15.4 Subtraction Phrase Translation

Subtraction Phrase Translation catalogs the verbal phrasings that signal a subtraction operation, with special attention to the order in which quantities must be written, since subtraction is not commutative and several common phrasings, such as "less than," place the quantities in an order that does not match the order they appear in the sentence.

Basic Difference Phrasing

Difference of Two Quantities

The phrase "the difference of" two named quantities translates into the first-named quantity minus the second-named quantity, with the order of the phrase matching the order in the symbolic subtraction.

Phrases Meaning "Decreased"

Quantity Decreased by a Number

The phrase "a quantity decreased by" a number translates into that quantity minus the stated number, with the quantity written first, matching the order suggested by the phrase.

Number Subtracted from a Quantity

The phrase "a number subtracted from" a quantity translates into that quantity minus the number, with the quantity being subtracted from becoming the first term of the subtraction, even though the number being subtracted is named first in the sentence.

The Reversing Phrase "Less Than"

Quantity Less than Another Quantity

The phrase "a number less than" a given quantity translates into that quantity minus the number, with the order of subtraction reversed relative to the order the words appear in the sentence, since the quantity described as having something removed becomes the first term.

Phrases Describing Remaining or Excess Amounts

Amount Remaining after Removal

The phrase "the amount remaining after" some quantity is removed from another translates into the starting quantity minus the removed quantity, following the same order as "decreased by."

Excess of One Quantity over Another

The phrase "the excess of" one quantity "over" another translates into the first-named quantity minus the second-named quantity, describing how much larger the first quantity is compared to the second.

Deficit Relative to a Quantity

The phrase describing a deficit, or how much a quantity falls short of another, translates into the target quantity minus the smaller quantity, expressing the gap between the two as a positive difference when the target is indeed larger.

Change from an Initial Quantity

The phrase "the change from" an initial quantity "to" a final quantity translates into the final quantity minus the initial quantity, following the convention that a change is measured as the later value minus the earlier value.

Grouping and Order Considerations

Grouped Difference Phrase

When a verbal statement describes a difference that will itself be used as a single quantity in a further operation, such as "three times the difference of a number and two," the difference is enclosed in parentheses in the symbolic translation to preserve it as one grouped unit.

Subtraction Operand Order

Because subtraction is not commutative, the order of the two quantities in the symbolic translation must exactly match which quantity is being subtracted from which, and reversing this order without justification changes the value of the translated expression.

More-Than and Less-Than Order Comparison

Comparing "more than," which preserves the word order in its translation, against "less than," which reverses it, highlights that these two phrasings require opposite handling despite their superficially similar grammatical structure, making careful, case-by-case attention to each phrase essential.

Ambiguous Difference Phrase Recognition

Some verbal statements describing a difference do not clearly indicate which quantity is being subtracted from which, and in such ambiguous cases, the surrounding context of the statement must be used to determine the correct order before a translation can be written with confidence.