**Question:**

A painter has a stepladder with two sets of steps. Both halves of the ladder have the same number of steps, but on one side of the ladder the steps are 27 centimeters apart and on the other side the steps are 21 centimeters apart. If a board is place on the third steps and the ladder is standing on horizontal ground, will the board be horizontal? Explain.

Answer:

No, the board will not be horizontal or leveled. The board will have an incline or slope of 7/9, 7 out of 9, or 7 in 2 chances. If the board was horizontal there will be a point of intersection for the two lines or sides of the ladder. See below.

No. of steps |
S |
S |

2 |
54 |
42 |

3 |
81 |
63 |

4 |
108 |
84 |

5 |
135 |
105 |

6 |
162 |
126 |

7 |
189 |
147 |

8 |
216 |
168 |

As you can see there is no point of intersection especially since the postulate is that there is more than one step because the question asks about the third step. In other words, an intersection at step zero (0) exists ergo, that is the only chance of being horizontal.

**Question:**

Use the fraction bar method for at least two iterations to determine the cost per minute for a $10 pre-paid cell phone card that gives 35 minutes. Round to the nearest millionth, then model the entire process or create an analytical expression for this fraction bar method and two iterations.

**Answer:
**

Fraction Bar

cost |
$10 |
$1.428571 |
$.285714 |

minutes |
35 minutes |
5 minutes |
1 minute |

Analytical Model

Let y_{t} = cost in t minutes,

Let t = time in whole minute increments where 0< t <_35, thus y_{1} = [(y_{35}/7)/5]

Or via 2 iterations say for instance: [eq1] y_{5} = y_{35}/7 [eq2] y_{1} = (y_{5}/5)

Or via generic amount of iterations y_{t} = [y_{35}/(35/t)]