15.9 Multi-Operation Verbal Expressions

Multi-Operation Verbal Expressions covers how a verbal statement describing more than one operation at once is broken into segments, translated from the correct starting point, and reassembled into a single symbolic expression that correctly reflects which operation applies first and which applies last.

Breaking the Statement into Pieces

Operation Phrase Segmentation

A verbal statement describing several operations is first divided into smaller phrases, each corresponding to a single operation, using punctuation and connecting words such as "and," "increased by," or "the result" as natural breakpoints between segments.

Main Operation Identification

Once the statement is segmented, the main, outermost operation is identified, typically the operation described last or the one that acts on the entire result of everything described before it.

Inner Operation Identification

Any operation described within a smaller segment, acting only on a portion of the full statement, is identified as an inner operation, to be resolved before it becomes an operand of the main operation.

Common Two-Operation Patterns

Sum of a Product and a Constant

A statement describing a product added to a stated number translates the inner product first, then adds the constant afterward, with no grouping required since the multiplication naturally binds more tightly than the following addition.

Difference between a Product and a Constant

A statement describing a product with a stated number subtracted, or subtracted from, translates the inner product first, then applies the subtraction in the order indicated by the specific phrasing used.

Product of a Number and a Sum

A statement describing a number multiplied by an entire described sum groups that sum in parentheses first, then applies the multiplication to the whole grouped quantity as the main operation.

Quotient of Two Compound Quantities

A statement describing the quotient of two compound quantities groups each described sum or difference separately, one as the numerator and one as the denominator, before the division is applied between the two completed groups.

Power Applied to a Compound Quantity

A statement describing a power applied to an entire described sum or difference groups that sum or difference first, then applies the exponent to the whole grouped quantity as the main operation.

Root Applied to a Compound Quantity

A statement describing a root applied to an entire described sum or difference places that sum or difference beneath a single radical bar, with the root applied to the fully grouped radicand.

Handling Three or More Layers

Three-Level Verbal Expression Structure

A verbal statement describing three nested operations, such as a root of a sum that itself contains a product, is resolved by identifying the innermost operation first, translating it, then wrapping that result in the middle operation's structure, and finally applying the outermost operation to the entire combined result.

Two Directions of Translating a Layered Statement

Translation from Outside to Inside

One approach begins by identifying the main, outermost operation, leaving a placeholder for whatever falls inside it, and then works inward to fill in that placeholder with the correctly translated inner phrase.

Translation from Inner Phrases to Outer Structure

The opposite approach begins by translating the smallest, most deeply nested phrases first into their own symbolic pieces, then assembles those pieces upward into the larger structure indicated by the surrounding operations; either direction, carried out carefully, produces the same correct final translation.