Introduction: Some Simple Math Problems

1. Suppose a man drives from Toronto to Ottawa (a distance, let’s say, of 500km) at 100km/h, then immediately turns around and drives back at 50km/h. What is his average speed?

2. Suppose a person rows two miles upstream and then two down a river that has a current of three miles per hour. Let us assume that the person can row five miles per hour in still water. How long should the entire trip take?

3. Suppose one adds a ½ litre of vermouth to a ½ litre of gin to make a martini. How much martini will there be?

4. Let us consider next a problem of time. A grandfather clock strikes (makes a bonging sound) six times in 5 seconds. How long should it take to strike twelve times?

5. A man some distance away from a tree notes that an apple is about to fall and he wishes to hit the apple with a rifle bullet. His acuity is so great that he will fire the rifle at the exact same moment that the apple begins to fall. He knows that by the time the bullet reaches the apple, the apple will have fallen some distance. Should he aim: i) slightly above the apple? ii) right at the apple? iii) slightly below the apple? ANSWERS:

1. Suppose a man drives from Toronto to Ottawa (a distance, let’s say, of 500km) at 100km/h, then immediately turns around and drives back at 50km/h. What is his average speed?

Answer: Intuitively, we might respond that since the same distance is being covered each way, that we can just halve the difference in the speeds of the two trips to get an average speed for the round trip of 75km/h. Makes sense, right?

WRONG! We must actually calculate the time involved and the distance. (500km at 100km/h = 5 hours) + (500km at 50km/h = 10 hours) = 15 hour journey to cover 1,000km. The average speed is thus 1,000/15 = 66.67 km/h.

2. Suppose a person rows two miles upstream and then two down a river that has a current of three miles per hour. Let us assume that the person can row five miles per hour in still water. How long should the entire trip take?

Answer: We might be inclined to think that it should take 4/5 of an hour (48 mins) since going with the current one way cancels out going against it the other way, and the distance covered was 4 miles and the rower can row at 5mph without a current. Sounds sensible…

MISTAKEN! Again, we must actually calculate the rate of travel each way and the distance covered. 2 miles going upstream at 2mph (5mph - 3mph current) = 1 hour. 2 miles going downstream at 8mph (5 mph + 3mph current) = ¼ hour. Total: 1 ¼ hours (75 mins).

3. Suppose one adds a ½ litre of vermouth to a ½ litre of gin to make a martini. How much martini will there be?

Answer: Okay, this one’s easy! ½ litre of an alcohol + ½ litre of another alcohol is 1 litre of martini. Thanks for the drink (shaken, not stirred)!

ERRONIOUS! Clearly one has had too much to drink… The two liquids actually have different properties and their molecules combine making a total of 900ml of martini. Think of adding sugar and coffee… the sugar melts into the liquid and takes up less volume…

4. Let us consider next a problem of time. A grandfather clock strikes (makes a bonging sound) six times in 5 seconds. How long should it take to strike twelve times? Answ: Yay! No rate of travel or chemistry in this one. 6 times in 5 seconds… 12 times must be double: 10 seconds!!

BZZZZZT! Have you learnt nothing yet?? Good-bye, you are the weakest link! Actually this one needs to be thought out a bit… There are 5 intervals between 6 bongs, so there are 11 intervals between 12 bongs. Or consider it another way: At the first bong time is at zero, and then the average of the rest is one per second, whether it be six or twelve. The answer will always be one less than the number of bonging sounds.

5. A man some distance away from a tree notes that an apple is about to fall and he wishes to hit the apple with a rifle bullet. His acuity is so great that he will fire the rifle at the exact same moment that the apple begins to fall. He knows that by the time the bullet reaches the apple, the apple will have fallen some distance. Should he aim: i) slightly above the apple? ii) right at the apple? iii) slightly below the apple?

Answ: Slightly above the able doesn’t makes sense, so that’s out. Right at the apple? But the apple will have fallen some distance by the time the bullet gets to it, so I’ll have to anticipate that a fraction and aim just below it so they’ll converge at the same point.

NOPE!! The force of gravity is consistent on the two objects (the apple and the bullet). They will fall to earth at the same rate, and cover the same vertical distance in the same amount of time. Thus, one should aim right AT the apple.

The lesson from all this?  Be prepared to abandon “common sense” or what “feels right.” Math will derive the right answers, even if they are not intuitive. Trust the math…