Distance Formula Word Problems

by Chuck
(Corning)

If a person can travel 20 miles upstream in 10 hours and the same distance downstream in 2.5 hours, find the rate of the current.


Karin from Algebra Class Says:

If a person can travel 20 miles upstream in 10 hours and the same distance downstream in 2.5 hours, find the rate of the current.

This problem is a little tricky.

First we want to use the distance formula to find his overall rate for both directions.

Upstream:
d = rt
20 = r10
20/10 = 10r/10
2 mi/h = r

His total rate, including the current is 2 mi/h.

Downstream:
d = rt
20 = r2.5
20/2.5 = 2.5r/2.5
8 mi/h = r

His total rate going downstream is 8 mi/hr.

Now we need to think about what is happening in this problem.

We have the swimmer who is swimming at x mph. We don't really know his swimming speed. Then if he's traveling downstream, we can add the speed of the current. If he's traveling upstream, then we subtract the speed of the current. (The speed of the current is y since we don't know that information.)

So,
Downstream:
Swimming speed (x) + current (y) = 8 mph
or
x+y = 8

Upstream:
Swimming speed (x) - current (y) = 2 mph
or
x - y = 2

Now we have a system of equations and we can solve to find the current (y).

Since I am solving for y, I am going to solve my first equation for x and substitute into the second equation.

x + y = 8
x + y - y = 8 - y
x = 8-y

Now substitute this into the second equation:
x - y = 2
(8 - y) - y = 2
8 - 2y = 2
8 - 8 - 2y = 2 - 8
-2y = -6
-2y/-2 = -6/-2
y = 3

Since y equals the current of the water, the current is 3 mph.

There are two steps. First you must use the distance the formula to find his overall rate in each direction. Then you must write a system of equations to find the rate of the current.

I hope this helps,
Karin

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Sep 30, 2013
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by: Anonymous

A passenger plane made a trip to Las Vegas and back. On the trip there it flew 432 mph and on the return trip it went 480 mph. How long did the trip there take if the return trip took nine hours?

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Karin from Algebra Class Says:

For this problem, we will use the distance formula to solve.

Distance = rate * time

We know that the distance to and from Las Vegas will be the same. So, since we know the rate and time we can find the distance.

D = rt r = 480 mph t = 9 hours

D = 480(9)
D = 4320 miles

We know that the distance to and from Las Vegas is 4320 miles. Now we can use this information to help us find the time to Las Vegas.

D = rt D = 4320 r = 432 t = ?
4320 = 432t

Now we can solve:

4320/432 = 432t/432
10 = t

The time to get to Las Vegas would be 10 hours.

I hope this helps!

Karin
Sep 30, 2013
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by: Chuck

A cattle train left Miami and traveled toward New York. 14 hours later a diesel train left traveling at 45 km/h in an effort to catch up to the cattle train. After traveling for four hours the diesel train finally caught up. What was the cattle train's average speed?

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Karin from Algebra Class Says:

Hi Chuck,

This is an interesting problem. Again, we are going to use the distance formula since we have information about rate and time.

The distance formula is: Distance = rate * time

We know more information about the diesel train, so let's start there.

It took 4 hours for the diesel train to catch up with the cattle train, so time is 4 hours. Rate of 45 km/h was given. So, with this information we can find the distance.

D = rt
D = 45 * 4
D = 180 km.

The distance was 180 km and since the diesel train caught up with the cattle train, the distance is the same for both trains.

So, now for the cattle train, we know that the distance is 180 km (same as diesel train) and we know that the time to get that distance is 18 hours. It's eighteen hours because the cattle train started 14 hours before the diesel train, plus the 4 hours that it took for the diesel train to meet up with the cattle train.

So, again we can use the distance formula to find the rate of the cattle train.

D = rt
180 km = r(18)

18r = 180 Switch the problem around
18r/18 = 180/18 Divide by 18
r = 10 km/h

The rate of the cattle train was 10 km/h

I hope this helps.

Karin
Sep 30, 2013
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by: Jay

It takes a passenger train 2 hrs less time than a freight train to make the trip from Central City to Clear Creek. The passenger train averages 96 km/h while the freight train averages 64 km/h. How far is it from Central City to Clear Creek?

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Karin from Algebra Class says:

You must use the distance formula to solve this problem. We will use the distance formula for the freight train and the passenger train.

freight: d= rt
passenger: d = rt

We know that the passenger train takes 2 less hours than the freight train. We don't know how long the freight train takes, so we are going to let our variable x = the number of hours for the freight train.

So we know:
freight: time = x, rate = 64km/hour
passenger: time = x -2 rate = 96 km/hour

We also know that the two trains are traveling the same distance, so we are going to set these two equations equal to each other.
freight distance = passenger distance
rt = rt
64x = 96(x-2)
64x = 96x - 192 Distribute
64x - 96x = 96x -96x -192 Subtract 96x from both sides.
-32x = -192
-32x/-32 = -192/-32
x = 6

Since x = 6 we know that the freight train takes 6 hours and the passenger train takes 4 hours.

So, let's figure out the distance - it should be the same for both trains.

freight: d= rt d = 64(6) d = 384 km
passenger: d = rt d = 96(4) d = 384 km

Therefore, the distance is 384 km.

Hope this helps!

All the best,
Karin
Mar 05, 2011
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by: Anonymous

Thank you!! A similar problem was driving me crazy on how to figure it out.

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