Algebra Word Problem

Q:  A jet plane traveling at 500 mph overtakes a propeller plane traveling at 200 mph that had a 2-hour head start. How far from the starting point are the planes?

A:  This problem has many ways of starting.  There is a method that I prefer, so we are going to use it… Seems pretty fair, it is my blog after-all 🙂

Whenever there is a “time delay” between two items, I like to adjust for that if possible.  Let me first define some terms so we can speak the same language:

Red Plane:  Traveling 200 mph (left 2 hours early)

Blue Plane: Traveling 500 mph

I want to start thinking about this problem at the time the blue plane took off.  When the blue plane took off, the red plane had already been traveling for 2 hours at 200 mph.  This means that the red plane is 400 miles ahead.

Look at the following “picture”.  This shows where the blue plane takes off (in blue).  Where the red plane is at that time (in red) and the point in the future where the red plane will be caught (in purple)

<blue> ———— 400 miles ———– <red> ————— ??? ———– <where red plane gets caught>

Main equation:  distance = rate * time: D = RT

Equation for the red plane until it reaches the point where it gets caught:

D = R*T

We know the rate is 200, and let’s call that unknown distance (???) the variable H.  So,

H = 200*T

Equation for the blue plane until it reaches the point where it catches the red plane:

D = R*T

We know the rate is 500, and we know that it has to travel 400 + H to get to that “purple catching zone”.. So,

400 + H = 500*T

Now, we need to combine these two equations… We start with:

400 + H = 500*T

H = 200*T

So, since H = 200*T, let’s substitute that in to the first equation:

400 + H = 500*T

400 + (200*T) = 500*T

400 + 200T = 500T

[subtract 200T from both sides]

400 = 300T

[divide both sides by 300]

400/300 = T

4/3 = T

Therefore, the red plane will be caught in 4/3 hours (1.3333 hours).  However, this is not what the question asked for *sigh*.  The question wanted us to find the distance from the starting point, so we need to find H.

Well, H = 200*T and T = 4/3… So,

H = 200*(4/3) = 800/3 (approximately 266.67 miles).

So, how far are both planes from the starting point?? 400 + H = 400 + 266.67 = 666.67 miles!

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