Stumped by overthinking what is probably simple: Lewis drove on the M6 @ 70mph

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Simonsky

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The question:

Lewis drove on the M6 (M=motorway in UK!) , averaging 70mph. He then left the M6 and drove on the M1 averaging 40mph. He drove a total of 163 miles in 2.5 hours. How far did he drive along the M6?

For some reason I seem to have got a 'brain freeze' with this question-rather embarrassing because I imagine it is simple. I can't seem to extract an equation for it. Clearly, to do the journey of 163 miles in 2.5 hours would require a specific ratio of driving at 70 mph to 40 mph.

The only thing I can think of is:

163/70 T + 163/40 T = 2.5 but I can't see how that will give me the result.

Nope-I'm just not able to see some simple relationship here - the cognitive wall has descended!
 
If you are trying to come up with an equation determine which value is the variable. I would start with time/speed/distance formula for the 2 segments of the trip, then see if you can combine them into a single equation.
 
Simonsky said:
The question:

Lewis drove on the M6 (M=motorway in UK!) , averaging 70mph. He then left the M6 and drove on the M1 averaging 40mph. He drove a total of 163 miles in 2.5 hours. How far did he drive along the M6?

For some reason I seem to have got a 'brain freeze' with this question-rather embarrassing because I imagine it is simple. I can't seem to extract an equation for it. Clearly, to do the journey of 163 miles in 2.5 hours would require a specific ratio of driving at 70 mph to 40 mph.

The only thing I can think of is:

163/70 T + 163/40 T = 2.5 but I can't see how that will give me the result.

Nope-I'm just not able to see some simple relationship here - the cognitive wall has descended!
Click to expand...

What does your variable T represent? Not thinking clearly about that is a common source of trouble.

Your equation should say, "Distance traveled at 70 mph + distance traveled at 40 mph = 163 mi". Your variable should probably represent the time (in hours) driving at 70. (Or you could use two variables, if you are comfortable with that.)

See what you can do with that.
 
Dr.Peterson said:
What does your variable T represent? Not thinking clearly about that is a common source of trouble.

Your equation should say, "Distance traveled at 70 mph + distance traveled at 40 mph = 163 mi". Your variable should probably represent the time (in hours) driving at 70. (Or you could use two variables, if you are comfortable with that.)

Indeed. I think another part of my problem is confusing the different units: miles and hours and being unclear how to represent these units in the equation.

After reviewing your suggestion and the suggestions of the others above, I'll try to think it through:

1) We've got two time periods t1 and t2
2) We've got two potential equations: one that adds up to miles and one that adds up to hours.
3) use simultaneous equations?

I'll try that:

Distance = speed x time so we can say:

70t1 + 40t2 = 163 and for hours: t1 + t2 =2.5 multiply the latter by 40 to eliminate one variable:

70t1 + 40t2 = 163
-
40t1 + 40t2 = 100

30t1 = 63 => t1 = 2.1 Therefore 70(2.1) = 147 which is the answer.

Wow! I made heavy weather of that and it disturbs me that yet again I've been handicapped by sloppy thinking that couldn't sort out variables with the units of measurement given. Anyway, got there with the help of prompting -thanks. At least I can go to bed tonight without the question going around in circles in my head.
Click to expand...
 
Didya draw a li'l diagram...like:

(u = hours @ 70mph, v = hours @ 40 mph)

@70.................70u...............>u : @40............40v...........>v

70u + 40v = 163
v = 2.5 - u
 
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As an alternative way, I always find these sorts of questions easier if you set up a table:

M6M1Total
Speed(mph)7040
Distance(miles)x163-x163
Time(hours) = distance/speed\(\displaystyle \frac{x}{70}\)
\(\displaystyle \frac{163-x}{40}\)​
2.5

where x is what is required in the question. (It makes sense to add distances and to add times, but obviously not speeds.)

Then you have the equation

\(\displaystyle \frac{x}{70}+\frac{163-x}{40}=2.5\)​

which yields x=147.
 
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