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  Distance, Rate, and Time  

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The formula Distance = Rate x Time expresses one of the most frequently used relations in algebra.

Since an equation remains true as long as you divide through by the same non-zero element on each side, this formula can be written in different ways:

  • To find rate, divide through on both sides by time:
         Rate = -----------
    Rate is distance (given in units such as miles, feet, kilometers, meters, etc.) divided by time (hours, minutes, seconds, etc.). Rate can always be written as a fraction that has distance units in the numerator and time units in the denominator, e.g., 25 miles/1 hour.

  • To find time, divide through on both sides by rate:
         Time = -----------
When using this equation, it's important to keep the units straight. For instance, if the rate the problem gives is in miles per hour (mph), then the time needs to be in hours, and the distance in miles. If the time is given in minutes, you will need to divide by 60 to convert it to hours before you can use the equation to find the distance in miles. Always make your units match: if the time is given in fortnights and the distance in furlongs, then the rate should be given in furlongs per fortnight.

You can see why this is true if you look carefully at how the units are expressed. Say a car is travelling at 30 mph and you want to figure out how far it will go in 2 hours. You can use the formula:

        Rate    x  Time    = Distance
    30 -------  x  2 hours = 60 miles
    The hours cancel, leaving only miles.

What if you want to calculate the number of miles a car travelling 30 mph goes in 120 minutes?

Since 120 minutes is equal to two hours (60 minutes in one hour x 2 hours = 120 minutes), we should get the same distance of 60 miles, but we will not get the answer this way:

        miles                        mile minutes
    30 ------- x 120 minutes = 3600 --------------
        hour                             hour
Now, 3600 mile minutes per hour isn't very helpful, since we'd like our answer in miles. We need to divide by 60 minutes per hour:
          mile minutes       1 hour
    3600 -------------- x ------------ = 60 miles
              hour         60 minutes
    The hours and the minutes cancel, leaving only miles.

Although we can find an answer this way in the correct units, a better method would be to convert minutes to hours before using the formula.

Remembering to be careful about units, let's look at a problem.

Superheroes Liza and Tamar leave the same camp and run in opposite directions. Liza runs 1 mile per second (mps) and Tamar runs 2 mps. How far apart are they in miles after 1 hour?

To begin, we can either convert rates to miles per hour, or we can convert the time to seconds. Let's convert from miles per second to miles per hour.

There are 3600 seconds in an hour, so if Liza runs 1 mile in a second, then she will run at 3600 x 1 = 3600 mph. Similarly, Tamar will run at 3600 x 2 = 7200 mph.

       miles      3600 seconds          miles
    2 -------- x -------------- = 7200 -------
       second       1 hour              hour
The seconds cancel, leaving miles per hour.

Back to the problem. How far does Liza run in one hour? We know her rate (3600 mph) and the time that she runs (one hour), so we can use the formula:

    3600 ------- x 1 hour = 3600 miles
This makes sense because, by definition, if Liza's speed is 3600 miles per hour, then she runs 3600 miles in an hour.

Tamar, whose speed is 7200 miles per hour, will run 7200 miles in an hour.

How far apart will the two runners be after an hour? The answer is simply the sum of the distance each runs in an hour: 3600 + 7200 = 10,800 miles apart.

Since the earth has a circumference of about 24,000 miles at its equator, that's a little less than halfway around the world!

These problems, however, can be tricky:

Karen can row a boat 10 kilometers per hour in still water. In a river where the current is 5 kilometers per hour, it takes her 4 hours longer to row a given distance upstream than to travel the same distance downstream. Find how long it takes her to row upstream, how long to row downstream, and how many kilometers she rows.

One of the best ways to start a problem like this is to make a table that uses all the information you have been given. Let's make one for the information we have about the distance, rate, and time Karen travels when she is going both upstream and downstream. We'll call the time it takes to row downstream x, which means that the time it takes to row upstream is x +4.

We'll start by calculating Karen's rates going upstream and downstream. When she is traveling against the current, she won't be able to row 10 kilometers/hour. Her speed relative to the shore will only be 5 kilometers per hour because the force of the current, which is flowing at 5 kilometers/hour, slows her rate by 5 km/hour. When Karen is rowing downstream, however, the current helps her go faster, so she moves 10 + 5 = 15 km/hour.

We can use the formula, written as Rate x Time = Distance:

                Rate (km/hr)   Time (hr)   Distance (km)
    Downstream      15            x            15x
    Upstream         5           x+4          5(x+4)
Because Karen goes the same distance upstream and downstream, we know that the two expressions of distance - for upstream and downstream - must be equal; we can set the upstream distance equal to the downstream distance. This produces the following equation, which we solve for x:
    Statement of original equation:  15x = 5(x+4)
    Distributing on right side:      15x = 5x+20
    Subtracting 5x from both sides:  10x = 20
    Dividing both sides by 10:         x = 2
x equals the time it takes Karen to row downstream, or 2 hours. Since it takes her four hours longer to row upstream, this time will be 2 + 4 = 6 hours.

How many kilometers does she row? Look at the distance column in the table. Since x is in hours, Karen's downstream distance is 15 x 2 = 30 kilometers.

The problem states that Karen rows the same distance upstream as down. Let's check our work... yes, 5(2+4) = 5 x 6 = 30 kilometers.

As is frequently the case with word problems, setting up the equations is the hardest part. Once that's done, the rest is relatively easy. Remember always to answer what the question asks - don't stop once you've solve for x, because that may be only part of what the question asked - and always check your answer.

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