Quantitative · 3 min read
Quantitative · Number Sense
The Quantitative section measures mathematical reasoning and problem solving at the level taught in grades 5 through 7. It appears twice on the test, as two separate 25-question sections of 30 minutes each, and both sections draw on the same content: arithmetic and number sense, pre-algebra, geometry, measurement, and data analysis. Calculators are not permitted, so accurate mental math, clean scratch work, and estimation are essential skills.
Number sense is the understanding of numbers, their relationships, and how they are used. Strong number sense is the foundation for every other quantitative skill on the test.
Place Value
In the base-ten system, each place represents a power of ten, and the value of a digit depends on its position. In the number 5,847,362:
| Millions | HundredThousands | TenThousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|---|
| 5(5000000) , , | 8(800000) , | 4(40000) , | 7(7000) , | 3(300) | 6(60) | 2(2) |
Place values continue to the right of the decimal point: tenths (0.1), hundredths (0.01), and thousandths (0.001).
Comparing, Ordering, and Rounding
To compare whole numbers, first count the digits: a number with more digits is greater. If the digit counts are equal, compare digit by digit from left to right. The symbols are > (greater than), < (less than), = (equal to), ≥ (greater than or equal to), and ≤ (less than or equal to).
Order 4,521, 4,512, 4,251, 4,125 from least to greatest.
Round 12.678 to the nearest tenth: the digit after 6 is 7 (5 or more), so round up → 12.7
To round, look only at the digit immediately to the right of the target place: 5 or greater rounds up; 4 or less rounds down.
Types of Numbers
| Type | Defi nition | Examples | ||
|---|---|---|---|---|
| Wholenumbers | Zeroandthecountingnumbers;nofractionsordecimals | 0 1 2 3 4 , , , , , … | ||
| Integers | Wholenumbersandtheirnegatives | −3 −2 −1 0 1 2 3 …, , , , , , , , … | ||
| Rationalnumbers Irrationalnumbers Evennumbers | Numbersthatcanbewrittenasafractionoftwointegers Numberswhosedecimalsneverendandneverrepeat Integersdivisibleby2 | 3/4 −2 0 .25 0 .333 , , , … √2 √3 π , , 0 2 4 6 8 , , , , , … | 1 |
| Type | Defi nition | Examples |
|---|---|---|
| Oddnumbers | Integersnotdivisibleby2 | 1 3 5 7 9 , , , , , … |
| Primenumbers | Greaterthan1withexactlytwofactors 1anditself , | 2 3 5 7 11 13 17 19 23 29 31 , , , , , , , , , , , … |
| Compositenumbers | Greaterthan1withmorethantwofactors | 4 6 8 9 10 12 , , , , , , … |
The number 1 is neither prime nor composite, and 2 is the only even prime number. Zero is an even number and a whole number, but it is neither positive nor negative.
Absolute Value and the Number Line
A number line shows numbers in order: values increase moving right and decrease moving left, and equal spaces must represent equal amounts. The absolute value of a number, written |x|, is its distance from zero, so it is never negative: |7| = 7, |−7| = 7, |0| = 0, and |−15| = 15.
Exponents, Squares, and Square Roots
An exponent shows how many times a base is multiplied by itself: 2⁴ = 2 × 2 × 2 × 2 = 16 and 3³ = 27. Any nonzero number raised to the power 0 equals 1 (5⁰ = 1), and any number raised to the power 1 equals itself (7¹ = 7). The square root undoes squaring: √25 = 5 because 5² = 25.
| ExponentRule(samebase) | Rule | Example |
|---|---|---|
| Multiplyingpowers | aᵐ×aⁿ=aᵐ⁺ⁿ | 2³×2² =2⁵=32 |
| Dividingpowers | aᵐ÷aⁿ=aᵐ⁻ⁿ | 5⁴÷5² =5² =25 |
| Powerofapower | (aᵐ)ⁿ=aᵐˣⁿ | (3²)³ =3⁶=729 |
Perfectsquarestomemorize(1²Through15²)
100 121 144 169 196 225
Scientific Notation
Scientific notation writes a number as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. Large numbers have positive exponents (5,400,000 = 5.4 × 10⁶); small numbers have negative exponents (0.00032 = 3.2 × 10⁻⁴). To convert back, move the decimal point right for a positive exponent and left for a negative exponent.
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