Quantity Words as Special Nouns

Quantity words are special nouns that refer to a numeric quantity in a generic, non-descriptive way.  Common examples include "number", "amount", "fraction", "quantity", "part", "portion", and "multiple".

When a noun phrase consists of only a quantity word, it is treated like a regular noun...  AutoMathic selects a variable to stand for it and continues normally:

> Half of what amount added to three gives twelve?

   Let 'A' stand for "AMOUNT"

   (Find A)

   Solving for A:

               A                     
3 + - = 12
2
                                   
A = 18

       A = 18

   18 is THE AMOUNT.

However, when a quantity word appears within a larger noun phrase, it is handled differently than a regular noun.  Note the difference between the structure of the following two translated equations, despite the fact that the only difference between them is that the second example uses a quantity word, "portion", instead of a regular noun, "length":

> The length of a beam is 6.

   Let 'L' stand for "LENGTH"

   Let 'B' stand for "BEAM"

   So...
               L
               - = 6
               B

> The portion of a beam is 6.

   Let 'P' stand for "PORTION"

   Let 'B' stand for "BEAM"

   So...
             P B = 6

"Percentage" is another common quantity word that can have a special meaning to AutoMathic.  Just like the other quantity words, "percentage" is treated like a regular noun when used by itself:

> The percentage is 33 and a third.

   Let 'P' stand for "PERCENTAGE"

                        1
               P = 33 + -
                        3

       P = 33.3333333333333 (or 100 / 3, or 33 & 1 / 3)

However, it too receives special treatment when used within a larger noun phrase.  It has the special mathematical meaning of dividing by a hundred and multiplying:

> What percentage of 73 is 25?

   Let 'P' stand for "PERCENTAGE"

   (Find P)

   Solving for P:

            73 P
            ---- = 25
            100

                   2500
             P = ----
                    73

       P = 34.2465753424658

   34.2465753424658 is THE PERCENTAGE.

Other than mathematical constants (e.g.  Pi, e, Golden Ratio, C), quantity words are the only class of nouns that have special meaning to AutoMathic.  All other nouns are essentially interpreted as-is.

Implicit Nouns

In general, starting a noun or noun phrase with "what" or "how much" always asks a question, and will cause that noun phrase's variable to be added to the list of things to solve for:

> Half of the total is what value?

   Let 'T' stand for "TOTAL"

   Let 'V' stand for "VALUE"

   (Find V)

   So...
               T
- = V
               2

However, sometimes input to AutoMathic uses phrases like "how much", "how many", or "what" in a generic way, without following it with a descriptive, explicit noun or noun phrase:

e.g.   How much added to 12 gives 21?
e.g.   Six times how many gives a dozen?
e.g.   What added to 12 gives 21?

The generic use of those kinds of phrases will be referred to as "implicit nouns" because they imply specific nouns where none exist.  When AutoMathic encounters an implicit noun that is part of a larger mathematical expression (and therefore must be represented as a variable) it will introduce an appropriately-named quantity word as a variable to use as a placeholder in the expression:

Implicit Noun Variable Example
"how much" "answer" How much added to 12 gives 21?
"how much." "amount" 21 is 12 added to how much?
"how many" "number" Six times how many is a dozen?
"what" "answer" What added to 12 gives 21?
"what." "amount" 21 is 12 added to what?

Since those implicit noun phrases always ask a question too, the automatically-generated variable is added to the list of things to solve for:

> Six times how many is a dozen?

   Let 'N' stand for "NUMBER"

   (Find N)

   Solving for N:

             6 N = 12

           N (6) = 12

              N = 2

       N = 2

   2 is THE NUMBER.

As will be shown in the "Mathematical Expressions" section, an implicit noun like "how many" that is not part of a larger expression is ignored during translation because it asks a "trivial" question...  Such standalone implicit nouns will not generate a placeholder variable:

Input Translation
How much is 12 from 21? 12 from 21
6 times a dozen is how many? 6 times a dozen
What is 12 added to 21? 12 added to 21