Ratio and Proportion

Multiplying the stated ingredients of a recipe by the same whole number makes a larger quantity able to serve more people. The same process is at the heart of several useful skills. The basic notion relies on division, as does solution to the herring puzzle. All that is needed is to write the words in the form of an equation:

a)           One and a half [herring] = One and a half [pennies]

This is the same as the numeric equivalent:

b)           1 and 1/2 [herring] = 1 and 1/2 [pennies]

Likewise, this is easily converted to a simple form. One way is to create the two equations that mean the same thing as b), namely:

c1)           1 [herring] = 1 [pennies]

c2)           1/2 [herring] = 1/2 [pennies]

From c1) the answer is clear: d)           6 [herring] = 6 [pennies].

An interesting variation is to go straight from b) to:

e)           3/2 [herring] = 3/2 [pennies]

The division of both sides of e) by just its left side gives:

f)           1 = 1 [pennies]/[herring]

What f) does is give a conversion factor, something that is equal to 1 but has different units on the other side of the equals sign. Here are some familiar conversion factors:

g)           1 = 25 [pennies]/[quarter] ... (from 1 [quarter] = 25 [pennies])

h)           1 = 2000 [pounds]/[ton] ... (from 1 [ton] = 2000 [pounds])

Conversion factors come from simple cancellation, beginning from an equation. To use the one we found, the factor in f) means that a herring costs a penny. Hence two cost two-cents; six, six-cents; and so forth.

Most of the ideas above are starting steps for topics in engineering, science, computing and mathematics. [*Items below indicate planned additions and are not yet in place.] Here are some of the terms that relate to ratios in those fields: