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### Find the Average Speed

```
Date: 11/27/97 at 08:59:37
From: ALI HAIDER

A motorist has to travel 160km. His average speed is 8km/h slower than
he anticipated and he takes 1 hour longer than intended. Find his
actual average speed.

I am unable to create an equation from this problem.
```

```
Date: 11/27/97 at 13:35:06
From: Doctor Luis

This problem isn't so bad if you keep track of what is happening
physically, and ask yourself what it means.

Speed is nothing more than the distance covered by something over a
period of time. Mathematically, you express this relation as follows,

distance covered
avg speed = ------------------------------------
time it took to cover that distance

Symbolically, you can write that as

v = d / t

Solving for the distance,

d = vt

So you see, the distance covered is the product of the speed times the
time. That makes sense because the faster you travel the more distance
you cover in the same amount of time. However, in this problem, the
distance to be covered is the same. That means that if you travel
faster (v is greater) you will get to where you want earlier (t is
less). Similarly, if you travel slower (v is less), it will take more
time to get where you want (t is greater). For example, to travel
160 km, you could travel at 80km/hr for 2 hours, or at 5 km/hr for
32 hours (1.33 days), but the product of the speed times the time
remains the same (because it's the same distance).

Now, the problem tells you that he expected to travel 160km at a speed
v (we don't know what it is yet, so we just represent it by a letter)
in a time of t hours. Since he traveled 8 km/hr slower than expected
(v-8 km/hr), the problem tells us that it took him 1 hour longer
(t+1 hours) to travel those same 160 km.

This means that, for his trip

anticipated trip                     actual trip

d = vt                            d = (v-8)(t+1)
160 = vt                          160 = vt - 8t + v - 8

Try to explain what the equations mean based on what I explained to
you. (Remember, v was his anticipated speed, and t was his anticipated
time)

The two equations above imply,

- 8t + v - 8 = 0

- 8(160/v) + v - 8 = 0      (because t = 160/v)

v^2 - 8v - 1280 = 0      (multiplying both sides by v)

(v - 40)(v + 32) = 0      (factoring)

Now, either

v - 40 = 0, or v + 32 = 0

which means that either

v = 40 or v = -32

Since the negative solution doesn't make sense (why?), we conclude
that he was expecting to travel at an average speed of 40 km/hr, so
his actual speed is 8km/hr less, or 32 km/hr.

I don't know how much math you've had, but if you have any questions
whatsoever, or there is anything you didn't understand, ask again and
we'll explain in more detail whatever it was you didn't get...

By the way, can you find out how long it actually took him to travel
those 160 km?

-Doctor Luis,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Basic Algebra

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