David W. Sabo (2003) Introduction to Unit Conversion: the SI Page 1 of 9 The Mathematics 11
Competency Test
Introduction to Unit Conversion:
the SI
In this and the next document in this series is presented and illustrated an effective and reliable approach to carrying out unit conversions – starting with a value expressed in one set of units of measurement, and calculating the equivalent quantity in another equivalent set of units of
measurement. In this particular document, we’ll develop the strategy of the method by illustrating conversion between units of measurement within the SI. In the next document, we’ll illustrate how the method works for conversions between compatible units in the SI and outside the SI. The strategy we will describe here is quite formal – it requires you to set your work up in a very specific manner. If you follow the procedure without any shortcuts, the only possible error you can make is in your arithmetic. As we pointed out in an earlier document, errors in unit
conversion can be very costly, so there is no justification for short-cuts to this method, or for using other less formal and less reliable methods (such as guess-and-hope-for-the-best, the most frequently used alternative).
Unit Conversion Factors
By unit conversion factor, we simply mean any true statement of the equivalence of two quantities, expressed with different units. Within the SI, these conversion factors come from the definition of the base units, the prefixes used to indicate multiples or fractions of base units, and definitions of the various supplemental units. So, for example, each of the following are
conversion factors implied by information we’ve given you about the SI: 1 m = 100 cm
1 min = 60 s 1 kg = 1000 g 1 MW = 1 000 000 W and so on.
It doesn’t really matter how these are stated. So, for instance 1 m = 100 cm
and
1 cm = 0.01 m
say exactly the same thing, and are equally useful in principle. One of the biggest advantages of the method we are about to describe is that if you know that
1 m = 100 cm
you never have to rearrange this to read 1 cm = 0.01 m
and thus one major potential source of error in these calculations is eliminated. We strongly recommend that you always use conversion factors in the metric system which involve whole numbers whenever possible.
Tabulations of commonly required unit conversion factors are readily available for units outside the SI. We include a short table in the next document in this series. Various publications (for example, the CRC Handbook of Chemistry and Physics) contain very extensive tables of unit
David W. Sabo (2003) Introduction to Unit Conversion: the SI Page 2 of 9 conversion factors. Also, there are a number of internet sites listing hundreds of unit conversion factors – to find them, use a search engine and keywords such as “unit conversion,” “conversion factors,” and similar.
The Method
The method consists of, if you like, four simple but essential steps. We will illustrate it with a fairly simple example.
Example 1: Convert the length 4.235 km to its equivalent in units of centimetres. solution:
This is kind of a silly request, it would seem, since you might wonder why we’d want to know what such a large length as 4.235 km is in units of centimetres. There’s probably no good answer to that question – but what you can easily demonstrate is that few people to whom this question is posed would be able to state an answer with confidence, and so it is a good example to use to illustrate the basic unit conversion method.
step 1: List a sequence of unit transitions for which you have unit conversion factors, and that will get each initial unit in the problem to its corresponding final unit. (This step is always absolutely essential. To skip it is an error because it means you are willing to accept the risk of a serious error that you could take steps to avoid.)
When working within the SI, use the definitions of the base units and prefixes. The sequence of conversions will almost always be simplest to implement if it contains the base units. When working with units outside the SI, you will need to consult an available and adequate table of unit conversion factors to develop the strategy which is the goal of this step. At this stage, no numbers are involved (though you will have to fill in the numbers eventually). This stage is for developing a plan or strategy only.
So, in this example, we need a sequence of units that will go from km to cm. Since we know that 1 km = 1000 m
and
1 m = 100 cm an appropriate path is km → m → cm
Note that since the initial and final units here are both in the SI, it has been easy to find a path from initial to final units that uses the base unit as an intermediate.
Step 2: Write out a product of
• the initial quantity to be converted, and
• a factor in the form of a fraction for each arrow that appears in all sequences of conversions developed in step 1. For the moment, each fraction contains just unit symbols in its numerator and denominator. These unit symbols are placed, top or
bottom, to cancel the unit at the tail of the arrow, leaving the unit at the head of the arrow. (DO NOT WRITE ANY NUMBERS in these fractions at this step, but leave room to write in numbers in the next step.)
For this example, our sequence from Step 1 has two arrows, so the formula we develop here will have two such fractions:
David W. Sabo (2003) Introduction to Unit Conversion: the SI Page 3 of 9
4.235
km
4.235
km
m
cm
km
m
=
×
×
Notice:
• the first fraction here must involve m and km (since the first arrow in the sequence from step 1 involves the conversion from km to m). The symbol ‘km’ is placed in the
denominator so that it will cancel the ‘km’ in the original value. This means that the symbol ‘m’ must be placed in the numerator.
• the second fraction must involve m and cm. The symbol ‘m’ is placed in the denominator since it must cancel the ‘m’ unit in the numerator of the previous fraction. This means that the symbol ‘cm’ must go in the numerator.
Again, do not write any numbers in at this stage. Here, we are simply setting up a template for the required calculation. By restricting our attention to the form of the template only, we avoid potential confusion and error due to trying to keep too many things in our minds at once.
Notice that if we just look at the unit symbols themselves, cancellation here will occur to produce a result with the desired final units:
4.235
km
=
4.235
km
×
m
km
cm
m
×
= …
cm
Step 3: Fill in numbers in the numerator and denominator of each fraction in the expression from step 2 so that the numerator and denominator represent the same physical quantity. Use your source/list of unit conversion factors to do this.
In this example, we get
1000
100
4.235
4.235
1
1
m
cm
km
km
km
m
=
×
×
since we know that 1 km = 1000 m and
1 m = 100 cm
Notice that you use the values in the unit conversion factors exactly as they are given. It is not ever necessary to rearrange something like
1 m = 100 cm into
1 cm = 0.01 m first.
Notice as well that we didn’t have to think too deeply about where each number goes, top or bottom. Since the first fraction already had the form
David W. Sabo (2003) Introduction to Unit Conversion: the SI Page 4 of 9
m
km
and since the top and bottom had to end up being identical physical quantities (in this case identical lengths), and since we know that
1 km = 1000 m
we know immediately that the ‘1’ goes with the ‘km’ in the denominator, and the ‘1000’ goes with the ‘m’ in the numerator. There is no puzzlement whatsoever by this stage of the problem over whether you should multiply by the 1000 or divide by the 1000 since the template developed in step 2 indicates precisely where the factor of 1000 should go in this fraction. This eliminates one more major source of error in unit conversion calculations – mistakenly multiplying when you should divide or dividing when you should multiply, a common error when people attempt to do unit conversion calculations using less systematic methods.
Step 4: Do the numerical arithmetic to get the final answer and verify once again that the units cancel or simplify to give the desired final units.
So, for this example, we have
4.235
km
=
4.235
km
×
1000 m
1 km
100
1
cm
m
×
(
4.235 1000 100 cm
)(
)(
)
=
423 500 cm
=
as the final answer.
This first example appears to be very lengthy because we described the method and discussed issues and strategies in considerable detail as we worked through the actual example. Now we’ll repeat this example without all of the discussion.
Example 1 (Repeat): Convert the length 4.235 km to its equivalent in units of centimetres. solution:
The required conversion here is from kilometres to centimetres. We know that 1 km = 1000 m
and
1 m = 100 cm.
So (step 1) a plan to accomplish the conversion from kilometres to centimetres is km → m → cm
David W. Sabo (2003) Introduction to Unit Conversion: the SI Page 5 of 9
4.235
km
=
(
4.235
km
)
m
km
cm
m
This template indicates that the plan will work, so insert the required numbers from the conversion factors to get (step 3)
4.235
km
=
(
4.235
km
)
1000 m
1 km
100
1
cm
m
(
4.235 1000 100 cm
)(
)(
)
=
423 500 cm
=
as the final answer.
You’ve probably noticed that what makes this strategy work is that we are simply multiplying the original value by fractions with equivalent numerators and denominators (hence they are equal to 1 and so the multiplication doesn’t change the real value of the original quantity). However, these fractions multiplied onto the original value are set up to cause undesired units to cancel and to leave the desired units in their place.
Example 2: Convert the speed of 550 cm/s to its equivalent in km/h. solution:
This is a more difficult problem than the first example. Obviously • we need to convert cm to km, and
• we need to convert s to h. Now since we know that
100 cm = 1 m and
1000 m = 1 km and, as well, we know that
1 min = 60 s and
1 h = 60 min,
then, strategies we can implement here are cm → m → km
and
David W. Sabo (2003) Introduction to Unit Conversion: the SI Page 6 of 9 This completes step 1. Since our strategies include four “arrows” altogether, step 2 will now require the formulation of four fractions, once for each arrow in the strategies:
min
550
550
min
cm
cm
m
km
s
s
s
cm
m
h
=
Make sure you understand exactly why this expression is the correct one to write down for step 2. Now, insert the numbers in these fractions appropriately
1
1
60
60 min
550
550
100
1000
1min
1
cm
cm
m
km
s
s
s
cm
m
h
=
550
cm
=
s
1 m
100 cm
1
1000
km
m
60 s
1 min
60 min
1h
(
)( )( )
(
)(
)
550 60 60
100 1000
km
h
=
19.8
km
h
=
Thus, the speed of 550 cm/s is equivalent to 19.8 km/h.
Example 3: The concentration of salt in a solution is 0.173 g/cm3. Convert this concentration to
units of kg/m3. You may use the facts that
1 m3 = 1000 litres and 1 litre = 1000 cm3.
solution:
In this conversion, we will be converting the mass units of g to kg, and the volume units of cm3 to m3. The most obvious strategies, given the conversion factors stated in the problem for volumes
and the fact that we know that 1 kg = 1000 g
are
g → kg and
cm3 → litre → m3.
Thus, step 2 gives the template
3 3 3 3
0.173
g
0.173
g
kg
cm
litre
g
litre
cm
cm
m
=
David W. Sabo (2003) Introduction to Unit Conversion: the SI Page 7 of 9 3
0.173
g
0.173
g
cm
=
cm
31
1000
kg
g
31000 cm
1 litre
1000 litre
1m
3
(
0.173 1000
)
=
(
)
(
1000
)
1000
(
)
3kg
m
3173 kg
m
=
as the final answer.
Example 4: Repeat the conversion in Example 3 above, but use a conversion factor relating centimetres to metres directly.
solution:
We are asked to do
3 3
0.173
g
?
kg
cm
→
m
as in Example 3. Now, however, instead of the strategy cm3 → litre → m3
we are asked to use simply cm → m.
As before, the strategy for the mass unit conversion will be simply g → kg
Now, in step 2, we may start out writing
3 3
0.173
g
0.173
g
kg
cm
g
m
cm
cm
=
But, if you check, you’ll see that this last fraction replaces only one of the three centimetre units by a unit of metres. In fact, we must repeat the last factor three times to complete the required unit conversion: 3
0.173
g
0.173
g
cm
=
cm
3kg
g
cm
cm
m
cm
m
3kg
m
m
=
David W. Sabo (2003) Introduction to Unit Conversion: the SI Page 8 of 9 3
0.173
g
0.173
g
cm
=
cm
3kg
g
cm
3m
keeping in mind that because of the exponent, the ‘cm’ in the numerator of the last fraction really amounts to ‘cm’ repeated three times.
Now, we can fill in the blanks with the appropriate numbers, and do the arithmetic to get the final answer: 3 3 3
1
100
0.173
0.173
1000
1
g
g
kg
cm
g
m
cm
cm
=
0.173
g
=
3cm
(
1
kg
)
(
100
3)
cm
31000 g
(
)
( )
1 m
3 3(
)
3 3 3100
0.173
173
1000
kg
kg
m
m
=
=
as obtained before.Remember that when a bracketed expression is raised to a power, every factor in the expression inside those brackets is raised to the same power. In detail, this means that
3 3 3 3 3 3 3 3
100
100
100
1
1
cm
cm
cm
m
m
m
=
=
Example 5: A rectangular field is 185 m long and 137 m wide. Compute its area and state your answer in units of hectares.
solution:
First, to get the area of a rectangular region, we need to multiply the length by the width. Here, this gives
A = LW = (185 m)(137 m) = 25345 m2.
Now we need to convert this value to units of hectares. Looking back in the preceding document in this series, we find that
1 hectare = 10 000 m2. Thus, 2 2
25345
m
=
25345
m
21
10 000
hectare
m
i
David W. Sabo (2003) Introduction to Unit Conversion: the SI Page 9 of 9
