Tutorial 1.4: Part 3

Basic conversions

To convert units we need a conversion factor.

For example, if you are asked how many minutes are equivalent to 120 s, what is the conversion factor?

To convert a measurement, we multiply by a conversion factor.

The next slide shows the conversion of cm to m and m to cm. Note the two ways that the conversion factor is written.

Example 1: 125 cm equals how many km?

First, write a conversion factor for the prefixes c and k.

Factor Number Name Symbol
1012 1 000 000 000 000 tera T
109 1 000 000 000 giga G
106 1 000 000 mega M
103 1000 kilo k
102 100 hecto h
101 10 deka da

Factor Number Name Symbol
10-1 0.1 deci d
10-2 0.01 centi c
10-3 0.001 milli m
10-6 0.000 001 micro µ
10-9 0.000 000 001 nano n
10-12 0.000 000 000 001 pico p

A km is five powers of ten larger than a cm, or $$1 \, km = 10^5 \,cm$$

The conversion factor can be written as $$\frac {1 \, km}{ 10^5 \, cm}$$ or $$\frac {10^5 \, cm}{1 \, km}$$

The first way of writing the conversion factor cancels the unit cm $$125 \,cm \times \frac {1 \, km}{ 10^5 \, cm}$$

The second way of writing the conversion factor does not cancel the unit cm $$125 \,cm \times \frac {10^5 \, cm}{1 \, km}$$

So the answer is: \begin{align*} 125 \, cm \times \frac{1 \,km}{10^5 \,cm} & = 125 \times 10^{-5} \, km \\\\ & = 1.25 \times 10^2 \times 10^{-5} \, km \\\\ & = 1.25 \times 10^{-3} \, km \\\\ \end{align*}

Example 2: 125 km equals how many cm?

The conversion factors are the same as in the previous example: $$\frac {1 \, km}{ 10^5 \, cm}$$ or $$\frac {10^5 \, cm}{1 \, km}$$

The first way of writing the conversion factor does not cancel the unit km $$125 \,km \times \frac {1 \, km}{ 10^5 \, cm}$$

The second way of writing does cancel the unit km $$125 \,km \times \frac {10^5 \, cm}{1 \, km}$$

So the answer is: \begin{align*} 125 \, km \times \frac{10^5 \,cm}{1 \,km} & = 125 \times 10^5 \, cm \\\\ & = 1.25 \times 10^2 \times 10^5 \, cm \\\\ & = 1.25 \times 10^7 \, cm \\\\ \end{align*}

Practicing One Step Conversions

Factor Number Name Symbol
1012 1 000 000 000 000 tera T
109 1 000 000 000 giga G
106 1 000 000 mega M
103 1000 kilo k
102 100 hecto h
101 10 deka da

Factor Number Name Symbol
10-1 0.1 deci d
10-2 0.01 centi c
10-3 0.001 milli m
10-6 0.000 001 micro µ
10-9 0.000 000 001 nano n
10-12 0.000 000 000 001 pico p