Question

How much longer (percentage) is a one-mile race than a 1500-mile race (“the metric mile”)?

Short answer

A one-mile race is 7.27% longer than a 1500-mile race.

Step-by-step solution

Step 1. Conversion factor

The unit of a quantity is changed either by multiplying or dividing the quantity with the conversion factor.

For example, if the length of a table is 7.00 inches and you want to measure it in centimeters, you can do this by multiplying the length of the table by its conversion factor. The conversion factor to convert inches into centimeters is $1 \mathrm{in}.=2.54 \mathrm{cm}$. Thus, the length of the table in centimeters will be $\left(7.00 \mathrm{in}.\right)\times \left(2.54 \frac{\mathrm{cm}}{\mathrm{in}.}\right)=17.8 \mathrm{cm}$

Step 2. Conversion of a mile into a meter

Here, you have to compare two quantities; i.e., a one-mile race and a 1500-mile race. The two quantities can only be compared when they have the same unit. Thus, you need to convert one mile into meters.

You know that there are 5280 feet in a mile, 12 inches in a foot, 2.54 cm in an inch, and 100 cm in a meter. Using these conversion factors, one mile can be written as follows:

$\begin{array}{c}1 \mathrm{mile}=\left(5280 \mathrm{ft}\right)\times \left(12 \frac{\mathrm{in}.}{\mathrm{ft}}\right)\times \left(2.54 \frac{\mathrm{cm}}{\mathrm{in}.}\right)\times \left(\frac{1 \mathrm{m}}{100 \mathrm{cm}}\right)\\ =1609 \mathrm{m}\end{array}$

Thus, there are 1609 meters in one mile.

Step 3. Comparison of a one-mile race and a 1500-m race

A one-mile race is equivalent to a 1609-meter race. Therefore, a one-mile race is longer than a 1500-meter race. Thus, the difference in lengths$=1609-1500=109 \mathrm{m}.$

The percentage difference in the lengths of a one-mile race and a 1500-m race is:

$\begin{array}{c}\mathrm{Diff}\%=\frac{109}{1500}\times 100\\ =7.27\%\end{array}$

Thus, a one-mile race is 7.27 % longer than a 1500-mile race.