Section Topics

 Fraction, Fraction with Units, Fractions &  Ratios; and Proportionality forwards & backwards.

Section Pages

1. Addition of Units
2.  Units and Equal Signs
3. Products with Units
2. Fractions with Units
4. Simplification of Fractions
5.  Fraction Reciprocals & Division
6. Converting Units in Fractions

Would you like to show yourself or others how to be  algebra power users? 

Converting or Changing Units in Fractions- Equivalent Rates 

A physical quantity may be described in different units.  Here we show how speed (it is written as a fraction with units) written in distance per hour may be expressed in distance per minute.  The trick of multiplying by a fraction with value 1 (a fraction with units which is equivalent to one) appears here. 

Fractions with units appear as proportionality constants or rates in daily life, chemistry and physics.. These proportionality constants or rates measure a relationship between two physical quantities.  The form of these fractions or rates depends on the choice of units. 

The measure or unit of slope, speed and velocity [may be given by the ratio [ km/min] or km per minute, or km per one sixtieth of an hour. But other units (or ratio of units) are possible. The question becomes: what units do you like for the expression of these quantities. It is possible to change units of time and distance.

Assumptions: When working with units,  Multiplying by the number 1 does not affect the value of an expression.  

 The number 1 can be written in many ways. Some, not all, are helpful, others not or inconvenient.  These different ways can sometimes help in changing the units used for the expression of a quantity. A few slope or velocity based examples follow. 

In the following,  [60min/1hr] = 1  since  60 minutes = 1 hour.

m1 = - 4 3 km min = - 4 3 km min ×1 = - 4 3 km min × 60 min hr = - 4 3 · 60 1 km min min hr = 4 1 · 20 1 km hr = -80 km hr m2 = 8 3 km min × 60min hr = 160 km hr m3 = 0 m4 = - 1 2 km min × 60min hr = -30 km hr m5 = - 3 2 km min × 60min hr = -90 km hr

Assumption: Multiplying by 1 = [60min/hr] does not change the speed or velocity m, but it does help to change the units from minutes to hours. Note to do the reverse change, multiply by

1 = hr 60min

instead. Note that care must be taken in selecting the ratio of units whose value is 1. A faulty choice will introduce more units instead of permitting some to cancel.

Remark.   An alternate method is to substitute 1 min = [1/60] hr into an expression. For example

m2 = 8 3 km min = 8 3   km 1 60 hr = 8 3 · 60 1 · km hr = 160 km hr

as before. Substitution replaces one unit by an expression for it in terms of another. With this method, there is little or no hazard of introducing units that don't cancel, but the algebra or arithmetic requires a little more thought. The choice of unit conversion method depends on where you would like to do the work or reasoning.

Remark (November 2009):   In the foregoing calculation practices, there are likely some hidden nuances or subtleties that should be exposed and explicitly cover.

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