Expert: Josh - 3/15/2006

Question
-------------------------
Followup To
Question -
An indian railway train has troble asending certain hills and has to ensure that the incline (gradient) of the hills it attempts to climb are not greater than 0.075 or 3/40.

the train driver comes to junction and has to determine which way he should go, on one hand he may turn left and face an incline of 3/35, alternativly he may proceed straght ahead and face an incline of 3/91 however he will incurr an extra 10 kilometers on his jouney by turning right.

A) which ways can the train driver travel?
B) which would be the quikest route?
C) which would be the easiest slope and why?

Thanks
Answer -
Jesse,

Let us take a visual approach to understand this problem. An incline is basically the vertical distance of a hill divided by the horizontal distance that it stretches. We often refer to this as "rise over run".

A
|
|
|
|
B-------------------------C

Let |AB| be the distance of line AB.
Let |BC| be the distance of line BC.
The incline is the ratio between |AB| and |BC|.

So, the various fractions you've been given in the question simply refer to how steep the terrain is.
The incline is just |AB|/|BC|.

The maximum incline the train can handle is 3/40.

It sounds like the train driver has 3 options.

Turn Left (not possible):  Slope is 3/35. Too steep to climb because 3/35 > 3/40.

Straight ahead: OK, the slope is very gentle since 3/91 is much smaller than the limit of 3/40 which the train can handle.

I haven't been told whether the train needs to climb if it turns right.

A) So, the train driver can keep going straight or turn right.
B) We don't have all the information needed to answer this. It depends on how costly an extra 10 km distance is relative to the train journey. If we go straight ahead, climbing a hill (even at a slope of 3/91) requires us to travel a longer distance. In the figure I have sketched, you can see that the length of the slope |AC| is always greater than the distance which appears on the map, |BC|. The longer the journey goes, the more extra distance we need to cover along AC, as opposed to the apparent distance, BC. This may (or may not) outweigh the 10km penalty if we were to take the option of turning right, travelling along flat terrain. Does this make sense?

Note: I don't think you would be considering another physical aspect (which is fair enough), but climbing an incline also slows down the train. So, this question is not as straight forward as it seems, since the data is incomplete. For instance, the remaining distance of the journey is unknown.

C) Once again, I'm not told about the slope if I had to turn right. So, I presume the land on the right is flat. Going straight ahead, the slope is easier.
In your answer, give reasons like
"slope(left) > slope(straight) because 3/35 > 3/90"

If you need further clarification, let me know.

Cheers.

Which would be bigger as a hill 3/91 or 5/72

Sorry i got the question wrong but the answer you gave helped me to answer
a and b

Answer
It's nice to know that some parts of the question makes more sense to you now.

Again, to make things really obvious, why don't we convert 3/91 and 5/72 into a decimal figure and compare them directly.

You can obviously do this on the calculator, but it's better to rely on mental arithmetic and perform some simple calculations. Actually, it doesn't have to be precise, a rough approximation will do.

3/91 is close enough to 3/90. Now, flipping the fraction up-side-down, we get 90/3=30 ...[*]
So, a slope of 3/91 is close enough to 1/30....From [*]
[mental note: 1/10=0.1, 1/30=(1/3)*(1/10)~=0.033]
In decimal, this is about 0.033. Great!

Now, let's look at 5/72. We can tell that 5/72 > 5/75.
Why? Because when we divide 5 by a larger number, it gets smaller.
Like before, 75/5=15. So, 5/75=1/15. Bingo!

3/91 is approximately (slightly smaller than) 1/30.
5/72 is approximately (slightly bigger than) 1/15. ...[#]

There's no doubt that a slope of 5/72 is a lot steeper.

The good thing is, we haven't used the calculator at all.
You should try this :)

Alternative view:
From [#], 1/15 is between 1/10 and 1/20.
i.e., 1/10 > 1/15 > 1/20
replacing with 1/10=0.1, 1/20=0.05, we see that the fraction "1/15" is bounded by 0.05 and 0.1 in decimal form.
This is definitely way bigger than 0.033.