Official Guide Explanation:
Problem Solving #D24

Background

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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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