Question

As part of an annual fundraiser to help raise money for diabetes research, Diane joined a bikeathon. The track she biked on was 1,920 yards long. Diane biked 38.5 laps. Her sponsors agreed to donate an amount of money for each mile she biked. How many miles did she bike? First fill in the blanks on the left side using the ratios shown. Then write your answer.
Given Ratios: 5280ft / 1 mi , 1 mi /5280 ft , 1,920 yards / 1 lap , 1 lap / 1,920 yards , 3 ft / 1 yard , 1 yard / 3 ft.


Blanks: 38.5 laps / 1 yard x (blank) x (blank) x (blank) = (blank) miles


I'm really confused on how to do this, and the explanations aren't exactly helping. If you could walk me through how to do this, it would be greatly appreciated.

Answer 1

Answer:

• $38.5\,\text{laps}\times\dfrac{1920\,\text{yd}}{1\,\text{lap}}\times\dfrac{3\,\text{ft}}{1\,\text{yd}}\times\dfrac{1\,\text{mi}}{5280\,\text{ft}}$
• 42 miles

Step-by-step explanation:

You know that fractions with the same value in numerator and denominator reduce to 1. This is true whether the value is a number, a variable expression, or some mix of those. That is ...

$\dfrac{1760}{1760}=1\\\\\dfrac{3\,\text{mi}}{1\,\text{mi}}=\dfrac{3}{1}\cdot\dfrac{\text{mi}}{\text{mi}}=3$

This example should show you that you can treat units as if they were a variable.

So, the unit conversion process is the process of choosing combinations of numerator and denominator units so that all the units you don't want cancel, leaving only units you do want.

You're starting with a number than has "laps" in the numerator. To cancel that, you need to find a conversion factor with "lap" in the denominator. On the list you are given, the one that has that is ...

  $\dfrac{1920\,\text{yd}}{1\,\text{lap}}$

Now, you have canceled laps, but you have yards. Also on your list of conversion factors is a ratio with yards in the denominator:

  $\dfrac{3\,\text{ft}}{1\,\text{yd}}$

This will cancel the yards in the numerator from the previous result, but will give you feet in the numerator. You want miles, so you look for a conversion factor between feet and miles, with miles in the numerator. The one you find is ...

  $\dfrac{1\,\text{mi}}{5280\,\text{ft}}$

These three conversion factors go into the blanks. When you form the product, you will get ...

  $\dfrac{38.5\cdot 1920\cdot 3}{1\cdot 1\cdot 5280}\cdot\dfrac{\text{laps \cdot yd \cdot ft \cdot mi}}{\text{lap \cdot yd \cdot ft}}=42\,\text{mi}$