During her last road trip, Sophia drove $402$ miles on $12$ gallons of gas. Sophia's car averages $37$ miles per gallon (mpg) on highways and $25$ mpg in cities. Which of the following best approximates the number of city miles she drove in her car on this trip?

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GCF and LCM: word problems

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Q. During her last road trip, Sophia drove

402

miles on

12

gallons of gas. Sophia's car averages

37

miles per gallon (mpg) on highways and

25

mpg in cities. Which of the following best approximates the number of city miles she drove in her car on this trip?

Calculate average mpg: Calculate the total average miles per gallon (mpg) Sophia's car achieved on the trip. $\newline$ Sophia drove $402$ miles on $12$ gallons of gas. $\newline$ Average mpg for the trip = Total miles driven / Total gallons of gas used $\newline$ Average mpg for the trip = $402$ miles / $12$ gallons $\newline$ Average mpg for the trip = $33.5$ mpg
Find mpg difference: Determine the difference between the average mpg on highways and in cities. $\newline$ Difference in mpg = Highway mpg - City mpg $\newline$ Difference in mpg = $37 \, \text{mpg} - 25 \, \text{mpg}$ $\newline$ Difference in mpg = $12 \, \text{mpg}$
Proportion of highway miles: Calculate the proportion of highway miles to city miles based on the average mpg. $\newline$ Since the average mpg for the trip ( $33.5$ mpg) is closer to the highway mpg ( $37$ mpg) than the city mpg ( $25$ mpg), we can infer that Sophia drove more highway miles than city miles. However, we need to find a way to estimate the number of city miles.
Set up equations: Set up a system of equations to represent the total miles driven. Let $x$ be the number of highway miles and $y$ be the number of city miles. We have two equations: $x + y = 402$ (total miles) $(37x + 25y) / 12 = 33.5$ (average mpg equation)
Solve for city miles: Solve the system of equations for $y$ (the number of city miles). $\newline$ First, we'll multiply the second equation by $12$ to get rid of the fraction: $\newline$ $37x + 25y = 12 \times 33.5$ $\newline$ $37x + 25y = 402$ $\newline$ Now we have two equations with the same total miles: $\newline$ $x + y = 402$ $\newline$ $37x + 25y = 402$
Identify mistake: Since we have two equations with the same total, we can set them equal to each other to find the relationship between $x$ and $y$ . $37x + 25y = x + y$ $36x = -24y$ $x = -\frac{24y}{36}$ $x = -\frac{2}{3}y$ This equation tells us that for every $3$ miles driven in the city, Sophia drove $-2$ miles on the highway, which doesn't make sense. There is a mistake in the algebraic manipulation.