&\frac{{\color{red}1}}{4} \\ \hline {\color{red}2} & {\color{red}1} & \frac{{\color{red}2}}{3} &\frac{{\color{red}1}}{2}&\frac{{\color{red}2}}{5} \\ \hline {\color{red}3} & {\color{red}1} & \frac{{\color{red}3}}{4} &\frac{{\color{red}3}}{5} &\frac{{\color{red}1}}{2} \\ \hline {\color{red}4} & {\color{red}1} & \frac{{\color{red}4}}{5}&\frac{{\color{red}2}}{3}&\frac{{\color{red}4}}{7} \\ \hline \end{array}$$

The different flavors are $$\{ {\color{red}0},\ \frac{{\color{red}1}}{4},\ \frac{{\color{red}1}}{3},\ \frac{{\color{red}2}}{5},\ \frac{{\color{red}1}}{2},\ \frac{{\color{red}3}}{5},\ \frac{{\color{red}4}}{7},\ \frac{{\color{red}2}}{3},\ \frac{{\color{red}3}}{4},\ \frac{{\color{red}4}}{5},\ {\color{red}1} \}$$

May 27, 2020