First, add all the quantities on each recipe to find the total amount using the proportions that are shown in the picture.
Popcorn mix
[tex]\begin{gathered} 4+1\frac{1}{2}+\frac{3}{4}=4+1+\frac{1}{2}+\frac{3}{4} \\ =5+\frac{1}{2}+\frac{3}{4} \\ =5+\frac{2}{4}+\frac{3}{4} \\ =5+\frac{5}{4} \\ =5+\frac{4}{4}+\frac{1}{4} \\ =5+1+\frac{1}{4} \\ =6+\frac{1}{4} \\ =6\frac{1}{4} \end{gathered}[/tex]Pretzel mix
[tex]\begin{gathered} 3+1+\frac{1}{3}=4+\frac{1}{3} \\ =4\frac{1}{3} \end{gathered}[/tex]Now, find the rate of cups of nuts to total of each recipe, by dividing the amount of nuts over the total.
Popcorn mix:
[tex]\frac{1\frac{1}{2}}{6\frac{1}{4}}=\frac{\frac{3}{2}}{\frac{25}{4}}=\frac{3\times4}{2\times25}=\frac{12}{50}=\frac{6}{25}[/tex]The rate of cups of nuts to total of the popcorn mix is 6:25.
Pretzel mix:
[tex]\frac{1}{4\frac{1}{3}}=\frac{1}{\frac{13}{3}}=\frac{1\times3}{13}=\frac{3}{13}[/tex]The rate of cups of nuts to total of the pretzel mix is 3:13.
If 6 cups of nuts are used in each recipe, divide 6 over the corresponding rates to find the total quantity of each mix:
Popcorn mix:
[tex]\frac{6\text{ cups of nuts }}{(\frac{6\text{ cups of nuts}}{25\text{ cups of popcorn mix}})}=25\text{ cups of popcorn mix}[/tex]Pretzel mix:
[tex]\frac{6\text{ cups of nuts}}{(\frac{3\text{ cups of nuts}}{13\text{ cups of pretzel mix}})}=26\text{ cups of pretzel mix}[/tex]Therefore, the pretzel mix has the greater quantity.
To find the specific quantities of each ingredient, notice that if we multiply every quantity on the popcorn mix by 4 and every quantity on the pretzel mix by 6, we will get a recipe which includes 6 cups of nuts and the right quantities of each ingredient.
Popcorn mix:
[tex]\begin{gathered} 4\text{ cups of popcorn }\times4=16\text{ cups of popcorn} \\ 1\frac{1}{2}\text{cups of nuts }\times4=6\text{ cups of nuts} \\ \frac{3}{4}\text{ cups of chocolate morsels}\times4=3\text{ cups of chocolate morsels} \end{gathered}[/tex]Pretzel mix:
[tex]\begin{gathered} 3\text{ cups of pretzels}\times6=18\text{ cups of pretzels} \\ 1\text{ cup of nuts}\times6=6\text{ cups of nuts} \\ \frac{1}{3}\text{cup of chocolate morsels}\times6=2\text{ cups of chocolate morsels} \end{gathered}[/tex]Therefore, for every 6 cups of nuts in the popcorn mix, there are 16 cups of popcorn and 3 cups of chocolate morsels, for a total of 25 cups of popcorn mix. For every 6 cups of nuts in the pretzel mix, there are 18 cups of pretzels and 2 cups of chocolate morsels, for a total of 26 cups of pretzel mix. So, the pretzel mix has the greater total quantity.
