As we have a recipe that is meant for 1/3 gallon of soup and we want to scale it to 4 gallons, we can finde the constant of proportionality dividing 4 by 1/3:
[tex]k=\frac{4}{\frac{1}{3}}=4\cdot\frac{3}{1}=12[/tex]
This means that the amount of ingredients of the 1/3-gallon recipe has to be multiplied by 12 in order to know the amount for a 4 gallon recipe.
Navy beans:
For a 1/3 gallon soup, we needed 1 1/4 cups, so for a 4 gallon soup we need:
[tex](1\frac{1}{4})\cdot12=12+\frac{12}{4}=12+3=15[/tex]
We need 15 cups of navy beans.
Carrots:
For a 1/3 gallon soup, we needed 3/4 cups, so for a 4 gallon soup we need:
[tex]\frac{3}{4}\cdot12=\frac{36}{4}=9[/tex]
We need 9 cups of carrots.
The same procedure (multiplying by 12) has to be applied to all the ingredients in order to make the grocery list for a 4 gallon soup.
