The other day, my wife showed me the new Rs 20 coin. To my surprise, it is the same size as the Rs 10 coin. Yet this has happened before. Not long ago, a new Rs 2 coin was unveiled, identical in size to the existing Rs 1 coin; the new Rs 1 coin was the same size as the existing 50 paise coin. (Anyone remember paise? We used to have paise).
Since then, there’s been a still newer Rs 2 coin that’s even smaller, but a new Rs 5 coin that’s larger than the earlier one. All of which is why you’ll see signs — at water dispensing outlets at our railway stations, for example — that say something like “Please use only new Rs 2 coins”. How new, though? Somebody’s having fun playing with our numismatic minds, I tell you.
Anyway, it got me thinking. Take a new Rs 20 coin and a Rs 10 coin, the ones that are the same size. Place them on a table side by side and touching. To make this easier, place them so that both numbers (10 and 20) are upright. Let’s say the 10 rupee coin is to the left.
Now hold the 20 rupee coin down with a finger so it doesn’t move. Revolve the Rs 10 coin around the Rs 20 coin, making sure they are constantly touching. Imagine they are two interlocked gears, so no slipping and sliding as it moves.
Question 1: When the Rs 10 coin has returned to where it started, to the left of the Rs 20 coin, how many rotations has it executed on its own? If you like, think of it like this: how many times was “10” upright again after it started moving?
Question 2: Now replace the Rs 10 coin with a Rs 5 coin. Assume the diameter of the Rs 5 coin is half that of the Rs 20 coin. Revolve the Rs 5 coin around the Rs 20 coin. How many rotations does it execute on its own?
Question 3: Now hold the Rs 5 coin down and move the Rs 20 coin around it. How many rotations does the larger coin make?
Question 4: Can you generalise this? For example, what if one coin was five times larger than the other? How many rotations would the smaller one make as it revolved around the larger? How many rotations would the larger one make as it revolved around the smaller?
Scroll down for the solution
Answer 1: It’s worth actually trying this with two identical coins, to see what happens. You’ll find that the moving coin actually makes two rotations.
The way to understand this is to imagine the coin rolling along a straight line that’s as long as the circumference of the stationary coin. If the coins are identical, that line is as long as the circumference of the rolling coin too, so clearly it will rotate once as it traverses that length. Add one more rotation because it is revolving around the stationary coin, to get two.
Answer 2: Use the same reasoning: the Rs 5 coin will rotate three times.
Answer 3: The Rs 20 coin will make half a rotation as it traverses the length of the Rs 5 coin’s circumference, plus one for the revolution. Thus one-and-a-half rotations.
Answer 4: The same reasoning: Six rotations for the smaller around the larger. 6 / 5 (or one and a fifth) rotations for the larger around the smaller. In general, if the larger coin’s diameter is “n” times the smaller coin’s, the smaller one will make n + 1 rotations; the larger one will make (n + 1) / n rotations.