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Brian mixed 8 cups of milk to make 6 dozen brownies. At this rate, how many cups of milk would he...

Question:

Brian mixed 8 cups of milk to make 6 dozen brownies. At this rate, how many cups of milk would he need to make 15 dozen brownies?

Proportions and Variation:

In any numerical ratio, this denotes a relationship between two numbers {eq}x {/eq} and {eq}y {/eq}. On the other hand, a proportion shows the similarity between two ratios. When two variables are dependent, variations in the magnitude of one variable will have a proportional effect on the other. When there is an increase or decrease of a variable {eq}x {/eq} with respect to another {eq}y {/eq}, for a ratio or constant K, variations are present. In the case that we have a direct variation, it happens that when one variable increases the other increases, which can also be written as: {eq}\frac{{{y_1}}}{{{x_1}}} = \frac{{{y_2}}}{{{x_2}}} {/eq}.

Answer and Explanation:

{eq}\eqalign{ & {\text{In this specific case }}{\text{,we have two proportional values }}\,x\,\left( {dozen{\text{ }}brownies} \right){\text{ }} \cr & {\text{and }}y\,\left( {cups{\text{ }}of{\text{ }}milk} \right){\text{ that have a variation in directly proportional form}}{\text{. }} \cr & {\text{So we have:}} \cr & \,\,\,\,{x_1} = 6\,dozen{\text{ }}brownies \cr & \,\,\,\,{y_1} = 8\,cups{\text{ }}of{\text{ }}milk \cr & \,\,\,\,{x_2} = 15\,dozen{\text{ }}brownies \cr & \,\,\,\,{y_2} = ?\,cups{\text{ }}of{\text{ }}milk \cr & {\text{Since}}{\text{, }}x{\text{ and }}y{\text{ vary directly}}{\text{, then}}{\text{, when }}x{\text{ increases it also }} \cr & {\text{increases }}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 15}}{6} = 20\,cups{\text{ }}of{\text{ }}milk \cr & {\text{Therefore}}{\text{, to make 15 dozen brownies we need }}\boxed{20\,cups{\text{ }}of{\text{ }}milk} \cr} {/eq}


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Ratios and Proportions: Definition and Examples

from Geometry: High School

Chapter 7 / Lesson 1
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