FIRST HALF

It's **December 25**. It's held at the **Olympic Stadium in Montreal** a **benefit match** in favor of the **Misunderstood Mathematicians**.

The initial **line-up** of one team consists of the following **players**:

Player | Age |
---|---|

Manuel Neuer | 26 |

Thomas Vermaelen | 27 |

Daniel Agger | 28 |

Fabio Coentrao | 24 |

Yaroslav Rakitskiy | 23 |

Jesús Navas | 27 |

Samir Nasri | 25 |

Urby Emanuelson | 26 |

Theo Walcott | 23 |

Mario Balotelli | 22 |

Robert Lewandowski | 24 |

The **mean age** of the team is exactly **25 years**.

In minute 60, the coach decides to **replace** **Nasri** (**25 years**) by another player.

After the substitution, the **average** age has **risen** **2 years**.

Could you estimate how **old** is the new **player**?

After this change, the coaches of both teams decide to **play** the last quarter of hour with **12 players**.

How old should be the new player entering, to **re-up** the **average** another **2 years**?

In the **first case**, for calculating the age of the player chosen to enter the pitch to increase the average age two years, the **easiest method** is as follows.

We will **add** the years of **all** the players. If the **average** age is **25 years**, and there are **11 players**, we multiply: 25 * 11 = 275.

**After** **replacement**, the players will total: 27 * 11 = 297 years.

As there is a **difference** of 297 - 275 = 22 years, this means that the **reserve** must have **22 years more** than the substituted player, in this case: 22 + 25 = 47 years

A nice **Christmas gift** to **Gheorghe Hagi**!

In case there are **12 players** on the pitch, the calculations would be as follows. Total **years before** substitution: 27 * 11 = 297

Total **years after** the change: 29 * 12 = 348. **Unlike** years: 348 - 297 = 51 years. The **incoming player** must be 51 years old.

Another **Christmas gift**, now to **Lothar Matthäus**!

Sometimes when we walk into a room where, for example, is held a competition of **ballroom dancing** for **over 80 years**, or a competition of **games consoles** for **under 10**, we use to say: 'I've just pushed up / down the average age 5 years'.

Actually, we control the concept of **average** in a relatively correct way when it comes to handling a few values, but we start to lose the global view of the average as we increase the population we study.

Thus, if we consider the second case, let's see what we need to **modify the mean age** of a group in a year, when we **add another element** to the group.

If we call x_{1},x_{2},...x_{n} to the different **ages** of the people from a **group of n individuals**, their **mean age** will be:

If we now **add one more person** (n+1), the **average age** should be:

If we want this **average** varies in a number of years respect to the previous average, we'll have:

So each year ( = 1) we want to **increase the age mean** of a population, we need to **introduce a person** with an age equal to the previous average age plus the total number of individuals plus 1.

Obviously, the larger n, that is, the **greater number** of the **individuals** who make up the group, the **harder** it will be to find someone old enough to **increase** that **average**.

And similarly, if we want to **reduce the average** by one half, ( = -1), we get that:

Which means that if **n** is **large** enough, it will be impossible to **bring down the mean** age in a year just by entering one person.

So the next time we visit a competition ballroom dancing competition or game consoles, we will not feel strange among the public, since the average age will barely suffer by our presence ...

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