Converse Equation Styles

In terms of word-problem solving, any assertion or statement of fact translates into an equation. Since equations can be formed by stating that two expressions are equivalent, and the formation of expressions has been covered, the discussion of equations can be characterized by how equivalence can be stated.

Depending on the style of input, equations can be given in a few different ways:

In Algebraic style, equations can be literally created by joining two mathematical expressions with an equals sign:

e.g. Alpha + beta = 3 x 7

In Simple Translation, equations are usually created using variations of simple synonyms or idioms for the equals sign (e.g. "equals", "is", "are", "was", "were", "make", "makes up", "gives", "yields", "results in"):

e.g. Alpha added to beta is 21.
e.g. Alpha added to beta equals 21.
e.g. Alpha added to beta makes 21.
e.g. Alpha added to beta results in 21.

In Natural Language, equations can come from sentences with almost any verb phrase based on the verb "to be":

e.g. The sum of alpha and beta would have been 21.
e.g. Alpha and beta must be 21.
e.g. Alpha added to beta should always be 21.

e.g. 21 is how many alphas and betas there should be.

e.g. There are 21 alphas and betas.
e.g. There must be as many alphas as betas.
e.g. There were seven times as many alphas as betas.

Statements of fact are not the only kind of natural language that create equations... In Natural Language, non-trivial questions can produce equations (themselves "statements-of-fact") that must be solved to answer the question.

> How much less than e is Pi over two?

   Let 'A' stand for "ANSWER"

   (Find A)

   Solving for A:

                   3.14159
     2.71828 - A = -------
                      2

                               3.14159
             - A = - 2.71828 + -------
                                  2

                               3.14159
         A (- 1) = - 2.71828 + -------
                                  2

                             3.14159
               A = 2.71828 - -------
                                2

       A = 1.147485

   1.147485 is THE ANSWER.

> There are how many times as many alphas as betas?

   Let 'N' stand for "NUMBER"

   (Find N)

   Let 'A' stand for "ALPHAS"

   Let 'B' stand for "BETAS"

   So...
                   A
               N = -
                   B

> How many unit cm longer than 6 unit inches is half a unit yard?

   Let 'N' stand for "NUMBER"

   (Find N)

   Let 'C' stand for "CENTIMETER"

   Let 'I' stand for "INCHES"

   Let 'Y' stand for "YARD"

   So...
           6   N    1
           - + - = ---
           I   C   2 Y

Regardless of their complexity or origin, the resulting equations are statements-of-fact about variables.