using alegra prove that 0.565656... / 0.124242424... is equal in value to 560/123

Guest Nov 12, 2017

#1**0 **

0.565656... / 0.124242424..... Multiply both top and bottom by 990.

**[990 x 0.56565656.... / 990 x 0.124242424.....] =560/123**

Guest Nov 12, 2017

#2**0 **

In order to prove algabraically that \(\frac{0.5656...}{0.12424...}=\frac{560}{123}\), let's try to convert the interminable decimals to fractions.

I'll start with \(0.5656...\)

**1. Set the Repeating Decimal equal to a Variable!**

This is a farily simple step. \(0.\overline{56}=x\). Now, you're good to go!

**2. Multiply Both Sides by 10 such until the Repeating Portion is the Whole Number**

In this case, if I multiply both sides by 100, which is 10^2, then the repeating portion will be to the left of the decimal point.

\(56.\overline{56}=100x\)

**3. Subtract your 2 Equations.**

\(56.\overline{56}\) | \(=100x\) |

\(\hspace{2mm}0.\overline{56}\) | \(=\hspace{7mm}x\) |

\(56\) | \(=\hspace{1mm}99x\) |

**4. Solve for x**

\(56=99x\) | Divide by 99 on both sides. |

\(\frac{56}{99}=x=0.\overline{56}\) | |

Great! Now, let's convert the next one.

\(0.1\overline{24}=y\)

Now, multiply by multiples of ten to get the repeating portion to the left until the repeating part lines up.

\(12.4\overline{24}=100y\)

Now, subtract the two equations.

\(12.4\overline{24}\) | \(=100y\) |

\(\hspace{3mm}0.1\overline{24}\) | \(=\hspace{7mm}y\) |

\(\hspace{1mm}12.3\) | \(=\hspace{2mm}99y\) |

Now, solve for y.

\(12.3=99y\) | Multiply by 10 on both sides to make the left hand side a whole number. |

\(123=990y\) | Divide by 990 to isolate y. |

\(\frac{123}{990}=y\) |

Now, let's calculate what x/y is.

\(\frac{x}{y}=\frac{560}{123}\) | Let's see if this is true. |

\(\frac{\frac{56}{99}}{\frac{123}{990}}\) | Multiply by 990/123 to eliminate the complex fraction. |

\(\frac{56}{99}*\frac{990}{123}\) | Notice that 990 and 99 can be simplified before any multiplication takes place. |

\(\frac{56}{1}*\frac{10}{123}\) | Simplify from here. |

\(\frac{560}{123}\) | |

Therefore, we have proven algabraically that \(\frac{560}{123}=\frac{0.\overline{56}}{0.1\overline{24}}\)

TheXSquaredFactor
Nov 12, 2017