15.10 Relative Quantity Relationships

Relative Quantity Relationships covers how a verbal statement describes one quantity in terms of a fixed amount or proportion relative to another, distinguishing additive comparisons, which describe a fixed amount more or less, from multiplicative comparisons, which describe a scaled or fractional relationship.

Additive Comparisons by a Fixed Amount

Quantity Greater by a Fixed Amount

A statement describing one quantity as greater than a reference quantity by a fixed amount translates into the reference quantity with that fixed amount added to it, representing the described quantity as the reference plus a constant.

Quantity Smaller by a Fixed Amount

A statement describing one quantity as smaller than a reference quantity by a fixed amount translates into the reference quantity with that fixed amount subtracted from it, representing the described quantity as the reference minus a constant.

Multiplicative Comparisons by a Factor

Quantity Equal to a Multiple of Another

A statement describing one quantity as a stated multiple of a reference quantity translates into the reference quantity multiplied by the stated whole number, representing the described quantity as a scaled version of the reference.

Quantity Greater by a Multiplicative Factor

A statement describing one quantity as greater than a reference quantity by a percentage or proportional factor translates into the reference quantity plus that same proportion of the reference quantity, combining into a single scaled expression.

Quantity Smaller by a Fractional Factor

A statement describing one quantity as smaller than a reference quantity by a stated fraction translates into the reference quantity minus that fraction of the reference quantity, combining into a single scaled expression representing the reduced amount.

A Quantity Compared to Itself

Quantity Increased by a Fraction of Itself

A statement describing a quantity increased by a fraction of itself, rather than by a fraction of some other reference quantity, translates the same way as a general fractional increase, but with the same variable serving as both the base quantity and the reference for the fractional amount.

Quantity Decreased by a Fraction of Itself

A statement describing a quantity decreased by a fraction of itself translates similarly, subtracting the stated fraction of the same variable from itself, using that one variable both as the base amount and the reference for the fractional decrease.

Which Quantity Serves as the Reference

Reference-to-Result Quantity Translation

Every relative comparison statement identifies one quantity as the reference and describes a second, resulting quantity in terms of that reference; correctly translating the statement requires clearly identifying which of the two named quantities is the fixed reference and which is the quantity being described relative to it.

Reference Quantity Order

Because comparisons like "greater than" and "less than" name the reference quantity after describing the amount of difference, careful attention to sentence structure, rather than simple word order, is needed to determine which quantity the described amount is measured against.

Distinguishing the Two Types of Comparison

Additive and Multiplicative Comparison Distinction

An additive comparison describes a difference in terms of a fixed amount added to or subtracted from the reference quantity, producing an expression like the reference plus or minus a constant, while a multiplicative comparison describes a difference in terms of a proportion or scale of the reference quantity itself, producing an expression like the reference multiplied by a factor; recognizing which type of comparison a statement describes is essential to choosing the correct operation during translation.