The table shows the distance covered by a runner at certain times. How far did the runner travel in the first 86 seconds?
And then we have a table which represents time in seconds at 20-second intervals and the distance traveled in meters. So, let’s look at what’s going on in the table. In the first 20 seconds, the runner traveled 50 meters. Then, after 40 seconds, they traveled 100 meters. That means, of course, that between the 20th and the 40th second, they’ve traveled another 50 meters. Then by 60 seconds, they traveled 150 meters, showing another additional 50 meters traveled in that final 20 seconds. Then by 80 seconds, they traveled 200 meters.
Now, after 100 seconds, they traveled 250 meters. So, our 86 seconds is somewhere between the 80 and 100. And to work out the distance that the runner traveled in the first 86 seconds, we have two techniques. Since we know that every 20 seconds the runner travels an additional 50 meters, we can calculate the average speed. Now we’re assuming that the runner travels at a consistent speed. But of course we don’t actually know what’s happening at, say, 10 or 11 seconds. And so the distance that the runner travels in the first 86 seconds will only ever be an estimate. So, let’s work out the average speed of our runner.
We could use the information that every 20 seconds the runner travels a total distance of 50 meters. And substituting this into the formula, we get speed equals 50 over 20 or 2.5. This is in meters per second since our time is in seconds and our distance is in meters. Note that since the distance traveled is directly proportional to the time taken, we could have used any one of these. We could, for example, used 200 divided by 80 or even 250 divided by 100. Either way, we find the average speed to be 2.5 meters per second.
Next, we rearrange our original equation. We multiply both sides of this equation by time, and when we do, we find that distance is equal to speed times time. Now, of course, we’ve just calculated the average speed of our runner to be 2.5 meters per second, and we’re looking to find how far they travel in the first 86 seconds, so distance is 2.5 times 86. And 2.5 times 86 is 215, so that runner must travel a total of 215 meters.
Remember though, this wasn’t the only technique we had. We saw that the distance traveled was directly proportional to the time taken. In other words, as the time increases, the distance increases at the same rate. And so, let’s consider purely what’s happening between 80 and 100 seconds. 100 minus 80 is 20. So there are 20 seconds in this time period. We want to know what’s happening at the 86th second, so that’s six seconds into this interval. As a fraction of the entire interval, we can say that’s six twentieths, which is equivalent to three-tenths of the interval. And so since we’re trying to work out the distance traveled three-tenths of the way through the interval, we need to work out three-tenths of the distance traveled in that interval itself.
Well, after 80 seconds, they’d traveled 200 meters. And after 100 seconds, they traveled 250 meters. The difference is 50 meters, so they travel 50 meters in that interval. We therefore need to calculate three-tenths of 50. Well, one-tenth is 50 divided by 10. It’s five. And so three-tenths will be three times this. Three times five is 15. So, three-tenths of 50 is 15. And they travel 15 meters in that interval. Since by 80 seconds, they’d reach 200 meters, we need to add 15 onto that. 200 plus 15 is 215. And so either way, we see that the runner has traveled a distance of 215 meters in the first 86 seconds.