Speed, rates and working together

Two trains rush toward each other. A snail climbs out of a well. Two painters share one fence. These are some of the oldest puzzles in school books, and they are still asked in exams and job interviews today. They look different, but most of them need only one idea: think about one hour (or one day, or one second) at a time. Once you see what happens in one hour, the rest is a short division.

Speed, distance and time

Speed tells you how far something goes in one unit of time. A car at 60 km/h (kilometres per hour) goes 60 km in every hour. Three numbers are linked: the speed, the time, and the distance (how far it goes).

A car at 60 km/h for 3 hours goes 60 × 3 = 180 km. To go 240 km at 60 km/h it needs 240 ÷ 60 = 4 hours. Every puzzle below uses this rule, sometimes in a clever way.

Two trains coming toward each other

When two things move toward each other, the gap between them shrinks by both speeds at once. In one hour, a train at 70 km/h covers 70 km and a train at 80 km/h covers 80 km, so the gap gets 150 km smaller. We call this the closing speed: the two speeds added together.

A harder version asks: how far apart are the trains one hour before they meet? You do not need the starting distance at all. In the last hour, the trains close the gap by the closing speed. So one hour before they meet, they are exactly that far apart. Trains at 40 km/h and 80 km/h are 120 km apart one hour before they meet, and 60 km apart half an hour before.

A train going through a tunnel

The train is not a dot. It has its own length. Watch the front of the train. It enters the tunnel, travels to the far end, and then keeps going until the back of the train comes out. By then, the front has gone the whole tunnel plus the whole train.

Average speed is not the average of the speeds

A car drives to a town at 60 km/h and comes back on the same road at 40 km/h. Most people say the average speed is 50 km/h. It is not. The average speed of a trip is the whole distance divided by the whole time.

The distance is not given, so choose an easy one. Pick a number that both speeds go into, such as 120 km. Any distance gives the same answer.

Why is it less than 50? The car spends more time at the slow speed (3 hours) than at the fast speed (2 hours). The slow part counts for longer, so it pulls the average down.

Working together

Ana paints a fence in 6 hours. Ben paints it in 3 hours. How long do they take together? Again, think about one hour. In one hour, Ana paints 1/616 of the fence and Ben paints 1/313 of it. This amount per hour is called a rate. Rates can be added; times cannot.

1. Write what each one does in one hour, as a fraction of the job.
2. Add the fractions. That is what they do together in one hour.
3. Turn it upside down to get the hours for the whole job.

Machines, cats and a hen and a half

Five machines make 5 toys in 5 minutes. How long do 100 machines take to make 100 toys? The fast answer is 100 minutes, and it is wrong. First find what one worker does. Each machine makes 1 toy in 5 minutes. So 100 machines, working side by side, make 100 toys in the same 5 minutes.

The famous hen and a half works the same way. A hen and a half lays an egg and a half in a day and a half. Keep the time the same and share the work: if 1.5 hens lay 1.5 eggs in 1.5 days, then 1 hen lays 1 egg in 1.5 days. In 6 days that hen lays 6 ÷ 1.5 = 4 eggs.

The snail in the well

A snail is at the bottom of a 10 m well. Each day it climbs 3 m, and each night it slides down 2 m. So it gains 1 m per day and night, and many people answer 10 days. But on the last day, the snail reaches the top and climbs out. It never slides back that night. After 7 days and nights it is at 7 m. On day 8 it climbs 3 m, reaches 10 m and is out. The answer is day 8.

1. The last day the snail climbs the full amount. So first it must reach the height of the well minus one climb.
2. Divide that height by the gain for one day and night. Round up if it is not a whole number.
3. Add 1 for the last day.

Doubling: think backwards

Water lilies cover a lake, and the area they cover doubles every day. The lake is full on day 48. When was it half full? Do not divide by 2. Go back one day: the day before, the lilies covered half as much. So the lake was half full on day 47. Two days back is a quarter (day 46), and three days back is an eighth (day 45).

The boat and the ladder

A rope ladder hangs from a boat, with 3 steps under water. The tide rises 1 m. The steps are 25 cm apart. How many steps are under water now? It is tempting to work out that 1 m is 4 steps and answer 7. But a boat floats. It rises with the water, and the ladder rises with it. Still 3 steps are under water. Before you calculate, ask what really moves.