## How to Deal with Venn Diagrams

This blog post is primarily for my niece Katya, who was asked this nice problem in her homework:

*In a camp, there are 79 kids; 27 of them are younger than twelve, 33 are girls, and 30 are boys that are twelve or older. Fill in this chart:*

Girls | Boys | All | |

Younger than twelve | |||

Twelve or older | |||

All |

The real question was to demonstrate that **this chart can only be filled in only one possible way**. So great, let's first enter in what we know:

Girls | Boys | All | |

Younger than twelve | 27 | ||

Twelve or older | 30 | ||

All | 33 | 79 |

Now, since the "All" must be equal to the sum of the parts, you have enough information to find out two of the cells, namely

Girls | Boys | All | |

Younger than twelve | 27 | ||

Twelve or older | 30 | I must be 79-27=52 | |

All | 33 | I must be 79-33=46 | 79 |

Now we're here:

Girls | Boys | All | |

Younger than twelve | 27 | ||

Twelve or older | 30 | 52 | |

All | 33 | 46 | 79 |

Then there are two more cells we know:

Girls | Boys | All | |

Younger than twelve | I must be 46-30=16 | 27 | |

Twelve or older | I must be 52-30=22 | 30 | 52 |

All | 33 | 46 |