Question

Cella is making two snack mixes for a party. Her recipes are shown in the table. If each snack mix contains 6 cups of nuts, which mix has the greater total quantity? Explain. Click the icon to view Celia's recipes. For every 6 cups of nuts in the popcorn mix, there are cups of popcorn and cups of chocolate morsels for a total of cups of popcorn mix. For every 6 cups of nuts in the pretzel mix, there are cups of pretzels and cups of chocolate morsels for a total of cups of pretzel mix. So, the has the greater total quantity.

Answer 1

First, add all the quantities on each recipe to find the total amount using the proportions that are shown in the picture.

Popcorn mix

`\begin{gathered} 4+1(1)/(2)+(3)/(4)=4+1+(1)/(2)+(3)/(4) \\ =5+(1)/(2)+(3)/(4) \\ =5+(2)/(4)+(3)/(4) \\ =5+(5)/(4) \\ =5+(4)/(4)+(1)/(4) \\ =5+1+(1)/(4) \\ =6+(1)/(4) \\ =6(1)/(4) \end{gathered}`

Pretzel mix

`\begin{gathered} 3+1+(1)/(3)=4+(1)/(3) \\ =4(1)/(3) \end{gathered}`

Now, find the rate of cups of nuts to total of each recipe, by dividing the amount of nuts over the total.

Popcorn mix:

`(1(1)/(2))/(6(1)/(4))=((3)/(2))/((25)/(4))=(3*4)/(2*25)=(12)/(50)=(6)/(25)`

The rate of cups of nuts to total of the popcorn mix is 6:25.

Pretzel mix:

`(1)/(4(1)/(3))=(1)/((13)/(3))=(1*3)/(13)=(3)/(13)`

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:

`\frac{6\text{ cups of nuts }}{(\frac{6\text{ cups of nuts}}{25\text{ cups of popcorn mix}})}=25\text{ cups of popcorn mix}`

Pretzel mix:

`\frac{6\text{ cups of nuts}}{(\frac{3\text{ cups of nuts}}{13\text{ cups of pretzel mix}})}=26\text{ cups of pretzel mix}`

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:

`\begin{gathered} 4\text{ cups of popcorn }*4=16\text{ cups of popcorn} \\ 1(1)/(2)\text{cups of nuts }*4=6\text{ cups of nuts} \\ (3)/(4)\text{ cups of chocolate morsels}*4=3\text{ cups of chocolate morsels} \end{gathered}`

Pretzel mix:

`\begin{gathered} 3\text{ cups of pretzels}*6=18\text{ cups of pretzels} \\ 1\text{ cup of nuts}*6=6\text{ cups of nuts} \\ (1)/(3)\text{cup of chocolate morsels}*6=2\text{ cups of chocolate morsels} \end{gathered}`

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.