SI - Ratios & Conversions

So far, we have used ratios to convert between units, however, we still might not understand how or why these strategies work. The ideas in this section will help take your knowledge of ratios a step further and become more comfortable with converting between units.

Equivalency:
So far, we have started all of our ratios with statements of equality such as:

$$ 1 hour = 60 minutes$$

and:

$$1 kilometer = 1,000 meters$$

We used these equalities to construct ratios such as:

$$\frac{60 min.}{1 hr.}$$

and:

$$\frac{1,000 m.}{1 km.}$$

But how did we get from an equality to a ratio?

If:

$$1 kilometer = 1,000 meters$$

Then:

$$ \frac{1}{1 kilometer} \times 1 kilometer = 1,000 meters \times \frac{1}{1 kilometer}$$

Cancel the units:

$$ \frac{1}{1 \cancel{kilometer}} \times 1 \cancel{kilometer} = 1,000 meters \times \frac{1}{1 kilometer}$$

Multiplying reveals the following equivalency:

$$ \frac{1}{1} = \frac{1,000 meters}{1 kilometer}$$

Which is the same as:

$$ 1 = \frac{1,000 meters}{1 kilometer}$$

Since the ratio of 1,000 m./1 km. is equivalent to 1, and anything multiplied by 1 is equal to itself, this and many other ratios can be used to convert between units without changing the magnitude of measurements.

Note:Ratios can only be created with units that measure the same type of magnitude. This makes sense when one considers that different types of magnitudes measure different attributes we associate with the physical world.