# Metric Units and Two Step Conversions

Unit Conversions/Dimensional Analysis Part 2
Metric Conversions, Scientific Notation and Two-Step Conversions
Dr. MJ Patterson

One of the advantages of the metric system is that in converting from one set of units to another, all you really do is move the decimal point.  The base of the number stays the same.  The only tricks are in determining how far to move the decimal, and how to enter the numbers into your calculator.

The following metric prefixes must be memorized - prefix, symbol, size and size in words.  We'll see how to use them shortly.

 Prefix Symbol Size Size in Words giga G 1 x 109 billion mega M 1 x 106 million kilo k 1 x 103 thousand centi c 1 x 10-2 hundredth milli m 1 x 10-3 thousandth

Using Metric Prefixes

The metric prefixes are used to eliminate writing a bunch of zeroes - just like scientific notation.  We typically pick the most convenient unit to report a value.  And, the prefix must accompany a basic unit - meter, gram, liter, second or mole.

Example 1:
If a textbook weighs 1,100 g, the value should be reported as 1.1 kg since 1 kg = 1 x 103 g or 1000 g.

Metric Conversion Factors

We frequently need to perform conversions from one metric prefix to another.  We can use the formalism outline in module 0 if we use the table above to create conversion factors.  Just remember that the size of the prefix substitutes for the prefix itself.

Example 2:
Write out all the conversion factors between meters and the prefixes combined with meters using the table above.

Solution 2:
1 Gm = 1 x 109 m  (1 x 109 substitutes for G)
1 Mm = 1 x 106 m (1 x 106 substitutes for M)
1 km = 1 x 103 m (1 x 103 substitutes for k)
1 cm = 1 x 10-2 m (1 x 10-2 substitutes for c)
1 mm = 1 x 10-3 m (1 x 10-3 substitutes for m)

Metric Conversions

Now that we have written out all of the conversion factors, we can perform conversions between the units with prefixes and the base units.  Recall from module 0 the basic procedure for conversions:

1. Write the quantity to be converted as a fraction divided by 1.
2. Write the conversion factor as a fraction so that the bottom units cancel with the units on the top of the original number in the previous step.

Example 3:
Convert 4.7 kg to grams.

Solution 3:
Remember that the units in red cancel!

 (4.7 kg) (1 x 103 g) = 4700 g (1) (1 kg)

Example 4:
Convert 0.125 meters to mm.

Solution 4:

 (0.125 m) (1 mm) = 125 mm (1) (1 x 10-3 m)

Conversions and Significant Digits

Conversion factors are assumed to have an infinite number of significant digits, or to be exact numbers.

Entering Conversion Factors into Your Calculator

When entering a number like 1 x 103 into your calculator, you must be careful to enter the 1 before hitting the EE or EXP key.

To enter 1 x 103, you should hit 1 EE 3
and the display should look like the calculator below.
(Leave the 10 out of it!)

 1   03 EE x2 log / 7 8 9 x 4 5 6 - 1 2 3 + 0 . +/- =

If you are consistently off by a power or two of ten in your calculations, you should go over this procedure carefully.  I'd bet you are sneaking in the x 10 somewhere along the way.

Two-Step Conversions

Frequently you will need to perform a conversion, but you do not have the particular conversion factor needed.  If you needed to convert from cm to mm, you would encounter this problem.

Let's try to convert this problem into two problems that we already know how to solve.  First, look at the conversion factors we know involving cm and mm.

1 cm = 1 x 10-2 m
1 mm = 1 x 10-3 m

Both factors involve meters, which means we can convert cm to m, and then m to mm.  In other words, this conversion will take two steps.

Example 5:
Convert 76 cm to mm.

Solution 5a:
First, convert 76 cm to m.

 (76 cm) (1 x 10-2 m) = 0.76 m (1) (1 cm)

Next, convert 0.76 m to mm.

 (0.76 m) (1 mm) = 760 mm (1) (1 x 10-3 m)

Solution 5b:
We could also perform this conversion in one equation by tacking on the second conversion factor to the first equation.  We will work a lot of problems this semester by stringing out conversion factors in this manner.  Just make sure that the bottom units of a step cancel with the top units of the previous step.

 (76 cm) (1 x 10-2 m) (1 mm) = 760 mm (1) (1 cm) (1 x 10-3 m)

Example 6:
Convert 5.4 Gbytes (gigabytes) to Mbytes (megabytes).

Solution 6:
Work it out as a two-step conversion: gigabytes to bytes, then bytes to megabytes.  Set up the conversion factors to cancel!

1 Mbyte = 1 x 106bytes
1 Gbyte = 1 x 109 bytes

 (5.4 Gbytes) (1 x 109 bytes) (1 Mbyte) = 5400 Mbytes (1) (1 Gbyte) (1 x 106 bytes)