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Jerry has a large car which holds $22$ gallons of fuel and gets $20$ miles per gallon. Kate has a smaller car which holds $16.5$ gallons of fuel and gets $30$ miles per gallon. If both cars have a full tank of fuel now and drive the same distance, in how many miles will the remaining fuel in each tank be the same? $\newline$ Choose $1$ answer: $\newline$ (A) $320$ $\newline$ (B) $325$ $\newline$ (C) $330$ $\newline$ (D) $335$

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Compound events: find the number of outcomes

Full solution

Q. Jerry has a large car which holds

22

gallons of fuel and gets

20

miles per gallon. Kate has a smaller car which holds

16.5

gallons of fuel and gets

30

miles per gallon. If both cars have a full tank of fuel now and drive the same distance, in how many miles will the remaining fuel in each tank be the same?

\newline

Choose

1

answer:

\newline

(A)

320

\newline

(B)

325

\newline

(C)

330

\newline

(D)

335

Calculate Total Distance: First, let's calculate the total distance each car can travel on a full tank. $\newline$ For Jerry's car: Total distance = Tank capacity $*$ Mileage per gallon $\newline$ = $22$ gallons $*$ $20$ miles/gallon $\newline$ = $440$ miles
Set Up Equation: For Kate's car: Total distance = Tank capacity $*$ Mileage per gallon $\newline$ = $16.5$ gallons $*$ $30$ miles/gallon $\newline$ = $495$ miles
Clear Fractions: Now, let's set up an equation to find the distance at which the remaining fuel in both tanks will be the same. Let $x$ be the distance driven by both cars. Jerry's car will have used $\frac{x}{20}$ gallons, and Kate's car will have used $\frac{x}{30}$ gallons. The remaining fuel in each car's tank will be their initial capacity minus the fuel used. $\newline$ For Jerry's car: Remaining fuel = $22 - \frac{x}{20}$ $\newline$ For Kate's car: Remaining fuel = $16.5 - \frac{x}{30}$
Distribute $60$ : We want to find the distance $x$ where the remaining fuel is the same for both cars: $\newline$ $22 - \frac{x}{20} = 16.5 - \frac{x}{30}$ $\newline$ To solve for $x$ , we'll first clear the fractions by finding a common denominator, which is $60$ (the least common multiple of $20$ and $30$ ). $\newline$ Multiplying both sides by $60$ gives us: $\newline$ $60*(22 - \frac{x}{20}) = 60*(16.5 - \frac{x}{30})$
Isolate x: Now, distribute the $60$ on both sides of the equation: $\newline$ $60\times22 - 3\times x = 60\times16.5 - 2\times x$ $\newline$ $1320 - 3\times x = 990 - 2\times x$ $\newline$ To isolate $x$ , we'll move the terms involving $x$ to one side and the constant terms to the other side: $\newline$ $1320 - 990 = 3\times x - 2\times x$ $\newline$ $330 = x$
Final Result: We found that $x = 330$ miles, which means that after driving $330$ miles, the remaining fuel in both Jerry's and Kate's cars will be the same.

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