Quantitative (Math) · 3 min read
Quantitative (Math) · Numbers and Operations
The Quantitative section of the Upper Level SSAT consists of two separately timed sections of 25 questions each, for 50 math questions in total, with 30 minutes per section. Questions test the concepts taught through the middle grades and early high school: arithmetic, number properties, ratios and percents, algebra, geometry, and data and probability. Calculators are not permitted, so accurate computation by hand and quick recall of facts and formulas are essential.
Types of Numbers
Each set of numbers contains the one before it. Natural numbers are the counting numbers 1, 2, 3, 4, …. Whole numbers add zero: 0, 1, 2, 3, …. Integers add the negatives: … −3, −2, −1, 0, 1, 2, 3 …, with no fractional or decimal parts. Rational numbers can be written as a fraction a/b of two integers with b ≠ 0; they include all integers, fractions, and terminating or repeating decimals. Irrational numbers cannot be written as fractions; their decimals neither terminate nor repeat (π, √2,
Properties of Operations
| Property | Rule | WhatItMeans | |
|---|---|---|---|
| Commutative | a+b=b+a;a×b=b×a | Orderdoesnotmatterforadditionandmultiplication | |
| Associative | (a+b)+c=a+(b+c);(a×b)×c=a×(b× c) | (notforsubtractionordivision) Groupingdoesnotchangeasumoraproduct |
| Property | Rule | WhatItMeans |
|---|---|---|
| Distributive | a(b+c)=ab+ac | Linksmultiplicationandaddition;essentialfor |
| simplifyingandsolving | ||
| Identity | a+0=a;a×1=a | 0istheadditiveidentityand1isthemultiplicative |
| identity | ||
| Inverse | a + ( −a) = 0; a × (1/a) = 1 | Anumberanditsoppositesumto0;anumberandits |
| reciprocalmultiplyto1 |
Order of Operations
➜
➜
➜
Grouping symbols first
Powers and roots
Left to right
Left to right
3 + 4 × 2² − (6 ÷ 2)
Parentheses: 6 ÷ 2 = 3 → 3 + 4 × 2² − 3
Exponents: 2² = 4 → 3 + 4 × 4 − 3
Multiplication: 4 × 4 = 16 → 3 + 16 − 3
Left to right: 3 + 16 = 19, 19 − 3 = 16
Factors, Multiples, and Primes
| Term | Meaning | Example |
|---|---|---|
| Factor | Dividesevenlyintoanumber withnoremainder , | Factorsof12:1 2 3 4 6 12 , , , , , |
| Multiple | Theproductofanumberandanyinteger | Multiplesof4:4 8 12 16 20 , , , , … |
| Primenumber | Exactlytwofactors:1anditself(2istheonlyeven | 2 3 5 7 11 13 17 19 23 29 , , , , , , , , , |
| prime;1isnotprime) | ||
| Compositenumber | Morethantwofactors;canbewrittenasaproduct | 4 6 8 9 10 12 , , , , , … |
| ofprimes | ||
| Primefactorization | Anumberwrittenasaproductofprimes | 60=2×2×3×5=2²×3×5 |
| Greatestcommonfactor(GCF) | Multiplytheprimefactorsthenumbersshare using , | 24=2³×3 36=2²×3² soGCF=2²×3 , , |
| thelowestpowers | =12 | |
| Leastcommonmultiple(LCM) | Multiplythehighestpowerofeveryprimethat | LCMof24and36=2³×3² =72 |
| appears |
| DivisibleBy | Rule | ||
|---|---|---|---|
| 2 | Thenumberendsin0 2 4 6 or8 , , , , | ||
| 3 4 5 | Thesumofthedigitsisdivisibleby3 Thelasttwodigitsformanumberdivisibleby4 Thenumberendsin0or5 | 1 |
| DivisibleBy | Rule |
|---|---|
| 6 | Thenumberisdivisiblebyboth2and3 |
| 8 | Thelastthreedigitsformanumberdivisibleby8 |
| 9 | Thesumofthedigitsisdivisibleby9 |
| 10 | Thenumberendsin0 |
Absolute Value
The absolute value of a number is its distance from zero on the number line, so it is never negative: |5| = 5 and |−5| = 5. When an equation involves absolute value, consider both the positive and negative cases: if |x| = 7, then x = 7 or x = −7.
Exponents and Roots
An exponent shows repeated multiplication: in 5³, 5 is the base and 3 is the exponent, so 5³ = 5 × 5 × 5 = 125.
| Law | Form | Example |
|---|---|---|
| Productrule | aᵐ×aⁿ=aᵐ⁺ⁿ | 2³×2² =2⁵=32 |
| Quotientrule | aᵐ÷aⁿ=aᵐ⁻ⁿ | 5⁴÷5² =5² =25 |
| Powerrule | (aᵐ)ⁿ=aᵐⁿ | (3²)² =3⁴=81 |
| Zeroexponent | a⁰=1(a≠0) | 7⁰=1 |
| Negativeexponent | a⁻ⁿ=1/aⁿ | 2⁻³ =1/8 |
| Producttoapower | (ab)ⁿ=aⁿbⁿ | (2x)³ =8x³ |
| Quotienttoapower | (a/b)ⁿ=aⁿ/bⁿ | (2/3)² =4/9 |
Square roots reverse squaring: √25 = 5 because 5² = 25. Know the perfect squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144. Cube roots reverse cubing: ∛27 = 3 because 3³ = 27; the perfect cubes include 1, 8, 27, 64, and 125. To simplify a square root, factor out a perfect square: √72 = √(36 × 2) = 6√2.
Scientific Notation
Scientific notation writes a number as a value from 1 to less than 10 multiplied by a power of 10: 45,000,000 = 4.5 × 10⁷ and 0.00032 = 3.2 × 10⁻⁴. To multiply, multiply the coefficients and add the exponents; to divide, divide the coefficients and subtract the exponents. Example: (3 × 10⁴)(2 × 10³) = 6 × 10⁷.
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