Four Topic Section:  Fraction, Fraction with Units, Fractions &  Ratios; and Proportionality forwards & backwards.

Section Pages

1. Two Term Ratios
2. New Fractions/Ratios from Old
3. Triple Term Ratios

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

Multiple Term Ratios - Three Term Ratios to be precise

We read the triple ratio  a : b :c as a to b to c. We further write 

a : b: c  ::  A: B: C

to  say two triple ratios  a : b: c  and   A: B: C are equal or equivalent when and only when 

The foregoing hold when and only when

The latter is equivalent to saying the three fractions 

all have the same value, call it k, which in turn is equivalent to 

A = k a,  B = k b and C = k c

for the common value k.  The foregoing shows the following statement (theorem) holds.

Theorem: Two triple ratios a : b: c  and   A: B: C are equivalent, that is,

a : b: c  ::  A: B: C

when and only when (i)  there is a single constant k such that 

A = k a,  B = k b and C = k c

 (ii) the three fractions 

A a

B b

and

C c

all have the same value; the value k above; 

(iii)

a b

=

A B

and

b c

=

B C

and (iv)  a:b  = A: B and b:c = B: C  (here we could write :: in place of = sign)

---

Remark:  The three fractions 

A a

B b

and

C k

all have the same value when and only when

A a

=

B b

=

C c

 

Exercise: Generalize the foregoing explanations to cover multiple term ratios  a:b:c:d: :: A:B:C:D  

Example:  If the sides of a triangle ABC have lengths a: b: c = 3: 4: 5 then the triangle ABC is similar to the 3-4-5 right triangle. For further examples of multiple term ratios, see the explanation of similar triangles in the geometry before coordinates site area.

Further examples of multiple (term) ratios are provided by  mixtures as is or scaled-up in cooking (cake production), in construction (concrete mixing) and in chemical reactions. 

Remark - A related concepts in advanced mathematics.  In the coordinization of rays and lines in  projective geometry,. two non-zero points (x,y,z) and (X,Y, Z) belong to the same ray  when and only when (X,Y,Z) = (kx, ky, kz) for some positive real number k while  twonon-zero points (x,y,z) and (X,Y,Z) belong to the same line  when and only when (X,Y,Z) = (kx, ky, kz) for some non-zero real number k.  That  is similar to the use of coordinates in projective geometry.

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