Combining units by multiplication and division

Mass and volume are quantities (properties that can be measured). Each quantity has a symbol, a number and a unit.
 
For example, mass has the symbol m and may have units g or kg or mg.
The mass of a cup of water would be reported as m(H2O) = 250 g (quantity = number × unit)

A quantity is meaningless without its unit, because, for example, 250 g is a much smaller mass than 250 kg. Therefore ,you must use numbers together with their units in all calculations.

Units can be multiplied and divided just like numbers.
Before entering the numbers into your calculator, it is helpful to check whether the units of the quantities that you have substituted into a mathematical relationship cancel appropriately.   See the examples below where the relationship between amount, mass and moles is used.
 
Formula Example 1:
m=n
M
Units of quantities in formula:
g=mol
g mol–1

Formula Example 2:
m = n x M
Units of quantities in formula:
g = mol x g mol–1
  • Units to the same power in the numerator and denominator cancel.
    Thus g cancels as shown in the Example 1 at the right.
     
  • The sign of the exponent on a unit in the denominator changes when that unit is written on one line.
    Thus mol–1 becomes mol1 = mol in Example 1 at the right.  The reason for this is that exponents are subtracted upon division as shown below.
     
    1=mol0=mol0–(–1) = mol1 = mol
    mol–1mol–1
     
  • Exponents add when the same unit is multiplied by itself.
    Thus mol × mol–1 = mol0 = 1 in Example 2 at the right.
    The mol unit is said to cancel because any number (unit) raised to the 0 power equals 1.

Significant figures
If you are preparing a solution and need to write the concentration on label on the bottle, it is important to report the concentration to the appropriate number of significant figures.  This tells the person who will use the solution how much care was taken in preparing the solution.  Depending on how you prepared a solution which has a calculated concentration of 0.1016
, the appropriate label may be 0.1
(1 signifcant figure) OR 0.10
(2 significant figures) OR 0.102
(three significant figures) OR 0.1016
(4 significant figures).

The appropriate number of significant figures in a number calculated by multiplying or dividing measurements is equal to the number of significant figures in the most inaccurate measurement.  If the mass in grams used in creating the solution above was measured as 1.023 g and the volume was measured as 25 mL, the appropriate number of significant figures to report is two (0.10
).
Dr Sheila Woodgate
© The University of Auckland 2002