equation with miles

Question

On a particular day, the wind added 4 miles per hour to Jaime's rate when she was rowing with the wind and subtracted 4 miles per hour from her rate on her return trip. Jaime found that in the same amount of time she could row 66 miles with the wind, she could go only 42 miles against the wind.What is her normal rowing speed with no wind?

Answer 1

Let x represent Jaime's normal rowing speed.  Make a chart. 

 

                           r     *          t         =     d   (Rate times Time equals distance, so Time is Distance divided by Rate)  

 

     with wind:   x + 4       66/(x + 4)         66                        

 

against wind:   x -  4       42/(x - 4)          42

 

 

Write the equation from the time column (since the times are equal)

 

       __ 66__    =       42     

         (x + 4)          (x - 4)

 

       66(x - 4)  =   42(x + 4)    cross multiply to clear the denominators

 

     66x - 264   =   42x + 168   distribute

 

     24x - 264   =   168            eliminate the variable term from the right side

 

           24x     =   432            isolate the variable term by adding 264 to both sides

 

               x    =    18

 

So, Jaime's normal rowing speed is 18 mph

 

Wow, that girl can row!

 

Answer 2

The key relationship in this problem is that Distance = Rate (aka Speed) multiplied by Time.  Units are important.  If speed is in miles per hour, then distance must be in miles and time in hours.

 

If I define my unknown (X) as Jaime's rowing rate with NO wind, then his rowing rate with the wind would be X + 4 and his rate against the wind would be X - 4. 

 

It helps to set up a table for these types of problems:

 

Scenario             Distance        =          Rate         x           Time

With wind               66                        X + 4                        T

Against wind           42                        X - 4                         T

 

We are not told anything about the time so I am leaving that as T for now.  We ARE told that she rowed "the same amount of time" both times--That will come in handy in a minute.

 

I can use the Distance = Rate x Time relationship to write two equations, one from with the wind and one from against the wind:

 

66 = (X + 4)T

42 = (X - 4)T

 

 

If I rearrange each equation to isolate T, I get:

 

66/(X+4) = T

42/(X-4) = T

 

Since we are told that she rowed the same amount of time both days, these Ts are equal, and we can use substitution to set these equal to each other:

 

66/(X+4) = 42/(X - 4)

 

Cross-multiplying gives you:

 

66(X-4) = 42(X+4)

 

Distribution gives you:

 

66X - 264 = 42X + 168

 

Add 264 to both sides to get:

 

66X = 42X + 432

 

Subtract 42X from both sides to get:

 

24X = 432

 

Divide both sides by 24 to get:

 

X = 18

 

So, Jaime's rowing speed with no wind is 18 mph.  With the wind, it is 22 mph.  Against the wind, it is 14 mph.