THE’riddle of the 99 Mancini starts from a very simple question:
If in a room there are 100 people and exactly the 99% is left -handedhow many left -handed ones should leave the room so that exactly the 98% is left -handed?
If you answered jet, you probably said “1”. Most people instinctively respond like this, but … it’s wrong! Let’s see why it is not enough get out a lonely person from the room and what is, however, the correct solution.
Riddle of the 99 Mancini: because “1” is the wrong answer
The situation is this: we are with 100 people in a room, of which the 99 are left -handed. There Percentage of Mancini On the total of people, therefore, it is 99/100 = 0.99, or 99%. Well, until here everything smooth. Now we want to understand how many Mancini they have to leave the room so that their percentage goes to 98%. On two feet the answer might seem obvious: one!
However, if we removed a single left -handed from the room, they would remain 99 people total of which 98 left -handed. On this new total of people, the percentage by Mancini is now 98/99 = 0.989, that is 98.9%. And not 98%! Remove only one left -handed makes the percentage drop, but it is not enough To get to 98%. We approach the solution, but not enough. To find the correct solution, let’s face the problem from another perspective.
The intuitive solution to the 99 Mancini’s abundance: change perspective
Instead of focusing on the left -handed ones, let’s look at to the steed.
At the beginning, there is only a dextriman person out of 100, which therefore represents the1% of the total. If we want the percentage of left -handed to 98%, the percentage of the steed owe go out al 2%. But the only way to do it, without letting new steed in the room enter is reduce The number total of people in the room.
How long? Here is the key to the solution: since the percentage of dextrimans must doubleThe total of the people must halve! In this way, a single right -handed person will correspond to 2% of the total. In fact, if I only take 50 people in all1/50 ago just 0.02, that is the 2%. So, in a room with 50 people of which 49 Mancini And 1 righti left -handed they are the 98% And destrimane is 2%.
The solution to the indovinel, in fact, is proper get 50 Mancini out.
The mathematical solution: we set the formula
If the intuitive solution does not fully convince you and you want a more precise explanation, we can write the reasoning that we have made above in mathematical language and solve the problem with … the proportions!
We want the room to be the 98% Of left -handed and the remaining 2% Of steed. Our only steed, therefore, must stay at the total of the people left in the room as 2 is to 100. By writing this relationship as a proportion, we have that:
1: number of remaining people = 2: 100
To resolve, we apply the property from the proportions And we multiply among them the terms close to the equal and those far from the equal. We therefore get:
2 × number of remaining people = 1 × 100
At this point, to find the number of remaining people, enough split both sides of the equation for 2:
number of remaining people = 100 /2 = 50
So, since 50 people have to remain in the room and initially we had 100, the solution of the indovinel is get 50 Mancini out.
An simple way to understand what we have just done without “wrapping” with the proportions, it can be this: we must think that 98% It can also be written as 98/100. If divide both numbers for 2 I get 49/50. So if Mancini’s 98% must remain in the room, this is equivalent to saying that they have to remain 49 left -handed out of 50 people. In both cases, letting 50 left -handed people go out allows me to get the percentage of Mancini to 98%.
