wRALLYE MATH PROBLEMS
Rallye math problems are being seen less often in recent years, mostly to the delight of contestants. Still, they do pop up occasionally, and we present here examples of most of the types we have seen. Problems are described logically and/or with an example. Algebraic details are avoided except when necessary. Contributions are from Dave Graubart, Stan Wantland, and Bill Jonesi.
Pauses, Gains, and staying on time: Many of the problems and solutions mention a PAUSE or GAIN. A PAUSE is simply time that you add to your elapsed time, most easily accomplished by stopping for the duration specified. A GAIN is a negative PAUSE; that is, time to be made up. Since you can't easily stop for a negative amount of time, you need another approach. If you have some kind of rallye computer or are doing detailed calculations, you subtract your GAIN time, driving faster until the computer or your calculations say you are back on time. Well, how about for the rest of us? There is a simple formula that can be followed: For each minute you need to make up, either because of a GAIN or time lost in traffic, traffic lights, etc., go 10% above your assigned speed for 10 minutes, or 20% above your assigned speed for 5 minutes, or 25% above your assigned speed for 4 minutes, etc.
PROBLEM 1: 'Travel the first half of the distance to RI #10 at 60 mph and the other half at 30 mph.' Solution: First, realize that you can't actually DO this as stated, since you don't know in advance what the total distance is, so you don't know at what mileage to make the speed change. At next glance, you might think that you could just travel the whole distance at the average speed, 45 mph, but this turns out to be incorrect. For example, if the total distance ends up being 10 miles, the correct elapsed time is 5 miles at 60 mph (5 minutes) plus 5 miles at 30 mph (10 minutes), for a total of 15 minutes. If you drove 45 mph for the 10 miles, you would take 13.33 minutes, and probably wonder why you were passing a couple of other rallye cars. The correct solution is not to average the speeds (miles per hour), but to average the inverse of the speeds (hours or minutes per mile). At 60 mph, your "factor" is 1 minute per mile. At 30 mph, it is 2 minutes per mile. If you go the average, 1.5 minutes per mile (better known as 40 mph), for the whole distance, you arrive with the correct elapsed time of 15 minutes.
PROBLEM 2: 'Travel the first half of the time to RI #10 at 60 mph and the other half at 30 mph.' Solution: In this case, traveling the whole distance at the average of the two speeds, 45 mph, does work. Suppose the total time ends up being 10 minutes. If you travel 5 minutes at 60 mph (covering 5 miles) and then 5 minutes at 30 mph (covering 2.5 miles), you travel a total of 7.5 miles in the 10 minutes. Driving the whole 10 minutes at 45 mph will also cover 7.5 miles. This becomes obvious if you imagine a section exactly two hours long. You would cover 60 + 30, or 90 miles total, which is exactly 45 mph.
PROBLEM 3: 'Travel the first half of the distance to RI #10 at 60 mph and the other half at 30 mph', and there is a 5 minute PAUSE before RI #10. Solution: The PAUSE here is only a distraction, unlike in the next problem. As with Problem 1, drive the average minute per mile factor (40 mph), and add a 5 minute PAUSE for a total elapsed time of 20 minutes.
PROBLEM 4: 'Travel the first half of the time to RI #10 at 60 mph and the other half at 30 mph', and there is a 5 minute PAUSE before RI #10. Solution: this one is more complicated because the PAUSE affects the total time, thus affects how much time you should spend at each speed. A reasonable rallyemaster will make it obvious which half contains the PAUSE, by putting it near the beginning or near the end of the section. The easiest way to solve this is to remove the PAUSE time from both halves, one at a stand-still (paused) and the other at speed, and then run the remaining distance at the average speed. So in this example, if the PAUSE was near the beginning, when you are "going 60 mph", then you should do a 5 minute PAUSE, 5 minutes at 30 mph (covering 2.5 miles), and the rest of the distance at 45 mph. If the PAUSE was near the end, when you are "going 30 mph", then you should do a 5 minute PAUSE, 5 minutes at 60 mph (covering 5 miles), and the rest of the distance at 45 mph.
PROBLEM 5: 'Change speed to 30 exactly one mile prior to the next Checkpoint.' Solution: At first glance, all that this requires is that you go one mile at 30 and the rest of the distance at your current speed. The easiest way to do this is change speed to 30 mph immediately. After you go one mile, revert to your original speed and drive on to the checkpoint. However, devious rallyemasters might place the Checkpoint less than one mile from this instruction in which case the speed change is never done. The only safe approach is to calculate the difference in the time at the two speeds, drive the mile, and then if you haven't found a checkpoint yet, PAUSE the difference. For example, if you had been going 40 mph, one mile at 40 mph takes 1.5 minutes while one mile at 30 mph takes 2.0 minutes. If you travel one mile without finding a checkpoint, PAUSE for 0.5 minutes. If you had been going slower than 30 mph before the instruction, you will have to make up time (a GAIN) rather than PAUSE after the mile.
PROBLEM 6: 'Divide your current speed by 1/2.' Solution: (Assume your current speed is 24.) The rallyemaster is trying to con you into dividing 24 by 2 (instead of 1/2) and coming up with a new speed of 12 mph. But they said to divide by 1/2, not by 2. And dividing by 1/2 is the same as multiplying by 2 (refer to your 5th grade arithmetic text). So the correct speed would be 48 mph! Another 'grammar' math problem would be 'Reduce your current speed by 1/2'. In this case the correct speed would be 23.5 not 12 mph.
PROBLEM 7: 'Decrease your speed by 20% for one mile then increase your speed by 20%.' Solution: This is another attempt at a con job. Assume your original speed is 50 mph. They are trying to get you to reduce your speed to 40 mph for one mile and then increasing it back to 50. Although reducing it to 40 is correct, you have to remember that 20% of 40 is only 8 so you should increase only back to 48 mph. It doesn't sound like much but it could make a difference over a long distance or on a rallye with low scores.
PROBLEM 8: 'Change speed to 31.4159' Solution: If you are not competing in a equipped class, forget the fractional part, it probably won't effect your score as much as the hassle of trying to calculate it. An easy way to deal with the fractional speed (if you think it matters) is to figure out the difference in speeds for say 10 miles, and then just adjust your time every couple of miles (31.4159 mph for 10 miles takes 19.098 minutes; 31 mph for 10 miles takes 19.354; the difference is 0.25 minutes per 10 miles. So every 2 miles just gain 5 hundredths (3 seconds). By the way, some fractional speeds that look difficult to calculate are really easier than some whole numbers, such as 33.3333... mph which is 1.80 minutes per mile compared to 35 mph which is 1.71428...
PROBLEM 9: 'Change speed to 35, then reduce your assigned speed continuously at a rate of 1 mph each mile.' Solution: This can only be solved precisely as a quadratic equation, but you can get very close by making an intelligent approximation, such as starting at 35 mph for a half mile, then 34 for a full mile, then 33 for a full mile, and so on. In this way, you're a little fast as much as you're a little slow, and you don't have to know or remember how to solve a quadratic equation.
PROBLEM 10: 'Change speed to 30, then reduce your assigned speed 3 mph at each "No parking any time", then five minutes after your out time change speed to 20.' Solution: Nine out of ten times that this occurs it will cause you to stop, and although the thought of traveling at such slow speeds panics new rallyists, coming to a stop is the easiest way to deal with the problem. If you come to a stop (0 mph) at the tenth sign it no longer matters what you have calculated up till then or where the other signs were, you will always depart this sign after the stated time.
Now that you're warmed up, here are the really tough ones:
PROBLEM 11: 'Left and change speed to 28. Exactly 4 minutes after you make this turn, a phantom Honda will make the turn and change speed to 104 mph. When the Honda passes you, change speed to 35.' Solution: This is called a phantom car problem and does require a little algebra. Remember that Distance = rate x time, and that we're interested in when the distance we traveled equals that of the Honda. Our rate x time is (60 minutes/hour / 28 miles/hour) x (4 minutes + ? minutes). The Honda's rate x time is (60 minutes/hour / 104 miles/hour) x (? minutes). Setting the two expressions equal and solving for ? gives you 7.429 minutes. So when you have gone a total of 7.429 + 4 or 11.429 minutes at 28 mph (or 5.33 miles), then change speed to 35.
PROBLEM 12: 'When you have traveled 1/3 of the distance between RI #20 and RI #25, change speed to 40.' Solution: Don't do this speed change while doing RIs #20-25. Keep a detailed log of the mileage at RI #20, at each speed change between #20 and #25, and at #25. Just after you do RI #25, inspect your log to determine what speed you were going when you were arrived at the 1/3 distance, and for what distance you continued until another speed change (or until now if there hasn't been another speed change). Then calculate the appropriate PAUSE or GAIN. For example, if the distance between #20 and #25 was 12.0 miles, the speed change would have been at 4.0 miles (1/3 into it). If you were going 45 mph at that mileage and continued at that speed for another 2.5 miles, this means you traveled 2.5 miles at 45 mph when you really wanted to be going 40 mph. 2.5 miles at 45 mph took you 3.33 minutes. 2.5 miles at 40 mph would take 3.75 minutes. So for this example, you should PAUSE for 0.42 minutes.
PROBLEM 13: 'When you have traveled 1/3 of the time between RI #20 and RI #25, change speed to 40.' Solution: This is much tougher than the preceding problem because changing the speed affects the time. Again, don't attempt the speed change while doing RIs #20-25, and again keep a detailed log of mileages and speed changes. Once you complete RI #25, you need to inspect your log and guess which section you were in when 1/3 of the time elapsed. This may or may not be obvious. If you guess wrong and there is a unique solution, the calculations will tell you. You'll need to remember lots of your high school algebra for this one. Suppose your log looks like the following.
If we guess that the speed change to 40 occurs in the first section, then we would set up the following equations where T is the total time in minutes including accounting for this new speed change, and X is the odometer where the speed change occurs:
T = (7.0 - X)(60/50) + (X - 3.0)(60/40) + (11.0 - 7.0)(60/30) + (15.0 - 11.0)(60/45)
1/3 T = (7.0 - X)(60/50)
Solving these simultaneous equations gives us X = 7.44 miles which is outside of the 3.0 to 7.0 range, so our guess that the speed change was in the first section was incorrect. Let's try again, guessing that the speed change to 40 occurs in the second section. The equations are:
T = (7.0 - 3.0)(60/50) + (X - 7.0)(60/30) + (11.0 - X)(60/40) + (15.0 - 11.0)(60/45)
1/3 T = (7.0 - 3.0)(60/50) + (X - 7.0)(60/30)
Solving these gives us X = 7.32 miles which is within the 7.0 to 11.0 range, so it is a solution!
We're almost done, needing to just calculate the PAUSE or GAIN. We traveled (11.0 - 7.32) or 3.68 miles at 30 mph, but wanted to travel at 40 mph. 3.68 miles takes 7.36 minutes at 30 mph or 5.52 minutes at 40 mph, so we have to GAIN 1.84 minutes. Whew!
A final word on using algebra. Be careful about setting up and simplifying equations if the instructions say anything about truncating or rounding numbers to a particular resolution. One rallye a few years ago had a contrived math problem where the algebra-savvy contestants confidently figured out that the problem simplified to something trivial, while other contestants followed the instructions literally, truncating numbers at the right time, and ending up with the right answer.