So when the hell did they meet anyway?

Okay, I have to admit it. I was supposed to post something this morning but unfortunately, I got hooked up into watching this Naruto Season 3 DVD (about 27 episodes, I think). So there I was, sitting in front of the TV for 7 hours, tirelessly enjoying every single episodes of Naruto (God I love the benefits of being jobless). And you know what this pattern leads to right? Yep, you guess that right.. anime fever. So as a consequence, my brain has now convinced me to download every single episode from seasons 1,2,4 and 5 and tell my subconcious that a 4GB download over a dial-up connection wouldn't take that long (yeah right! oh the things that I would do for cartoons). And yeah, I think that another consequence of me watching cartoons for 7 hours is not having any goodnight hugs and kisses from my girlfriend at all. I think she got irritated coz it took me long to reply to one of her messages (damn those cartoons!!). But I couldn't blame her for getting mad though coz it's my fault anyway. If you can hear me out there, I'm sorry honey.

Yesterday, while I was working on the math book, I encountered this problem that costed me a lot of pain in the neck and almost forever to solve. Its funny how simple math problems look so damn hard when you do not know how to solve them and how very elementary it is once you've gotten the hang of it. Anyway, the problem had something to do about motion. Here is the problem:

A commercial jet took off from Kansas City for San Francisco, travelling a distance of 2550 km at a speed of 800 km/h. At the same time a private jet left San Francisco, travelling at 900km/h and bound for Kansas City. How long after takeoff will the jets pass each other?

Okay, so the guy who wrote the problem is asking us at what hour from their initial departure will the two planes meet given their rate of movement. So when the hell did they meet anyway?! Lets find out.

Here are the facts in the given problem:

Kansas - San Francisco
Distance (d1): 2550 km
Rate (r1): 800 km/h
Time (t1): 2550 / 800 h or 3 hours and 3 /16 minutes

San Francisco - Kansas
Distance (d2): 2550 km
Rate (r2): 900 km/h
Time (t2): 2550 / 900 h or 2 hours and 15 /18 minutes

To compute for distance, we multiply its travelling rate by the time it spent travelling. Hence, we get the general formula:

Distance (d)= Rate (r) * Time (t)

and so to compute for the distance from Kansas to San Francisco and vice versa, we have:
d1 = 800 km/h * (51/16) ->2550 km
d2 = 900 km/h * (51/18) ->2550 km

But that's a given. The real problem is how the hell do we use the formula in order to figure out when the two freaking planes met. Take a look at this:

We say that the total distance from Kansas - San Francisco is equivalent to the total distance from San Francisco - Kansas. And so, we have the relationship:

Distance from Kansas - San Francisco (d1) = Distance from San Francisco - Kansas (d2)
d1 = d2
r1 * t1 = r2 * t2

And so, looking from the previous equation, we instinctively write:
800 km/h * (51 / 16) = 900 km/h * (51 / 18)

Now everything seems in order, right? No, that's wrong! We will only end up having 2550 = 2550 as an answer! What this equation suggests is the distance from San Francisco - Kansas and vice versa, not the distance from Kansas - Location of Plane B and San Francisco - Location of Plane A given given a certain time. So what do we do now? Take a look at the following crappy diagram:

Point of meeting:
+----------------2550 KM--------------------+
Plane A------------->o<----------------Plane B

Time to destination:
+----------------2550 KM--------------------+
-----------------------------------> 51/16 h (A)
51/18 h (B)<-----------------------------------

Time when the two planes will meet:
+----------------2550 KM-------------------+
---------x hrs-----+o+---(51 /16) - x hrs--- (A)
---------x hrs-----+o+---(51 /18) - x hrs----(B)

Okay, I know those diagrams are not very convincing but hey! don't blame me, Im trying my best to make all of these clear! Anyway, the diagram suggested that Plane A will take x hours to reach Plane B's "will-be" location and Plane B will take (51/18) - x hrs to reach Plane A's "will-be" location - all happening under constant motion. And so, we have:

t1 = x
t2 = (51 / 18) - x

and:
d1 = 800km/h * x
d2 = 900km/h * (51 / 18) - x

Now that we have those relationships in place, lets go back to our original equation:
r1 * t1 = r2 * t2
800km/h * x = 900km/h * ((51 / 18 h) - x)

So what does this equation tells us? It simply states that x hours after leaving the airport, Plane A would have travelled a distance of 800km/h * x and Plane B 900km/h * x in the opposite direction and most importantly, in that x hour, they will have meet!! Now that's a lot more convincing than our previous equation!! Now that we know we're on the right track, let's solve for x and get this over with:

800km/h * x = 900km/h * ((51 / 18 h) - x)
800km/h x = 2550km x - 900km/h x
1700km/h x = 2550km

x = 1.5 h

And so, 1.5 hours after the two planes left, they will meet. Hence, Plane A would have travelled 1200km in 1.5 hours and Plane B 1350km in the same duration as Plane A (Plane B is faster by 100km/h than Plane A) before they meet!!

Now that wasn't complex was it? So what's the moral lesson here? Well, I've learned that unless you have a very high IQ or unless you were born with a superhuman ability to process math problems, you should always and I mean ALWAYS try to understand the problem. Don't get upset when you didn't understand the problem the first time (not everybody does!) Try to read the problem over and over again until you have a good grasp of what it is really saying. Think of creative ways (e.g. drawing, asking a Chinese guy) that will help you understand the problem at hand (honestly, if I had 2 matchbox cars with me while I was trying to solve that problem, I would have probably used it to see if they really will meet - but of course, I could never simulate something moving around 900km/h using a matchbox). Always find possible relationships and facts that are stated in the problem as they are your clues in solving it (and they are all that you will have!). And well, perhaps the most important thing of all is that you should never give up! It's all about determination baby!! No matter how hard or complex the problem is, one way or another, it has a solution and it is up to you to find that out! Once you give up, it's all over! Remember, nobody does it perfect the first time, or the second time, or the third time, or even how many times you would have done it!! The important thing here is that no matter how many times you've failed, you should always get up and learn from your failures and try to solve the problem all over again. Everything is possible through hard work and perseverance. Nobody is born being great with something. We all must work hard in order to gain something. The reason why those Chinese guys are great in math is because of their dedication to learn!

Okay, enough about that. I'll end my post here and continue downloading Naruto :) I might as well continue on solving another ten numbers from the math book so that I can go on with the next chapter. Oh, and Detroit beat Miami just now. Good for them.