Official Guide Explanation:
Problem Solving #D24




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Solution and Metadata

Question: D24
Page: 23
Difficulty: 7 (Very Difficult)
Category 1: Word Problems > Rate Problems > Speed
Category 2: Algebra > Equations >
Category 3: Algebra > Linear Equations-Two Unk >

Explanation: This is a tricky question. It can spiral out of control whether you attack it algebraically or by plugging numbers in for the variables. I prefer to do it algebraically.

The question assigns variables for the two speeds at which Aaron travels, as well as the total time he spends. Let's call the one - way distance d and the time he spends jogging t1.

Thus, the rate formula for Aaron's jogging trip is:

d = xt1

Since he travels the same distance in each direction, he walks the same distance. The total time is t, so his time spent walking is t - t1. Thus, the rate formula for his walking trip is:

d = y(t - t1)

It may be tempting to immediately set the two formulas equal to each other, since both have a d isolated on one side. However, that would be a mistake, because d is what we're looking for. If we combine the equations in that way, we'll never find d. The other common variable is t1, so we'll have to substitute using that.

The first equation can be altered to become:

t1 = ((d)/(x))

We can substitute that into the second equation:

d = y(t - ((d)/(x)))

Now, it's just a matter of dealing with the algebra to isolate d:

d = yt - ((dy)/(x))

dx = xyt - dy (multiply everything by x)

dx + dy = xyt (get all the d's on one side)

d(x + y) = xyt (factor out the d's)

d = ((xyt)/(x + y)), choice (C).

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