Previously we had a look at different measurements systems. Now let's see how to convert between them. You already know that for example, 1 pound = 16 ounces. Dimensional analysis, also known as the unit factor method, lets us use this information to easily convert between specific values of each measurement.

The way it works is simple. Let's say we have 30 ounces of a powder & we'd like to know how many pounds that is. Well, since we know that 1 pound is the same thing as 16 ounces we can actually multiply the 30 ounces directly by 1 pound / 16 ounces to get the answer! Cool, right?! Let's see how that works:

$$\frac{\text{30 ounces}}{1} \times \frac{\text{1 pound}}{\text{16 ounces}} = \text{1.875 pounds}$$

The reason this works is because we're simply multiplying fractions! This allows the ounces to cancel & leaves us with pounds. As shown in the example above, we can write any single value like 30 ounces as a fraction: 30 ounces / 1.

Dimensional analysis does not work if the measurement in the denominator of one fraction does not match with the numerator of the other. For example, we would NOT write:

$$\frac{\text{30 ounces}}{1} \times \frac{\text{16 ounces}}{\text{1 pound}}$$

because the ounces wouldn't cancel each other out. Similarly, if we wanted to convert 2 pounds to ounces we would write

$$\frac{\text{2 pounds}}{1} \times \frac{\text{16 ounces}}{\text{1 pound}} = \text{32 ounces}$$

## Examples

25 pounds to kilograms (notice the two step process)

$$\frac{\text{25 lb}}{1} \times \frac{\text{453.592 g}}{\text{1 lb}} = \text{11,339.8 g}$$

$$\frac{\text{11,339.8 g}}{1} \times \frac{\text{1 kg}}{\text{1,000 g}} = \text{11.3 kg}$$

16 quarts to gallons

$$\frac{\text{16 qt}}{1} \times \frac{\text{1 gal}}{\text{4 qt}} = \text{4 gal}$$

500 mg to mcg

$$\frac{\text{500 mg}}{1} \times \frac{\text{1k mcg}}{\text{1 mg}} = \text{500k mcg}$$

## More Examples

5 teaspoons to tablespoons

$$\frac{\text{5 tsp}}{1} \times \frac{\text{1 tbsp}}{\text{3 tsp}} = 1\,\frac{2}{3} \text{ tbsp}$$

250 mg to g

$$\frac{\text{250 mg}}{1} \times \frac{\text{0.001 g}}{\text{1 mg}} = \text{0.25 g}$$

2,000 mL to L

$$\frac{\text{2,000 mL}}{1} \times \frac{\text{1 L}}{\text{1,000 mL}} = \text{2 L}$$

## Practice

Try converting these:
250 g to mg
10 tbsp to cups
7 pints to quarts
500 mcg to mg
100 m to cm
20 km to m
80 mL to L
3 L to cc
800 mg to g
2 cups to mL
1,100 mcg to mg
11 m to mm
15 cups to quarts
50 mcg to g
20 cc to gallons
12 ounces to cc