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? 

The first part of this lesson  shows how to add subtract numerical multiplies of them in a way that resembles the forward and backward use of a distributive law for counting and measuring.   Carries and borrows in addition and subtraction provide a unit-free form of conversion. In counting and arithmetic, conversions are further present in expressing counts or numbers in groups of ones, tens, hundreds and so on, and in adding, subtracting and multiplying such counts.  The second part of this lesson talks about changing and converting the units used to measure or keep track of quantities.  Your fortune may be measured in pennies. or in dollars? The third part show how to form products of quantities. The operations here are similar to work with monomials where the product of 4xy with  5 xy³z is  20 x²y⁴z. Instead of using letters x, y and z, we use measurement units and counting units as well, and could be more meaningful for students.

A.  Product of  Numbers with Units. 

A quantity in the first instance is given by a number of units, a number times a unit, that is a product of a number with a unit. In measurement, such products appear to count many units are present. The count may be fractional. The count is called a coefficient. 

Three times the amount  7 dishes is  7 dishes + 7 dishes + 7 dishes.  So we write

3 x  (7 dishes)  = (3 x 7) dishes = 21 dishes.

Here multiplication by 3 is just repeated addition. 

In general, if a and b are whole number, fraction or decimals, we assume

 a (b units)  = (ab) units

and use this equality to compute the left hand side a (b units)  -- read as a times b units. 

Moreover, if a is nonzero, we assume

For instance a quarter of 12 cups, all identical, is given by 3 cups where 3 is a quarter of 12.  In symbols

(¼) 12 cups =  (12/4) cups = 3 cups.

Like wise 12 cups divided by 4 is again three cups:

Division by four (4) is the same as multiplying by a quarter (¼).

B. Changing and Converting Units

Recall   1 metre = 100 centimetres and  1 decimetre = 10 centimetres.  Here the unlike units metre, centimetre and decimetre are of the same type [L] for length.  So 

 9 metres + 8 decimetres + 3 centimetres 

=  9 x 100 centimetres + 8 x 10 centimetres + 13 centimetres 
= (900 + 80 + 3) centimetres
 = 983 centimetres.

Alternatively  983 centimetres = 98.3 decimetres = 9.83 metres.

In calculations involving lengths, we can replace 

•  metres by 100 centimetres or 10 decimetres; 
• decimeters by 0.1 metres or 10 centimetres; and 
• centimetres by 0.01 metres or 0.1 decimetres.

Here I have use decimal notation for the fractions (1/10) and (1/100) for convenience while typing this page. That is, we could use fraction notation and mixed number notation as well in the foregoing. 

Similarly, we can use the equations

1 minute = 60 seconds, 
1 hour = 60 minutes,
1 day = 24 hours

to go back and forward between unlike units of measurement of time [T]. 

Further examples could follow using units of mass and force or weight.  

C. Products of Quantities 

The operations here are similar to work with monomials where the product of 4xy with  5 xy³z is  20 x²y⁴z. Instead of using letters x, y and z, we use measurement units and counting units as well.  The monomials below involve units of measure or counting in place of "variables" x, y, z and so on. 

The following examples illustrate the multiplicative computation conventions for units. 

(10 cm)(5 sec) = 10x5 (cm)sec = 50  cm sec

(3.4 cm)(2.0 kg) = 6.8  cm kg 

(4 hours)(3 hours) = 12 hour²

So we take 

(a unit₁)(b unit₂) = (ab) (unit₁)(unit₂)

We take or declare a product of units  (unit₁)(unit₂) to be new unit, a compound unit. 

All the foregoing considerations with simple units of measurement work with compound units.   Division of units yields more compound units and more compound fractions. Examples follow to show how.  The appearance of units alone, in products and in quotients (fractions) in the description of rates and further proportionality constants - exact or approximate - will provide a context for these operations. Have patience.

Here we multiply the units and their numerical coefficients separately. Unit multiplication behaves like monomials. We take

hour²hour⁵ = hour^{2+7  =} hour⁷

Saying how to write the product of  units or quantities of like and unlike kind defines the operation. The operation here is a notational convenience. In general we take

(a unit^m)(b unitⁿ) = (ab) unit^m+n

So the multiplication rule for quantities  is to add exponents and add coefficient of identical units.    

Operations with units and their numerical coefficient represent exercises on paper  in the first instance, but again  they are useful later in the discussion of rates,  proportionality constants and computations with physical quantities. 

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