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?
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 \(\{ {\color{red}0},\ \frac{{\color{red}1}}{4},\ \frac{{\color{red}1}}{3},\ \frac{{\color{red}2}}{5},\ \frac{{\color{red}1}}{2},\ \frac{{\color{red}3}}{5},\ \frac{{\color{red}4}}{7},\ \frac{{\color{red}2}}{3},\ \frac{{\color{red}3}}{4},\ \frac{{\color{red}4}}{5},\ {\color{red}1} \} \)