Word Problems Again..

Question

I always have such a hard time figuring out word problems!

 

Help is greatly appreciated:

 

An airplane leaves Los Angeles for Denver at a speed of 430 mph. Thirty minutes later another plane going from Denver to Los Angeles leaves Denver, which is 850 miles from Los Angeles, at a speed of 510 mph. When they meet, how far are they from Denver?

 

So confusing! Step by step would be wonderful so I can figure this out.

Answer 1

Hi Addie, I hated word problems too.  The best way I found to approach them is to write down the information in as few words as possible.

 

plane 1: Los Angeles to Denver at 430 mph

plane 2: Denver to Los Angeles at 510 mph

 

plane 1 leaves 30 mins before plane 2

 

LA and Denver 850 miles apart

 

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So to begin with plane 1 got a 30 minute head start, so since it was going at 430 miles per hour and 30 minutes is half an hour, it has traveled 215 miles when plane 2 takes off.

 

Now since the locations are 850 miles apart, at the time plane 2 leaves, they are only 850-215=635 miles apart.

 

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Now lets see what we have to do, the planes are 635 miles apart one is traveling at 430 mph and the other 510 mph.

 

Now you can write an equation to find the amount of time it will take for the trains to meet.

430x+510x=635

940x=635

x=635/940 hrs until the trains meet from the time the second one left (keep it as this fraction to avoid rounding errors)

 

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Now the final stage!!!! (this is a very involved math problem)

 

Find the distance from denver.  I would work off the second plane numbers now.

 

So plane 2 is going from Denver to LA at 510 mph and has traveled for 635/940 hrs, so to calculate the distance it has covered would be (510mi/hr)(635/940 hrs) = 344.5 miles.

 

 

I hope I did all the math correctly, but thats the idea.  I'm going to do a check of my work and i'll edit if I find any issues.

 

 

 

 

Answer 2

 

430t + 510(t - .5) = 850

 

430t + 510t - 255 = 850

 

940t = 850 + 255

 

940t = 1105

 

940t/940 = 1105/940

 

t ≈ 1.176 hours

 

510(1.176 - .5) = 344.76 miles

 

Answer 3

Hi Addie;

LOS ANGELES - DENVER...430 miles/hour, x hours

DENVER - LOS ANGELES...510 miles/hour, (x-0.5) hours

Please note, that while you present the unit of 30 in minutes, we require hours.  This is because the speed is in miles/hour.

TOTAL DISTANCE...850 miles

First, let's understand when they meet.

[(430 miles/hour)(x hours)]+[(510 miles/hour)((x-0.5) hours)]=850 miles

Let's cancel units.  Hours as a numerator cancels hours as a denominator...

[(430 miles/hour)(x hours)]+[(510 miles/hour)((x-0.5) hours)]=850 miles

[(430 miles)(x)]+[(510 miles)(x-0.5)]=850 miles

Miles are a numerator in all facets of the problem.  These cancel...

[(430 miles)(x)]+[(510 miles)(x-0.5)]=850 miles

(430x)+[510(x-0.5)]=850

430x+510x-255=850

940x-255=850

Let's add 255 to both sides...

255+940x-255=850+255

940x=1105

Let's divide both sides by 940...

(940x)/940=1105/940

x=1.18

In 1.18 hours from the time the first plane departed, these planes will meet.

In 0.68 hours from the time the second plane departed, these planes will meet.

The second plane departed from Denver.  The question asks how far they are from Denver.

Distance=(rate)(time)

Distance=(510 miles/hour)(0.68 hours)

Distance=346.8 miles