Official Guide Explanation:
Problem Solving #149

Background

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Solution and Metadata

Question: 149
Page: 173
Difficulty: 6 (Moderately Difficult)
Category 1: Word Problems > Rate Problems > Speed
Category 2: Arithmetic > Percents > other

Explanation: Any time you're faced with a rate question, start with the rate formula. To find Francine's average speed, we'll eventually need:

((\text{total distance})/(\text{total time}))

The first part of the trip--x percent of the total distance at a speed of 40 miles per hour--can be represented like this:

40 = (((((x)/100))d)/(t₁))

In other words, the rate is 40, the distance is x percent of d (the total distance), and the time is t₁.

The second part of the trip can be represented like this:

60 = (((1 - ((x)/100))d)/(t₂))

Here, 1 - ((x)/100) represents the remaining percent of the distance.

Going back to our initial expression to find average speed, we'll need total distance and total time. Total distance we've already called d. Total time is the sum of t₁ and t₂, which we can find (in terms of d and x) from the two equations.

First:

40t₁ = ((dx)/100)

t₁ = ((dx)/4000)

Next:

60t₂ = d - ((dx)/100)

t₂ = ((d)/60) - ((dx)/6000)

Finally, we can set the average speed:

((\text{total distance})/(\text{total time})) = ((d)/(((dx)/4000) + ((d)/60) - ((dx)/6000)))

There's a d in every term, so if the top and bottom are divided by d, we're left only with x:

(1/(((x)/4000) + (1/60) - ((x)/6000))) = (1/(((3x)/12000) + (200/12000) - ((2x)/12000)))

= (1/(((x + 200)/12000))) = (12000/(x + 200)), choice (E).

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