# Martha is following a recipe for 13 liters of punch that uses 8 liters of pineapple juice and the...

## Question:

Martha is following a recipe for 13 liters of punch that uses 8 liters of pineapple juice and the rest lemonade. how much pineapple juice is used for every liter of lemonade?

## Proportions and Variation:

In real life, there are several examples of proportional relationships between two quantities. We can see that in recipes they are excellent examples of this type of problem. In this specific case, we have a recipe for a beverage in which two ingredients are mixed. So if we want to prepare a portion of the recipe, we must mix the ingredients in the same proportion that they have in the recipe of the whole beverage.

{eq}\eqalign{ & {\text{According to the problem the punch recipe consists of the following ingredients:}} \cr & \,\,\,\,13{\text{ }}liters{\text{ }}of{\text{ }}punch = 8{\text{ }}liters{\text{ }}of{\text{ }}pineapple{\text{ }}juice + 5\,liters{\text{ }}of{\text{ }}lemonade \cr & {\text{In this specific case}}{\text{, we have two proportional values }}\,y\,\left( {liters{\text{ }}of{\text{ }}pineapple} \right){\text{ }} \cr & {\text{and }}x\,\left( {liters{\text{ }}of{\text{ }}lemonade} \right){\text{ that have a variation in directly proportional form}}{\text{. }} \cr & {\text{So we have:}} \cr & \,\,\,\,{x_1} = 5\,liters{\text{ }}of{\text{ }}lemonade \cr & \,\,\,\,{y_1} = 8\,liters{\text{ }}of{\text{ }}pineapple \cr & \,\,\,\,{x_2} = 1\,liters{\text{ }}of{\text{ }}lemonade \cr & \,\,\,\,{y_2} = ?\,\,liters{\text{ }}of{\text{ }}pineapple \cr & {\text{Since}}{\text{, }}x{\text{ and }}y{\text{ vary directly}}{\text{, then}}{\text{, when }}x{\text{ decreases it also }} \cr & {\text{decreases }}y{\text{. For this reason}}{\text{, it must be satisfied that:}} \cr & \,\,\,\,\frac{{{y_2}}}{{{x_2}}} = \frac{{{y_1}}}{{{x_1}}} \cr & {\text{So if we do cross - multiplying:}} \cr & \,\,\,\,{y_2} \cdot {x_1} = {y_1} \cdot {x_2} \cr & {\text{Now}}{\text{, solving for }}\,{y_2}{\text{:}} \cr & \,\,\,\,{y_2} = \frac{{{y_1} \cdot {x_2}}}{{{x_1}}} \cr & {\text{So}}{\text{, substituting the given values:}} \cr & \,\,\,\,{y_2} = \frac{{8 \times 1}}{5} = 1.6\,\,liters{\text{ }}of{\text{ }}pineapple \cr & {\text{Therefore}}{\text{, Marta will need }}\boxed{1.6{\text{ }}liters{\text{ }}of{\text{ }}pineapple{\text{ }}juice}{\text{ for every liter of lemonade}} \cr} {/eq}