# A recipe for a home-made disinfectant calls for adding one-half cup of Clorox to one gallon of...

## Question:

A recipe for a home-made disinfectant calls for adding one-half cup of Clorox to one gallon of water. Mrs. Jenkins wants to make this solution in a 32-ounce spray bottle. How many fluid ounces of Clorox should be added to 32 fluid ounces of water to have the same Clorox-to-water ratio as the original recipe?

## Proportions and Variation:

In everyday life, there are many different scenarios of why two ingredients are in proportion. A typical example is recipes for substances that are common in our daily lives, such as disinfectants. In this particular case, we have a recipe for a product 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 product.

{eq}\eqalign{ & {\text{According to the problem the desinfectant recipe consists of the following ingredients:}} \cr & \,\,\,\,disinfectant = \frac{1}{2}{\text{ }}cup{\text{ }}of{\text{ }}clorox + 1\,gallon\,\,of\,\,water \cr & {\text{We have that:}} \cr & \,\,\,\,\,\,1\,cup = 8\,ounces{\text{ and }}\,1\,gallon = \,128\,ounces \cr & {\text{Then:}} \cr & \,\,\,\,disinfectant = 4\,ounces{\text{ }}of{\text{ }}clorox + 128\,ounces\,\,of\,\,water \cr & {\text{In this specific case}}{\text{, we have two proportional values }}\,y\,\left( {ounces{\text{ }}of{\text{ }}clorox} \right){\text{ }} \cr & {\text{and }}x\,\left( {ounces{\text{ }}of{\text{ }}water} \right){\text{ that have a variation in directly proportional form}}{\text{. }} \cr & {\text{So we have:}} \cr & \,\,\,\,{x_1} = 128\,ounces\,\,of\,\,water \cr & \,\,\,\,{y_1} = 4\,ounces{\text{ }}of{\text{ }}clorox \cr & \,\,\,\,{x_2} = 32\,ounces\,\,of\,\,water \cr & \,\,\,\,{y_2} = ?\,\,ounces{\text{ }}of{\text{ }}clorox \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{{4 \times 32}}{{128}} = \boxed{1\,\,ounce{\text{ }}of{\text{ }}clorox} \cr & {\text{Therefore}}{\text{, 1 fluid ounce of Clorox should be added to 32 fluid ounces of water}}{\text{.}} \cr} {/eq} 