15.6 Division and Ratio Phrase Translation

Division and Ratio Phrase Translation catalogs the verbal phrasings that signal a division operation, with careful attention to which quantity becomes the numerator and which becomes the denominator, since division is not commutative and reversing the two operands changes the value entirely.

Basic Quotient Phrasing

Quotient of Two Quantities

The phrase "the quotient of" two named quantities translates into the first-named quantity divided by the second-named quantity, with the order of the phrase matching the order they are written in the symbolic quotient.

Quantity Divided by a Number

The phrase "a quantity divided by a number" translates directly into that quantity as the numerator and the stated number as the denominator, matching the word order in the sentence.

Number Divided by a Quantity

The phrase "a number divided by a quantity" reverses the roles: the stated number becomes the numerator and the quantity becomes the denominator, since the object of "divided by" is always the divisor.

Ratio and Rate Phrasing

Ratio of One Quantity to Another

The phrase "the ratio of" one quantity "to" another translates into the first-named quantity divided by the second-named quantity, following the standard convention that a ratio written "a to b" corresponds to the fraction a over b.

Quantity per Unit Quantity

The phrase describing a quantity "per" another unit quantity, such as "distance per hour," translates into the first quantity divided by the second, reflecting the standard meaning of "per" as division into equal units.

Named Fractional Divisions

One-Half of a Quantity

The phrase "one-half of a quantity" can be translated either as that quantity divided by two or, equivalently, as one-half multiplied by that quantity, since dividing by two and multiplying by one-half produce the identical value.

One-Third of a Quantity

The phrase "one-third of a quantity" similarly translates as that quantity divided by three, or equivalently as one-third multiplied by that quantity, following the same reasoning as one-half.

Reciprocal of a Quantity

The phrase "the reciprocal of a quantity" translates into one divided by that quantity, representing the multiplicative inverse of whatever value the quantity takes.

Grouping within Fraction Translation

Grouped Numerator Phrase

When the numerator of a translated fraction consists of more than one term, such as "the sum of a number and three, divided by five," the numerator's terms are grouped together above the fraction bar as a single unit before the division is applied.

Grouped Denominator Phrase

When the denominator of a translated fraction consists of more than one term, such as "a number divided by the sum of two and three," the denominator's terms are grouped together beneath the fraction bar as a single unit.

Final Structural Reminders

Division Operand Order

Because division is not commutative, correctly identifying which quantity plays the role of dividend and which plays the role of divisor is essential, and reversing this order without justification produces a translated expression with a different value than the original statement intended.

Fraction-Bar Scope in Translation

When translating a verbal statement into a fraction, the fraction bar must be understood to group everything in the numerator separately from everything in the denominator, exactly as though each were enclosed in its own parentheses, ensuring the full extent of each phrase is captured correctly on its proper side of the bar.