How To Calculate The Distance, Rate And Time

The word ​rate​ can be defined as the amount that something measurable—such as money, temperature or distance—changes over time. ​Speed​ is the rate at which distance changes over time. Students in math and physical science classes are often asked to solve rate problems, the first of which usually deal with speed. Problems may involve calculating speed itself or rearranging the equation for speed to solve for time or distance.

All rates have equations associated with them. The equations relate the change being measured and the amount of time that has passed. The equation for speed is the rate equation that relates distance and time. Speed is mathematically defined as distance divided by time. In this equation, ​s​ stands for speed, ​d​ stands for distance and ​t​ stands for time:

\(s=\frac{d}{t}\)

One way to use the equation for speed is to calculate the speed of a traveling object. For example, a car travels 400 miles in seven hours and you want to know how fast, on average, the car traveled. Using the equation, plug in the distance of 400 miles for ​d​ and time of seven hours in for ​t​:

\(s=\frac{400\text{ miles}}{7\text{ hours}} = 57.1\text{ miles per hour}\)

To solve for distance instead of speed, imagine the car travels at 40 miles per hour for 2.5 hours. To find the distance the car traveled, you must rearrange the rate equation to solve for ​d​. Start by multiplying both sides by ​t​. Once you've done that, ​d​ will be by itself on the right side. The equation now looks like this:

\(d=s\times t\)

Now just plug in your values for speed and time to solve for distance:

\(d=(40\text{ miles per hour})\times (2.5\text{ hours}) = 100\text{ miles}\)

Like solving for distance, solving for time involves rearranging the speed equation. But this time there are two rearranging steps instead of one. To get ​t​ alone, you must first multiply both sides by ​t​, then divide both sides by ​s​. Now ​t​ will be alone on the left side of the equation:

\(t=\frac{d}{s}\)

Imagine the car travels 350 miles at an average speed of 65 miles per hour and you want to know how long the trip took. Plug the values for distance and speed into the newly rearranged equation:

\(t=\frac{350\text{ miles}}{65\text{ miles per hour}}=5.4\text{ hours}\)