Quantitative Reasoning · 4 min read
Quantitative Reasoning · Operations and Number Sense
Quantitative Reasoning is the foundation of the TSIA2 Mathematics test. It measures how accurately you compute with numbers and how well you reason about quantities: operations with integers and rational numbers, properties of exponents, conversions among fractions, decimals, and percents, ratios, rates, and proportions, and linear expressions and equations. The official skill list stresses calculating ratios, proportions, and percents and identifying, manipulating, and interpreting linear equations and expressions, so expect many questions to be set in real-world contexts such as prices, wages, recipes, and rates.
Properties of the Four Operations
Addition combines quantities (the numbers are addends, the result is the sum); subtraction finds a difference; multiplication finds the product of factors; division splits a dividend by a divisor to give a quotient. Knowing which properties apply to which operation lets you rearrange expressions safely.
| Property | Statement | Example |
|---|---|---|
| Commutative | Orderdoesnotmatterforadditionandmultiplication | 5+3=3+5;4×6=6×4 |
| Associative | Groupingdoesnotmatterforadditionand | (2+3)+4=2+(3+4) |
| multiplication | ||
| Distributive | Multiplicationdistributesoveradditionand | a(b+c)=ab+ac |
| subtraction | ||
| Identity | Adding0ormultiplyingby1leavesanumber | 7+0=7;7×1=7 |
| unchanged | ||
| Inverse | Anumberplusitsoppositeis0;anumbertimesits | 5+( −5) =0;5×⅕=1 |
| reciprocalis1 | ||
| Notcommutativeor | Subtractionanddivision | 7 − 3 ≠ 3 − 7; (10 − 5) − 2 ≠ 10 − (5 − 2) |
| associative |
Order of Operations
When an expression contains more than one operation, evaluate it in this order (PEMDAS): Parentheses and other grouping symbols, Exponents and roots, Multiplication and Division from left to right, then Addition and Subtraction from left to right. Multiplication does not automatically come before division; they share one step and are done in the order they appear.
3 + 4 × 2² − (6 ÷ 2)
= 3 + 4 × 2² − 3 parentheses
= 3 + 4 × 4 − 3 exponents
= 3 + 16 − 3 multiplication
= 16 addition and subtraction, left to right
A fraction bar is a grouping symbol: in (12 − 4) ÷ (6 − 2), evaluate the numerator and denominator first.
Operations with Integers
| Operation | Rule | Examples |
|---|---|---|
| Add samesigns , | Addtheabsolutevaluesandkeepthesign | ( −5)+( −3) =−8 |
| Add differentsigns , | Subtractthesmallerabsolutevaluefromthelarger; | ( −7)+4=−3;9+( −2) =7 |
| usethesignofthelarger | ||
| Subtract | Addtheopposite thenfollowtheadditionrules , | 8 − ( −3) = 11; ( −4) − 6 = −10; ( −5) − ( −9) = 4 |
| Multiplyordivide | Same signs positive; different signs negative → → | ( −6)( −4) =24;( −8)(3) =−24;36÷( −6) =−6 |
Absolute Value
The absolute value of a number is its distance from zero on the number line, so it is never negative: |7| = 7, |−7| = 7, and |0| = 0. Treat absolute value bars as grouping symbols: evaluate the expression inside first, then take the absolute value. For example, |−8 + 3| = |−5| = 5, while |−4| + |−6| = 4 + 6 = 10.
Multiplication and division share a step (left to right), as do
Subtracting a negative is adding a positive: 8 − (−3) = 11.
addition and subtraction.
(−3)² = 9, but −3² = −9, because the exponent applies only
A negative sign in front of parentheses changes the sign of
to 3.
every term inside: −(x − 5) = −x + 5.
Properties of Exponents
An exponent tells how many times a base is used as a factor. In bⁿ, b is the base and n is the exponent, so 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The rules below apply only when the bases are the same; 2³ × 3² cannot be combined with the product rule.
| Rule | Form | Examples | ||
|---|---|---|---|---|
| Product | aᵐ·aⁿ=aᵐ⁺ⁿ | x³ ·x⁴=x⁷;2² ·2⁵=2⁷=128 | ||
| Quotient | aᵐ÷aⁿ=aᵐ⁻ⁿ | y⁸÷y³ =y⁵;5⁶÷5² =5⁴=625 | ||
| Powerofapower | (aᵐ)ⁿ=aᵐⁿ | (x²)⁴=x⁸;(3²)³ =3⁶=729 | ||
| Powerofaproduct | (ab)ⁿ=aⁿbⁿ | (2x)³ =8x³;(3y²)² =9y⁴ | ||
| Powerofaquotient Zeroexponent | (a/b)ⁿ=aⁿ/bⁿ a⁰=1(a≠0) | (x/3)² =x²/9;(2/5)³ =8/125 5⁰=1;( −3)⁰=1 | 1 |
| Rule | Form | Examples |
|---|---|---|
| Negativeexponent | a⁻ⁿ=1/aⁿ | 2⁻³ =1/8;(3/4)⁻² =(4/3)² =16/9 |
| Fractionalexponent | a^(m/n)=(ⁿ√a)ᵐ | 8^(1/3) =2;16^(3/4) =2³ =8;27^(2/3) =9 |
| WORKED EXAMPLE: COMBINING RULES (2x³y⁻²)² 3xy⁵ · = 4x⁶y⁻⁴ · 3xy⁵ power of a product = 12x⁷y¹ product rule: multiply = 12x⁷y | coefficients add exponents , |
Handle coefficients and each variable separately, and write the final answer with positive exponents.
Radicals
A square root undoes squaring: √49 = 7 because 7² = 49. Simplify a radical by factoring out perfect squares: √72 = √(36 · 2) = 6√2, and √50 + √8 = 5√2 + 2√2 = 7√2. Only like radicals (same number under the root) can be added. Products work across radicals: √3 · √12 = √36 = 6.
Scientific Notation
Scientific notation writes a number as a × 10ⁿ with 1 ≤ |a| < 10. Move the decimal point to make a number between 1 and 10 and count the places moved: large numbers get a positive exponent (45,000,000 = 4.5 × 10⁷), and numbers between 0 and 1 get a negative exponent (0.00032 = 3.2 × 10⁻⁴).
(3 × 10⁴)(2 × 10³) = 6 × 10⁷ multiply coefficients, add exponents
(8 × 10⁶) ÷ (2 × 10²) = 4 × 10⁴ divide coefficients, subtract exponents
(5 × 10³)(4 × 10⁵) = 20 × 10⁸ = 2 × 10⁹ adjust so the coefficient is less than 10
Add exponents when multiplying, subtract when dividing,
(a + b)² is not a² + b²; exponents do not distribute over
multiply when raising a power to a power.
addition.
A negative exponent means reciprocal, not a negative
Check that a scientific-notation answer has a coefficient
number: 2⁻³ = 1/8.
between 1 and 10.
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