Converting from One Metric Unit to Another
Skills you need to do this include:
1) memorize the metric prefixes names and symbols
2) determine which of two prefixes represents a larger amount
3) determine the exponential "distance" between two prefixes
4) significant figure rules
5) scientific notation
Here are two typical metric conversion problems:
1) Convert 2.50 mg to picograms.
2) Convert 0.080 cm to km.
A slightly more complex one is:
Convert the speed of light (3.00 x 108 m/sec) to km/year.
We will concentrate on the first type in this tutorial.
The key skill in solving these problems is to construct a conversion factor. This
conversion factor will make the old unit go away (micrograms and km in the top two
examples) and create the new unit (pm and cm) in its place. Along with this change,
there will be a change in the value of the number.
Let's focus on the first example: Convert 2.50 mg to picograms
STEP ONE: Write the value (and its unit) from the problem, then in order write: 1)
a multiplication sign, 2) a fraction bar, 3) an equals sign, and 4) the unit in the
answer. Put a gap between 3 and 4. All that looks like this:
The fraction bar will have the conversion factor. There will be a number and a unit
in the numerator and the denominator.
STEP TWO: Write the unit from the problem in the denominator of the
conversion factor, like this:
STEP THREE: Write the unit expected in the answer in the numerator of the
conversion factor.
STEP FOUR: Examine the two prefixes in the conversion factor. In front of the
LARGER one, put a one.
There is a reason for this. I'll get to it in a second.
STEP FIVE: Determine the absolute distance between the two prefixes in the
conversion unit. Write it as a positive exponent in front of the other prefix.
Now, multiply and put into proper scientific notation format. Don't forget to write
the new unit. Sometimes, the exponential number is in the denominator. You must
move it to the numerator and when you do so, remember to change the sign. Also,
DO NOT move the unit with it. That unit has been cancelled and is no longer there.
Here are all five steps for the second example, put into one image:
Note that the old unit cancels, since it appears in the numerator and denominator
of two parts of a multiplication problem.
Why a one in front of the larger unit? I believe it is easier to visualize how many
small parts make up one bigger part, like 1000 m make up one km. Going the other
way, visualizing what part a larger unit is of one smaller unit, is possible, but
requires more sophistication. For example, how many meters are in one nanometer?
The answer is 0.000000001 or 10¯9. You may be able to handle the conversion and
that is just fine. I'm just trying to make it simple.
What I have been discussing above is sometimes called a "unitary rate." Unitary in
this case simply means the number one. I have created a system above where the
one can be in either the numerator or denominator of the conversion unit. You may
have a teacher that forces the one to be only in the denominator. What that means
is that you will have to decide if you are going from a large unit to a small unit
(making the numerator exponent positive) or going from a small one to a large one
(making the numerator unit negative).
I think my way is better.
Two Comments
1) If you do the conversion correctly, the numerical part and the unit
will go in opposite directions. If the unit goes from smaller (mm) to
larger (km), then the numerical part goes from larger to smaller.
There will never be a correct case where number and unit both go
larger or both go smaller.
2) A common mistake is to put the one in front of the SMALLER unit.
This results in a wrong answer. Put the one in front of the LARGER
unit.
Practice Problems
1. 0.75 kg to milligrams
2. 1500 millimeters to km
3. 2390 g to kg
4. 0.52 km to meters
5. 65 kg to g
6. 750 micrograms to g
7. 0.25 megameters to cm
8. 23.8 fg to kg
9. 2.77 kg to mg
10. 2.90 cm to terameters
11. 45.6 microliters to megaliters
12. 1.08 kg to mg
13. 9.57 x 10¯8 mm to nanometers
14. 2.00 L to mL
15. 35.28 mL to L
Answers to Practice Problems
1. 0.75 kg to milligrams
Answer = 7.5 x 105 mg
2. 1500 millimeters to km
Answer = 1.5 x 10¯3 km
3. 2390 g to kg
Answer = 2.39 kg
4. 0.52 km to meters
Answer = 520 m
5. 65 kg to g
Answer = 6.5 x 104 g
6) 7.5 x 10¯4 g
7) 2.5 x 107 cm
8) 2.38 x 10¯17 km
9) 2.77 x 106 mg
10) 2.90 x 10¯14 Tm
11) 4.56 x 10¯11 ML
12) 1.08 x 109 mg
13) 9.57 x 10¯2 nm
14) 2.00 x 103 mL (2000 mL shows only one sig. fig.)
15) 0.03528 L
Answers to First Two Practice Problems
1. 0.75 kg to milligrams
Answer = 7.5 x 105 mg
2. 1500 millimeters to km
Answer = 1.5 x 10¯3 km