Word Problem on Linear Equations: Problem Type 2

 Uniform Motion Poblems

 These problems involve motion and require use of the formula rate x time = distance (rt=d)

 It may be helpful to organize the information in a table with column headings: rate, time, and distance.  The above formula tells you that for a given row whatever you put in a given rate column times whatever is in the time column must equal what is in the distance column.

 These problems vary in that you sometimes solve for rate, sometimes for time, and sometimes for distance, but the table can help no matter which variable you are looking for.

 Example:

The rates (speeds) for each train are given.  We don’t know the times, but we do know that train 1, which left at 5:20, has been moving 2 hours and 10 minutes longer than train 2, which left at 7:30. The time must be expressed in hours because the rate is in miles per hour.  10 minutes is 10/60  or 1/6 of an hour.  Thus, train 1 traveled for 2 1/6 hour plus how ever long train 2 traveled.  We don’t know this so we let the time train 2 traveled be x.

Now, we have the rate and time blocks filled in.  We can represent the distance traveled by Train 1 by multiplying its rate (70 mph) times its time (x + 2 1/6).  We can represent the distance traveled by Train 2 by multiplying its rate (161 mph) times its time (x).

Next we need to get an equation.  This is where these types of problems vary.  In this example the trains leave from the same city and travel in the same direction until one catches up with the other.  At that point they will both be exactly the same distance from where they started, city A, so we set our two expressions representing the distance each train traveled equal and then solve for x.

 70(x + 2 1/6) = 161 x       Use the distributive property to get rid of the parentheses

70x + 151 2/3 = 161x       You might want to multiply both sides by 3 to get rid of the fraction.

 210x + 455= 483x           Subtract 210x from both sides

 455 = 273x                        Divide both sides by 273

x = 1.666667 or 1 2/3 hours

 Train 2 will catch up to train 1 in 1 and 2/3 hour or 1 hour and 40 minutes.