ATI TEAS Mathematics Practice Guide Version 7

Basic Addition and Subtraction

• Standard Arithmetic Operations: The fundamental process for subtraction involves subtracting digits column-by-column from the rightmost digit. When the top digit is smaller than the bottom digit, the "borrowing" method is applied (e.g., in $1,234 - 578$, borrow from the tens place to make the ones place $14 - 8 = 6$).
• Addition with Carrying: Summing numbers like $493 + 208$ requires adding columns from right to left; if a sum exceeds 9, the tens digit is carried over to the next column (e.g., $3 + 8 = 11$, carry the 1).
• Negative Number Rules:     * Subtracting a Negative: Subtracting a negative number is equivalent to adding its positive counterpart ($x - (-y) = x + y$). For example, $243 - (-157) = 243 + 157 = 400$.     * Adding a Negative: Adding a negative number is equivalent to subtracting that value ($x + (-y) = x - y$). For instance, $625 + (-489) = 625 - 489 = 136$.     * Negative Result Logic: Subtracting a larger number from a smaller one results in a negative value ($3 - 7 = -4$).
• Time and Distance Calculations:     * Marathon Times: When adding times (e.g., $2 \text{ hours } 27.5 \text{ minutes} + 3 \text{ hours } 35.2 \text{ minutes}$), add minutes first ($62.7$). Since $60 \text{ minutes} = 1 \text{ hour}$, the total is $6 \text{ hours and } 2.7 \text{ minutes}$.     * Total Journey Distance: Distance problems often involve summing legs of a trip ($1,200 \times \text{ City A to B} + 800 \times \text{ City B to C} = 2,000 \text{ km}$). Stops do not affect total travel distance.     * Direct Return Distance: Calculated by subtracting completed legs from the total trip distance (e.g., $500 \text{ total} - (145 + 230) = 125 \text{ km}$).

Basic Multiplication and Division

• Order of Operations (PEMDAS):     * Parentheses: Operations inside parentheses must be solved first.     * Exponents: Apply power rules before further operations.     * Multiplication and Division: Perform these from left to right as they appear in the expression.     * Addition and Subtraction: Perform these last from left to right.     * Example: 50 \text{ % } 5 \times 3 + 10 \rightarrow 10 \times 3 + 10 \rightarrow 30 + 10 = 40.
• Rules for Signed Numbers:     * Negative multiplied/divided by Negative = Positive.     * Positive multiplied/divided by Negative = Negative.     * Example: -36 \text{ % } 4 \times 3 + 15 \rightarrow -9 \times 3 + 15 \rightarrow -27 + 15 = -12.
• Calculations with Scenarios:     * Production Increase: To find total yearly production after an efficiency increase, multiply the monthly rate by the increase factor, then by 12 months (e.g., $250 \times 4 \times 12 = 12,000$).     * Team Scoring: Final score = $(Wins \times \text{ Points/Win}) - (Losses \times \text{ Points/Loss})$.     * Rate and Time: Net distance is the difference between opposite directions (e.g., $50 \text{ km/h} \times 10 \text{ hours north} - 50 \text{ km/h} \times 3.4 \text{ hours south} = 330 \text{ km north from start}$).

Decimals and Fractions

• Conversions:     * Decimal to Fraction: Terminate decimals are written over a power of 10 and simplified (e.g., $0.75 = \frac{75}{100} = \frac{3}{4}$; $0.125 = \frac{125}{1000} = \frac{1}{8}$).     * Fraction to Decimal: Divide the numerator by the denominator (\frac{7}{8} = 7 \text{ % } 8 = 0.875).     * Percentage to Decimal: Divide the percentage by 100 (75\text{%} = 0.75).     * Decimal to Percentage: Multiply the decimal by 100 (0.125 \times 100 = 12.5\text{%}).
• Comparison and Ordering: When comparing mixed formats, convert all numbers to decimals (e.g., to compare $0.25$, 25\text{%}, and $\frac{1}{4}$, recognize all are $0.25$).
• Proportion Logic:     * Complimentary Sets: In a class of 30 where 18 are girls, the fraction of girls is $\frac{18}{30} = \frac{3}{5}$. The remaining students are $\frac{12}{30} = \frac{2}{5}$.     * Fiction Ratios: If 1,500 of 10,000 books are fiction, the ratio of non-fiction is \frac{8,500}{10,000} = 85\text{%}.

Operations with Fractions and Mixed Numbers

• Addition and Subtraction:     * Common Denominator: To add or subtract (e.g., $\frac{3}{4} + \frac{1}{6}$), find the least common multiple of the denominators ($12$). Result: $\frac{9}{12} + \frac{2}{12} = \frac{11}{12}$.     * Mixed Number Conversion: Convert mixed numbers to improper fractions before calculating ($2 \frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{9}{4}$).
• Multiplication and Division:     * Multiplication: Multiply numerators and denominators independently ($\frac{3}{4} \times \frac{5}{6} = \frac{15}{24}$, simplified to $\frac{5}{8}$).     * Division: Multiply by the reciprocal of the divisor (\frac{7}{8} \text{ % } \frac{2}{3} = \frac{7}{8} \times \frac{3}{2} = \frac{21}{16} = 1 \frac{5}{16}).
• Relative Quantity Problems:     * Ratio Factors: If a recipe needs $3 \frac{1}{3}$ times as much sugar as salt and you have $2 \frac{1}{2} \text{ cups}$ of salt, multiply: $\frac{5}{2} \times \frac{10}{3} = \frac{50}{6} = 8 \frac{1}{3} \text{ cups sugar}$.     * Unit Breakdown: A container with $3 \frac{1}{2} \text{ liters}$ can fill glasses holding $\frac{1}{4} \text{ liter}$ by dividing: \frac{7}{2} \text{ % } \frac{1}{4} = \frac{7}{2} \times \frac{4}{1} = 14 \text{ glasses}.

Ratios, Proportions, and Percentages

• Equivalent Ratios: Ratios like $3:4$, $0.75$, and 75\text{%} are identical proportions. Ratios like $6:9$ simplify to $2:3$.
• Percentage of a Value: Calculate as $(\text{Percent} \times \text{Value})$ (e.g., 40\text{% of } 250 = 0.40 \times 250 = 100).
• Finding the Whole: If a portion is known (e.g., 120 \text{ is } 25\text{% of } x), solve using x = 120 \text{ % } 0.25 = 480.
• Solving Proportions: Use cross-multiplication for $a:b :: c:x$ (e.g., 9:16 :: x:144 \rightarrow (9 \times 144) \text{ % } 16 = 81).
• Financial Applications:     * Compound Change: Successive increases/decreases are multiplicative. A 20\text{%} price increase followed by a 20\text{%} decrease results in 1.20 \times 0.80 = 0.96 \text{ (96% of original cost)}.     * Tips and Gratuities: Calculated on the pre-tax amount. Total = $\text{Bill} + (\text{Bill} \times \text{Tax Rate}) + (\text{Bill} \times \text{Tip Rate})$.     * Markups: Retail Price = $\frac{\text{Cost}}{1 - \text{Markup percentage as decimal}}$ or \text{Cost} \times (1 + \text{Markup %}) (e.g., 120 \text{ sale price with } 20\text{% markup } \rightarrow 120 \text{ % } 1.2 = 100 \text{ cost price}).
• Concentration Problems: A solution involving 60% Solvent A in 500 mL means 300 mL is Solvent A. To reach 50% concentration without changing Solvent A, total volume must become 600 mL, requiring 100 mL additional Solvent B.

Equations with One Variable

• Algebraic Isolation: Solving $3x - 7 = 2x + 8$ involves moving $x$ to one side ($3x - 2x = x$) and constants to the other ($8 + 7 = 15$).
• Inequality Rules:     * Directional Sign: When multiplying or dividing both sides by a negative number, the inequality sign must be reversed.     * Example: $-5x > -4 \rightarrow x < \frac{4}{5}$.     * Simplification: Distribute constants first (e.g., $2(x-3) \rightarrow 2x-6$), combine like terms, then isolate the variable.
• Fractional Equations: Eliminate denominators by multiplying all terms by the Least Common Multiple (LCM) of the denominators.

Solving Real-World Mathematical Problems

• Cost Equations: Total Cost = $(\text{Daily Fee}) + (\text{Rate per mile} \times \text{Miles})$. For a bill of $95$ with a $50$ daily fee drive $70 \text{ miles}$, calculating (95-50) \text{ % } 70 = 0.64 \text{ per mile}.
• Mixture Math: Use weighted averages for mixture costs (e.g., almond/cashew mix) and concentrations (e.g., alcohol solutions).
• Work Rates: For two pipes with filling times of $2 \text{ and } 3 \text{ hours}$, the combined rate is $\frac{1}{2} + \frac{1}{3} = \frac{5}{6} \text{ tank/hour}$. Time to fill is 1 \text{ % } \frac{5}{6} = 1.2 \text{ hours (1 hr 12 min)}.
• Break-even Analysis: Break-even point occurs when $\text{Total Revenue} = \text{Fixed Costs} + \text{Variable Costs}$. (e.g., Unit sale $250x = 100x + 5000$).

Powers, Exponents, Roots, and Polynomials

• Exponent Laws:     * Product Rule: $a^m \times a^n = a^{m+n}$.     * Power of a Power: $(a^m)^n = a^{mn}$.     * Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$.     * Zero Exponent: $a^0 = 1$.     * Negative Exponents: $a^{-n} = \frac{1}{a^n}$.     * Fractional Exponents: $a^{\frac{1}{n}} = \text{The } n\text{-th root of } a$.
• Radical Simplification: Factor the number into prime components to remove perfect squares (e.g., $\text{Factor } 1152 \rightarrow 2^7 \times 3^2 \rightarrow \text{Pull out } 2^3 \times 3 \rightarrow \text{Result } 24\text{√}2$).
• Polynomial Characteristics:     * Degree: The highest power of the variable present in the polynomial.     * Leading Coefficient: The coefficient of the term with the highest degree.
• Polynomial Operations:     * Multiplication (FOIL): Multiply binomials term by term.     * Subtraction: Distribute the negative sign to all terms of the polynomial being subtracted before combining like terms.     * Factoring: Difference of squares ($x^2 - 9 = (x-3)(x+3)$) and perfect squares ($x^2 + 4x + 4 = (x+2)^2$).     * Division: Synthetic or long division. The Remainder Theorem states the remainder of f(x) \text{ % } (x-c) is $f(c)$.

Standards of Measure

• Imperial Units:     * $1 \text{ pound (lb)} = 16 \text{ ounces (oz)}$.     * $1 \text{ foot (ft)} = 12 \text{ inches (in)}$.     * $1 \text{ yard (yd)} = 36 \text{ inches} = 3 \text{ feet}$.     * $1 \text{ mile (mi)} = 5,280 \text{ feet}$.     * $1 \text{ gallon (gal)} = 4 \text{ quarts (qt)} = 128 \text{ fluid ounces}$.     * $1 \text{ quartile (qt)} = 2 \text{ pints (pt)}$.     * $1 \text{ cup} = 16 \text{ tablespoons} = 236.6 \text{ mL}$.
• Metric Units:     * $1 \text{ kilogram (kg)} = 1,000 \text{ grams (g)}$.     * $1 \text{ kilometer (km)} = 1,000 \text{ meters (m)}$.     * $1 \text{ meter (m)} = 100 \text{ centimeters (cm)} = 1,000 \text{ millimeters (mm)}$.     * $1 \text{ liter (L)} = 1,000 \text{ milliliters (mL)}$.     * $1 \text{ cubic meter (m}^3\text{)} = 1,000,000 \text{ cm}^3 = 1,000 \text{ Liters}$.
• Conversions & Temperature:     * $1 \text{ inch} = 2.54 \text{ cm}$.     * $1 \text{ mile} \text{ ≈ } 1.609 \text{ km}$.     * $1 \text{ kg} \text{ ≈ } 2.205 \text{ lb}$.     * Celsius to Fahrenheit: $F = \frac{9}{5}C + 32$.     * Fahrenheit to Celsius: $C = \frac{5}{9}(F - 32)$.     * Celsius to Kelvin: $K = C + 273.15$.     * Note: $-40^{\text{o}}F \text{ and } -40^{\text{o}}C$ represent the same temperature.

Mean, Median, Mode, Range, and Probability

• Statistics Definitions:     * Mean: The sum of all values divided by the count ($N$).     * Median: The middle value in an ordered set. If $N$ is even, average the two middle numbers.     * Mode: The number that appears most frequently. If no number repeats, there is no mode.     * Range: The difference between the highest and lowest values in the set.
• Probability Theory:     * Calculation: $\frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}}$.     * Independent Events: If $A \text{ and } B$ are independent, the probability of both occurring is $P(A) \times P(B)$.     * Non-replacement: When drawing multiple items without replacement, denominators must decrease for subsequent events (e.g., drawing two green balls from a bag of 15 starts with $\frac{6}{15} \times \frac{5}{14}$).     * Complementary Probability: The probability of an event not happening is $1 - P(\text{Event})$.

Geometric Quantities

• Perimeter Formulas:     * Square: $4s$.     * Rectangle: $2(L + W)$.     * Triangle: Sum of three sides.     * Parallelogram: $2(\text{base} + \text{side})$.     * Trapezoid: Sum of all sides (bases + non-parallel sides).
• Area Formulas:     * Rectangle: $L \times W$.     * Square: $s^2$.     * Triangle: $\frac{1}{2} \text{base} \times \text{height}$.     * Parallelogram: $\text{base} \times \text{height}$.     * Trapezoid: $\frac{1}{2}(b_1 + b_2) \times h$.     * Circle: ̀r^2.
• Volume Formulas:     * Rectangular Prism: $L \times W \times H$.     * Triangular Prism: $\text{Base Area} \times \text{Prism Height}$.     * Pyramid: $\frac{1}{3} \text{Base Area} \times \text{Height}$.     * Cylinder: ̀r^2h.     * Cone: \frac{1}{3} ̀r^2h.     * Sphere: \frac{4}{3} ̀r^3.
• Circles and Semicircles:     * Circumference: C = 2̀r = ̀d.     * Semicircle Perimeter: ̀r + 2r \text{ (Half-circumference + diameter)}.     * Semicircle Area: \frac{1}{2} ̀r^2.
• Composite Figures: To calculate the perimeter of combined shapes, add the remaining outer sides and subtract the length of the shared interior side where the shapes connect.