Sometimes, increasing something by 10% and then decreasing it by 10% does not take you back to where you started. This riddle comes from one of our readers, who proposed it from a very everyday scene: a car, a highway, and a cruise-control button that looks more innocent than it is.
Cruise Control
Riddle statement
The trip to the beach is long, and Diego decides to let cruise control do its job. He sets it to 100 km/h.
His car's control has a particular feature: each press upward increases the car's current speed by 10%; each press downward decreases the car's current speed by 10%.
At one point in the trip, Diego presses upward once. The car then goes to 110 km/h.
Now he wants to return to the original 100 km/h without touching the accelerator or the brake, using only that same control. He may press up or down as many times as he wants, in any order.
Is there any finite sequence of presses that returns the car exactly to 100 km/h?
Show solution
Solution
There is no such finite sequence.
Each upward press multiplies the speed by
and each downward press multiplies it by
Suppose that, starting from the original 100 km/h, Diego presses upward a total of u times and downward a total of d times. Since he has already pressed upward once, we have u \geq 1.
The order of the presses does not matter, because all of them are multiplications. To return exactly to 100 km/h, we would need
Canceling 100,
That is,
But this is impossible: the left-hand side is odd, while the right-hand side is a power of 10, and therefore even.
So Diego cannot return exactly to 100 km/h by any finite sequence of button presses.
He could get very close, and the speedometer might even round the value and display 100. But under the exact rules of the problem, he never reaches it exactly.
To undo a 10% increase, he would need to multiply by
which is equivalent to decreasing by approximately 9.09%, not by 10%.