Lesson Using proportions to solve some nice simple Travel and Distance problems

This Lesson (Using proportions to solve some nice simple Travel and Distance problems) was created by by ikleyn(52752)

: View Source, Show
About ikleyn:

Using proportions to solve some nice simple Travel and Distance problems

Problem 1

Justin can run 10 kilometers in the same amount of time that Leo can run 12 kilometers.
If Leo can run 1 kilometer per hour faster than Justin, how fast can Leo run?

Solution

Let x be Leo' speed, in kilometers per hour.

Then Justin' speed is (x-1) km/h.


From the problem, the time equation is

     =   (Justin's time for 10 km is the same as Leo's time for 12 km).


This equation is a proportion. To find x from this equation/proportion, first cross-multiply

    10x = 12(x-1).


Now simplify, step by step

    10x = 12x - 12

    12 = 12x - 10x

    12 = 2x

    x  = 12/2 = 6.


ANSWER.  Leo' speed is 6 km/h.

Problem 2

A and B are running a race. A has an 84 feet head start when they begin running.
B runs 12 feet while A covers 8 feet. Determine distance A runs before B overtakes A.

Solution

Let d be the distance A runs before B overtakes A.

Then the distance that B will cover at that time moment is (d+84) feet.


Their speeds are in the ratio  =  = .

They spend the same time - THEREFORE, the ratio of distances is the same 
as the ratio of their speeds.


So, we write this proportion

     = .


Cross-multiply

    3d = 2*(d+84).


Simplify and find d

    3d = 2d + 2*84

    3d - 2d = 2*84

       d    = 168.


ANSWER.  The distance A runs before B overtakes A is 168 feet.

Problem 3

In the 1984 Olympics, C.Lewis of the United States won the gold medal in the 100-meter race
with a time of 9.99 seconds. In the 1896 Olympics, Thomas Burke, also of the United States,
won the gold medal in the 100-meter race in 12.0 seconds.
If they ran in the same race repeating their respective times, by how many meters would Lewis beat Burke?

Solution

It can be solved in several different ways; but because we are in the Proportion section,
I will show you how to solve the problem using proportions.

Burke ran 100 meters in 12 seconds. We have to determine the distance "d" he would have covered in 9.99 seconds.


His rate is a constant, so the distance is proportional to the elapsed time.


Based on it, we write this proportion

     = 


Each side of this proportion is the Burke's rate, in meters per second.


From the proportion, we find d, the distance Burke covered in 9.99 seconds

    d =  = 83.25 meters.


Thus Burke would cover 83.25 meters in 9.99 seconds; hence, 
Lewis would beat Burke by 100 - 83.25 = 16.75 meters.        ANSWER

Problem 4

Three runners, Dirk, Edith, and Foley all start at the same time for a 24 km race,
and each of them runs at a constant speed.
When Dirk finishes the race, Edith is 4 km behind, and Foley is another 4 km behind Edith.
When Edith finishes the race, how far behind is Foley, in kilometers?

Solution

From the problem, when Dirk completed the race, Edith ran  24-4 = 20 kilometers;
                                                Foley ran  20-4 = 16 kilometers. 


From it, we conclude that Edith rate to Foley rate ratio is 20/16 = 5/4.


So, when Edith will complete 24 kilometers, the ratio of distances Edith and Foley run
will be in the same proportion 5/4.  So we write this proportion


    24          5
  ---------- = ---.
  Foley ran     4


From the proportion

    Foley run (when Edith finished) =  =  = 19.2 kilometers.


Hence, when Edith finishes the rate, Foley is  24-19.2 = 4.8 kilometers behind (rounded value).

My other lessons on proportions in this site are
    - Proportions
    - Using proportions to solve word problems
    - Using proportions to solve word problems in Physics
    - Using proportions to solve Chemistry problems
    - Typical problems on proportions
    - Using proportions to estimate the number of fish in a lake
    - HOW TO algebreze and solve these problems using proportions
    - Using proportions to solve word problems in Geometry
    - Advanced problems on proportions
    - Problems on proportions for mental solution
    - Selected problems on proportions from the archive
    - Entertainment problems on proportions
    - OVERVIEW of lessons on proportions

This lesson has been accessed 1092 times.