Amount of Miles

Question

Gary has two cars: His fairly new van and his old clunker station wagon. the wagon currently has 16 times as many miles on it as the van had when the wagon had 3 times as many miles as the van has now. Between now and then, each vehicle has not yet been driven 100,000 miles. All mileage amounts are in thousands. How many miles has the van been driven now?

I am confused on how to set this problem up. I know if x= miles of van , than station wagon would equal 16x=3x? Not sure how to set the opposite side up? Or exactly find the miles on the van if I am not given a set number.

Answer 1

Let $w$ = mileage of wagon now, $u$ = mileage of wagon before, $x$ = mileage of van before, and $y$ = mileage of van now. Then:

$w = 16x$

$u = 3y$

$w - u = 100$

$y - x = 100$

$\longrightarrow 16x - 3y = 100$

$\longrightarrow 16(y - 100) - 3y = 100$

$\longrightarrow 13y - 1600 = 100$

$\longrightarrow 13y = 1700$

$\longrightarrow y = \frac{1700}{13} = 130.76$

So the van now has $130,760$ miles on it.

Answer 2

Hint: Define the variables

1. $x(t)=$ the miles the new van has at time $t$,
2. $y(t)=$ the miles the station wagon has at time $t$,

and consider two times: $t_0$ "then" (the past one) and $t_1$ "now" (the current one). Then, you have that: $$y(t_1)=16x(t_0)$$ and $$y(t_0)=3x(t_1)$$ and you want to determine $x(t_1).$ But, according to you have written there is unsufficient information to proceed (at least to my understanding).

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We are also given that $$y(t_1)-y(t_0)<100.000$$ and similarly $$x(t_1)-x(t_0)<100.000$$ but as long as these are not given as equalities we cannot proceed.