5.1 Number & Quantity / Modeling

Why Number & Quantity Matters

Number and Quantity (N-Q) carries only 4%–10% of Algebra I Regents credits, but its ideas are woven into modeling questions across every part of the exam. When a problem asks whether an answer is reasonable, what units a result should carry, or whether a sum must be rational, it is testing N-Q. Mastering it protects easy credits and stops careless unit errors on harder problems.

The real number system is everything you plot on a number line. It splits cleanly into two camps: rational and irrational numbers. Within the rationals sit the integers (…, −2, −1, 0, 1, 2, …) and within those the whole numbers and natural numbers. Knowing where a value lives in this hierarchy is the foundation for every closure question.

Rational vs Irrational Numbers

A rational number can be written as a fraction p/q of two integers (q ≠ 0). Its decimal form either terminates (0.75) or repeats (0.333…). Integers, fractions, and percents are all rational.

An irrational number cannot be written as such a fraction. Its decimal never ends and never repeats. Classic examples are √2 ≈ 1.41421…, π ≈ 3.14159…, and √3. A square root is irrational only when the radicand is not a perfect square: √9 = 3 is rational, but √10 is irrational.

Closure: What Happens When You Combine Them

Regents items love asking what kind of number results from an operation. Memorize these rules:

• rational + rational = rational (e.g., 1/2 + 1/3 = 5/6).
• rational × rational = rational.
• nonzero rational + irrational = irrational (e.g., 3 + √2 cannot become a clean fraction).
• nonzero rational × irrational = irrational (e.g., 2√5 is irrational).

Worked example. Is the sum 4 + √9 rational or irrational? First simplify the radical: √9 = 3, which is rational. So 4 + 3 = 7, a rational number. The trap is treating every square root as irrational — always simplify first. By contrast, 4 + √8 is irrational because √8 = 2√2 stays irrational.

One subtlety: the “nonzero” condition matters. 0 × √2 = 0, which is rational, and irrational + irrational is not always irrational — for instance √2 + (−√2) = 0. Regents items usually use a clean nonzero rational with an irrational, where the result is reliably irrational, but watch for these edge cases in the answer choices.

Units and Dimensional Analysis

Dimensional analysis converts units by multiplying a quantity by unit ratios equal to 1, arranged so unwanted units cancel. Because each ratio equals 1, the quantity's value does not change — only how it is labeled.

Worked example. A car travels at 60 miles per hour. How many feet per second is that? Use 1 mile = 5,280 ft and 1 hour = 3,600 s:

60 mi/hr × (5,280 ft / 1 mi) × (1 hr / 3,600 s) = (60 × 5,280) / 3,600 ft/s = 316,800 / 3,600 = 88 ft/s.

The “mi” cancels with “mi” and “hr” cancels with “hr,” leaving ft/s. Set conversions up so the unit you want to remove is on the opposite side of the fraction bar from where it started.

The same technique handles two-step conversions and area or volume units. To convert 3 square yards to square feet, square the linear ratio: 3 yd² × (3 ft / 1 yd)² = 3 × 9 = 27 ft². Forgetting to square (or cube) the conversion factor for area and volume is one of the most common N-Q errors on the exam.

Accuracy, Precision, and Interpreting Quantities

The appropriate level of accuracy depends on the data, not on your calculator's display. If lengths were measured to the nearest tenth of a meter, reporting an area as 12.4863 m² overstates precision — round to a sensible 12.5 m². Reporting too many digits is a common Regents mistake.

In modeling contexts, every quantity has a unit and a meaning. If t is “hours since noon” and C(t) is “cost in dollars,” then C(3) = 47 means “3 hours after noon the cost is $47.” Always:

1. Define the variable and its unit before computing.
2. Track units through each step so the final unit is correct.
3. Check reasonableness — a person's height cannot be 200 ft; a sale price cannot exceed the original.

Rates are the structural heart of word problems: a rate like “$0.12 per kilowatt-hour” becomes the slope of a linear cost model, and a starting fee becomes the y-intercept. Recognizing the rate and the fixed amount turns a word problem into y = (rate)x + (start). The unit of the rate is a free hint about the structure: “dollars per hour” must multiply hours, while a plain “dollars” amount is added once. Reading units carefully tells you which number is the slope and which is the constant before you even pick variables.