General Operations with Fractions
Examples of Algebra of Units Calculations
General operations with Fractions
Be sure you understand the principles of multiplying and dividing fractions:
Multiplication of two fractions:
(a / b) * (c / d) = (a * c) / (b * d)
Multiplication of a fraction by a number:
c is the same thing as c / 1, since division by 1 doesn't change a quantity. Note also that multiplication by 1 doesn't change a quantity so that b * 1 = b; this fact is used below:
(a / b) * c = (a / b) * (c / 1) = (a * c) / ( b * 1) = a * c / b
Division of a fraction by a number:
(a / b) / c means (a / b) divided by c.
Division by c is the same as multiplication by 1 / c. Thus
(a / b) / c = (a / b) * (1 / c) = (a * 1) / (b * c) = a / (b * c)
or more briefly
Division of a fraction by a fraction:
(a / b) / (c / d) means (a / b) divided by (c / d).
Division by (c / d) is the same as multiplication by d / c.
Thus
(a / b) / (c / d) = (a / b) * (d / c) = (a * d) / (b * c)
or more simply
Examples of calculations using units
These examples are given in typewriter notation, then in standard notation.
#1
a / b * c = a * b / c, so if a is in meters, b in seconds and c in seconds we have
meters / second * seconds = (meters * second) / second = meters * (second/second) = meters
The meaning:
meters / second is the standard unit of velocity; when multiplied by a time interval in seconds we obtain a result which is measured in meters
The units of the calculation [ average velocity * time interval ] turn out to be meters.
#2
(a / b) / c = a / (b * c), so if a is in meters, b and c both in seconds the we have
(meters / second) / (seconds) = (meters / second) * (1 / second) = meters / (second * second) = meters / second^2
The meaning:
meters / second is the standard unit of velocity; when dividedby a time interval in seconds we obtain a result which is measured in meters / second^2.
The units of the calculation [ change in velocity / change in clock time ] are (meters / second) / seconds = meters / second^2.
Meters / second^2 is the unit of acceleration, or rate of change of velocity.
#3
Newtons·meter^2/kilogram^2·(kilogram·kilogram/meter^2) = Newtons·meter^2·kilogram^2/(kilogram^2·meter^2) = Newtons·(meter^2/meter^2)·(kilogram^2/kilogram^2) = Newtons
#4
Newtons·meter^2/kilogram^2·(kilogram·kilogram/meter) = Newtons·meter^2·kilogram^2/(kilogram^2·meter) = Newtons·(meter^2/meter)·(kilogram^2/kilogram^2) = Newton * meters
#5
(Newtons / (amp * meter) )* meter^2 = Newtons * meter^2 / (amp * meter) = (Newtons / amp) * (meter^2 / meter) = (Newtons / amp) * meter = Newton * meters / amp
#6
Newtons/(coulombs/second·meter)·meter^2 = Newtons·meter^2/(coulombs/second·meter) = Newtons·(meter^2/meter)·(seconds/coulomb) = Newtons·seconds/coulombs·meter = Newtons·seconds·meters/coulomb
#7
kilograms * (meters / second) / (seconds) = kilograms * (meters / second) * (1 / second) = kilograms * meters / (seconds * seconds) = kilogram * meters / second^2
