Video Transcript
Given that the table shows the relation between the distance a runner can run and the time it would take him to run that distance, determine how long it would take him to cover a distance of 336 meters.
And then we have a table which gives time in seconds at two-seconds intervals and the distance in meters. We’re given information about the relationship between the distance traveled after a given time. And we’re asked to find the time taken given a distance of 336 meters. And so we begin by noticing something quite interesting about the numbers in our table. The time increases by two seconds each time. We go zero to two, two to four, four to six, and so on. But the distance also increases by a set value each time. Between zero and two seconds, it increases by eight. Between two and four seconds, it also increases by eight. And that pattern continues all the way up through our table. We’re therefore able to say that the distance traveled must be directly proportional to the time taken. As one increases, so does the other at the same rate.
Now, what we’re going to assume from this, then, is that as the time increases, the distance also increases at the same rate. Now of course, in reality, this is much less likely to be true. By the time the runner has run a distance of 300 meters, for example, they’re likely to be traveling a little bit slower than if they’d sprinted at the beginning. But what we are able to do is make that assumption and calculate the average speed of the runner. Once we know the average speed, we can then use that to calculate the time it would take to run much longer distances.
Now, since these are directly proportional, we can use any pair of values to work out the average speed. For instance, if we use the second column, we can say that the distance is eight and the time is two. So the speed is eight divided by two, which is four or four meters per second. We could, alternatively, have used the next pair of values; 16 divided by four is also four, as is 24 divided by six, and so on. Either way, we find the average speed of the runner to be equal to four meters per second.
As we mentioned before, we are going to assume that this average speed remains unchanged even up to much longer distances. And what we’re going to do next is rearrange our earlier formula. We’re going to multiply each side of the equation by time and divide each side by speed. When we do, we find that we can calculate the time taken by dividing the distance traveled by the average speed. And of course, we’re interested in a distance of 336 meters. We just calculated the average speed to be equal to four meters per second. So we can say that the time must be 336 divided by four.
And in the absence of a calculator, we could use the bus stop method to calculate this. Three divided by four is zero. So we carry this unused three and we instead divide 33 by four. Eight fours make 32. So 33 divided by four is eight with a remainder of one. Then we know that four fours make 16. And so 336 divided by four must be 84. Since we’re working in seconds and meters and our speed was in meters per second, the time must be in seconds. And we can say that it would take the runner 84 seconds to cover a distance of 336 meters.
