# Shandy's car can travel 128 miles on 4 gallons of gas. How far can her car travel on 15 gallons...

## Question:

Shandy's car can travel 128 miles on 4 gallons of gas. How far can her car travel on 15 gallons of gas?

## Proportions and Variation:

A numerical ratio tells us the relationship between two values. That is, a numerical value can be expressed in terms of another value. For example, when a value {eq}x {/eq} is increased by another value {eq}y {/eq} is also increased, in this case, we say that there is a direct proportion and it is represented by a ratio with a constant value {eq}K: {/eq} {eq}K = \frac{y}{x} {/eq}.

{eq}\eqalign{ & {\text{In this specific case }}{\text{,we have two proportional values }}\,y\,\left( {miles} \right){\text{ and }} \cr & x\,\left( {gallons{\text{ }}of{\text{ }}gas} \right){\text{ that have a variation in directly proportional form}}{\text{. }} \cr & {\text{So we have:}} \cr & \,\,\,\,{y_1} = 128\,miles \cr & \,\,\,\,{x_1} = 4\,gallons \cr & \,\,\,\,{x_2} = 15\,gallons \cr & \,\,\,\,{y_2} = ?\,\,miles \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{{128 \times 15}}{4} = 480\,miles \cr & {\text{Thus}}{\text{, the car will travel }}\boxed{{\text{480}}\,miles}. \cr} {/eq} 