# numerator and denominator increased by same amount

Let the positive fraction $\frac{a}{b}$ be altered by adding a positive number $\delta $ to both $a$ and $b$. Then

$$ |

$$\frac{a}{b}>\frac{a+\delta}{b+\delta}\mathit{\hspace{1em}}\text{if}\mathit{\hspace{1em}}a>b.$$ |

The asserted inequalities^{} follow from the identity

$$\frac{a}{b}-\frac{a+\delta}{b+\delta}=\frac{(a-b)\delta}{{b}^{2}+b\delta}.$$ |

Accordingly, we have for example

$$ |

Title | numerator and denominator increased by same amount |
---|---|

Canonical name | NumeratorAndDenominatorIncreasedBySameAmount |

Date of creation | 2013-03-22 19:36:29 |

Last modified on | 2013-03-22 19:36:29 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Result |

Classification | msc 26D07 |

Classification | msc 11A99 |

Related topic | AdjacentFraction |