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# Think of Two Numbers Poster

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Age 11 to 14

Challenge Level

- Problem
- Student Solutions

I subtract 9 from your answer, and the digits of the number that I'm left with, are your two numbers.

For example, if your final answer is 74, when I subtract 9, I get 65, so I know your numbers were 6 and 5.

**How does it work?**

If the two numbers are $a$ and $b$,

Take one of your numbers |
$a$ |

Add $1$ |
$a+1$ |

Multiply by $5$ |
$5(a+1) = 5a+5$ |

Add $1$ again |
$5a+5+1 = 5a+6$ |

Double your answer |
$2(5a+6) = 10a+12$ |

Subtract $1$ |
$10a+12-1 = 10a+11$ |

Add your second number |
$10a + 11 + b$ |

Add $2$ |
$10a + 11 + b + 2 = 10a + b + 13$ |

Double again |
$2(10a + b + 13) = 20a + 2b + 26$ |

Subtract $8$ |
$20a + 2b + 26 - 9 = 20a + 2b + 18$ |

Halve this number |
$\frac12(20a + 2b + 18) = 10a + b + 9$ |

Final answer is $10a + b + 9$, so I subtract $9$ to leave $10a+b$

$10a + b$ is $a$ tens and $b$ ones

Tens | Ones |

$a$ | $b$ |

Since $a$ and $b$ are both less than $10$, they fit in the tens and ones columns and so they are the digits of the number.