When a certain object travels at a uniform rate of $\mathbf{r}$ miles per hour for a time of $\mathbf{t}$ hours, then if $\mathbf{d}$ miles is the distance traveled, the formula to derive $\mathbf{d}$ is:

\[ d = r \cdot t \]
An algebraic problem involving the use of this formula is called uniform-motion problem. This is because in Physics, the object is assumed to move with a constant rate in every second along the way.

In applying the formula, we should always remember that the units of measurement of the rate $\mathbf{r}$, time $\mathbf{t}$, and distance $\mathbf{d}$ MUST be consistent. It means that if the distance is measured in mile and time in hour, then the rate must be expressed in miles per hour.

Problem:
One runner took 3 min 45 sec to complete a race and another runner required 4 min to run the same race. The rate of the faster runner is 0.4 m/sec more than the rate of the slower runner. Find their rates.

Solution:
Since the given data for time in the problem is seconds, we choose second as a measurement of time. Because we want to determine the rates of the runner, we have the following definition:

    $\mathbf{r:}$  the number of the meter per second in the rate of the rate slower runner
    $\mathbf{r + 0.4:}$  the number of the meters per second in the of the faster runner

Since each runner is in the same race, they travel equal distances.  This fact will be used to obtain an equation needed to solve the problem. Using the rate formula, we can make a table of data below.

 From the last column in the table, we observe that the number of meters in the m distance can be represented by either $\mathbf{240r}$ or $\mathbf{225(r + 0.4)}$.

Thus, we have the following equation:

$240r = 225(r + 0.4)$
$240r = 225r + 90$
$15r = 90$
$r = 6$
$AND$
$r + 0.4 = 6 + 0.4$
$= 6.4$

Therefore, the rate of the slower runner is 6 m/sec, and the rate of the faster runner is 6.4 m/sec.

Check:
In 240 seconds the slower runner travels 1440 meters ($6 \cdot 240 = 1440 $). In 225 seconds, the faster runner travels 1440 meters ($ 6.4 \cdot 225 = 1440 $).