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## Official Guide Explanation:

Problem Solving #D24

**Background**

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Review (12th ed.). Click the links on the question number, difficulty level, and categories to find explanations for other problems.

These are the same explanations that are featured in my "Guides to the Official Guide" PDF booklets. However, because of the limitations of HTML and cross-browser compatibility, some mathematical concepts, such as fractions and roots, do not display as clearly online.

Click here for an example of the PDF booklets. Click here to purchase a PDF copy.

**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 t_{1}.

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

d = xt_{1}

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 - t_{1}. Thus, the rate formula for his walking trip is:

d = y(t - t_{1})

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 t_{1}, so we'll have to substitute using that.

The first equation can be altered to become:

t_{1} = ((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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