ATI TEAS Version 7 Math Comprehensive Review

Fundamental Principles of Fractions

Fractional components are categorized into two primary parts: the numerator and the denominator. The numerator occupies the position above the fraction line and signifies the specific number of parts possessed relative to a whole. The denominator is located below the fraction line and represents the total number of equal parts into which the whole is divided. In the fraction $\frac{3}{4}$, the digit $3$ serves as the numerator while the digit $4$ acts as the denominator. The denominator establishes the total parts considered, whereas the numerator indicates the specific quantity focused upon.

Visualizing these concepts through a pizza model provides clarity. A pizza typically consists of $8$ uniform slices. Mathematically, the whole pizza represents the value $1$, and dividing it into $8$ parts means each slice constitutes $\frac{1}{8}$ of the pizza. Consuming $2$ slices equates to $\frac{2}{8}$. If a pizza is instead sliced into quarters, each piece represents $\frac{1}{4}$. Because $\frac{2}{8}$ and $\frac{1}{4}$ represent the same quantity, they are termed equivalent fractions. Dividing both the numerator and denominator of $\frac{2}{8}$ by the common factor of $2$ yields $\frac{1}{4}$, demonstrating that fractions can express identical values with different denominators once simplified.

Understanding Place Values and Decimals

Decimal numbers are structured around the position of digits relative to a decimal point. Using the number $1234$ as a baseline, the rightmost digit before the potential decimal point occupies the ones place. The digit $4$ in this position represents its face value. Moving westward, the tens place is found, where each digit is $10$ times greater than the ones place; here, the digit $3$ signifies $30$. Progressing left again introduces the hundreds place, where digit $2$ represents $200$, followed by the thousands place, where digit $1$ equals $1000$.

Introducing a decimal point to the right of the ones place allows for the representation of values smaller than one. Transitioning rightward from the hundreds to the tens place, or the tens to the ones place, involves dividing by $10$. Continuing this pattern, dividing $1$ by $10$ results in the tenths place, located immediately to the right of the decimal point. Pushing further right results from dividing a tenth by $10$, creating the hundredths place ($\frac{1}{100}$). A third step to the right yields the thousandths place ($\frac{1}{1000}$), which is the result of dividing a hundredth by $10$. In the number $1,234.567$, the digit $5$ represents five tenths ($0.5$), digit $6$ represents six hundredths ($0.06$), and digit $7$ represents seven thousandths ($0.007$).

Percentages as Ratios

The term percent originates from the Latin word centum, meaning $100$. Therefore, percent translates to "per $100$" or "out of $100$". This method expresses quantities by scaling them to a standard benchmark of $100$. For instance, achieving a score of $90\%$ implies a ratio of $90$ correct answers for every $100$ questions. This proportionality remains consistent across different scales, such as $9$ out of $10$, $18$ out of $20$, or $450$ out of $500$. benchmark numbers against $100$ provides a convenient standard for comparative analysis.

Percentages are fundamentally basic fractions. For example, $80\%$ translates to $\frac{80}{100}$. By dividing both the numerator and denominator by the common factor of $20$, the fraction simplifies to $\frac{4}{5}$. Similarly, $50\%$ is $\frac{50}{100}$, which simplifies to $\frac{1}{2}$ by dividing both numbers by $50$. Some percentages, such as $3\%$, are already in their simplest form as $\frac{3}{100}$ because there is no common factor for further reduction.

Numerical Conversion and Interchangeability

Fraction, decimal, and percentage representations are interchangeable using specific methods. To convert a fraction to a decimal, divide the numerator by the denominator (e.g., $\frac{3}{4} = 0.75$). To transition from a decimal to a percentage, multiply the value by $100$ and shift the decimal point two places to the right. Conversely, to move from a percentage to a decimal, divide by $100$ and shift the decimal point two places to the left. To convert a decimal to a fraction, divide by the appropriate place value; for instance, $0.3$ signifies the tenths place, resulting in $\frac{3}{10}$.

When converting a percentage like $30\%$ to a fraction, it starts as $\frac{30}{100}$. A practical simplification trick involves removing trailing zeroes, converting $\frac{30}{100}$ into $\frac{3}{10}$. Writing this as a decimal involves placing the digit $3$ in the tenths position, resulting in $0.3$. This demonstration highlights that $\frac{3}{10}$, $0.3$, and $30\%$ all express the same numerical quantity.

Sequence of Operations: PEMDAS

Handling multiple mathematical operations simultaneously requires adherence to a specific protocol known as PEMDAS. This acronym stands for Parentheses, Exponents, Multiplication and Division (moving left to right), and Addition and Subtraction (moving left to right). Initially, all calculations within parentheses must be addressed. Subsequently, exponents, powers, and square roots are calculated. Multiplication and division are treated with equal priority, meaning whichever operation appears furthest to the left is performed first. Addition and subtraction similarly share priority and are resolved from left to right.

Consider the expression: $4 + (3 \times 2) - 8 \div 2$. First, resolve the parentheses: $3 \times 2 = 6$. The equation becomes $4 + 6 - 8 \div 2$. Next, address the division: $8 \div 2 = 4$. Now, resolve addition and subtraction from left to right: $4 + 6 = 10$, and finally $10 - 4 = 6$. In a more complex example such as $15 + (3 + 2)^2 - 9 \times 6 + 2^3$, the process starts with parentheses ($3 + 2 = 5$), then exponents ($5^2 = 25$ and $2^3 = 8$), then multiplication ($9 \times 6 = 54$). The final string is $15 + 25 - 54 + 8$. Working left to right, $15 + 25 = 40$, $40 - 54 = -14$, and $-14 + 8 = -6$.

Rational and Irrational Numbers

A rational number is defined as any value that can be expressed as a ratio of two integers where the denominator is not zero. Integers like $1$ are rational because they can be written as $\frac{1}{1}$, $\frac{-2}{-2}$, or $\frac{500}{500}$. Negative integers like $-7$ are rational via $\frac{-7}{1}$ or $\frac{-28}{4}$. Non-whole numbers such as $3.75$ are also rational because they can be converted to fractions ($\frac{375}{100}$ or $\frac{15}{4}$). Repeating decimals, such as $0.333...$ (represented with a bar over the repeating digit), are rational because any decimal with a repeating sequence can be expressed as a fraction.

Irrational numbers cannot be expressed as a simple ratio of two integers. These numbers feature digits that extend indefinitely without termination and without repeating patterns. Prominent examples include $\pi$ (the ratio of a circle's circumference to its diameter), $e$ (Euler's number), and the square root of non-perfect squares like $\sqrt{2}$, $\sqrt{3}$, or $\sqrt{15}$. Combining any irrational number with a rational number through addition or multiplication instinctively results in an irrational outcome.

Comparative Analysis and Ordering of Numbers

Rational numbers can be sequenced from smallest to greatest using tools like a number line or vertical stacking. The smallest values are positioned furthest to the left (negative infinity direction) and the largest to the right. When arranging a set of numbers, it is most effective to convert all fractions to decimals first. For example, to compare $\frac{-3}{4}$, $0.5$, $-1.5$, $\frac{2}{3}$, $\frac{-1}{2}$, and $1.75$, the conversions are $-0.75$, $0.5$, $-1.5$, $0.666...$, $-0.5$, and $1.75$. Arranged from least to greatest, the order is $-1.5$, $-0.75$, $-0.5$, $0.5$, $0.6$, and $1.75$.

In the stacking method, numbers are aligned vertically by their decimal points to compare place values from left to right. This is particularly useful for positive numbers. When dealing with negative numbers, the value further from zero is the smallest. For instance, comparing $5.347$ and $5.582$, the ones place is identical ($5$), but in the tenths place, $5$ is greater than $3$, making $5.582$ the greater number. Inequality symbols facilitate these comparisons: less than ($

Algebraic Foundations: Vocabulary and Symbols

Algebraic expressions are composed of four essential components: variables, constants, coefficients, and terms. Variables represent unknown quantities and are symbolized by letters such as $x$. Constants are fixed numbers that stand alone without an attached variable, such as the $5$ in $15x + 5$. Coefficients are numbers paired with variables through multiplication; they always precede the variable (e.g., $15$ in $15x$). Terms are the individual building blocks of the expression separated by addition or subtraction operators ($15x$ and $5$ are two distinct terms).

Inverse arithmetic operations determine what must be applied to a number to result in zero or one. The additive inverse of a positive value $x$ is $-x$, as $x + (-x) = 0$. The multiplicative inverse, or reciprocal, of a whole number $b$ (represented as $\frac{b}{1}$) is $\frac{1}{b}$, as $\frac{b}{1} \times \frac{1}{b} = \frac{b}{b} = 1$. To find the reciprocal of any fraction, simply invert the numerator and the denominator.

Solving Single-Variable Equations and Proportions

To solve equations with one variable, perform the inverse operation on both sides to isolate the unknown. If the equation is $x - 6 = 13$, add $6$ to both sides of the equal sign to find $x = 19$. For $3x + 9 = 0$, first subtract $9$ ($3x = -9$), then divide by the coefficient $3$ to find $x = -3$. When a variable is divided by a number, such as $11 = \frac{w}{3}$, multiply both sides by that number to isolate the variable, resulting in $w = 33$.

Proportions involving two fractions with an equal sign between them are solved using cross-multiplication. For $\frac{x}{3} = \frac{4}{6}$, multiply the diagonals: $6 \times x = 12$. Dividing by $6$ yields $x = 2$. For more complex proportions like $\frac{7}{x} = \frac{14}{28}$, cross-multiplying results in $14x = 196$, and dividing by $14$ gives $x = 14$. This method effectively bridges gaps in understanding proportionate relationships.

Estimation of Metric Length, Weight, and Capacity

Estimation skills are vital for practical applications in healthcare. Length measurements in the metric system include the meter ($1\,m$), which is roughly the height of a doorknob. A centimeter ($1\,cm$) is approximately the width of a paper clip. A millimeter ($1\,mm$) is the tiny diameter of a grain of salt, while a kilometer ($1\,km$) is roughly the length of an airport runway. For weighted measurements, one gram ($1\,g$) is similar to the weight of a paper clip, one milligram ($1\,mg$) is the weight of a pinch of salt, and one kilogram ($1\,kg$) is about the weight of a bag of rice ($2.2\,lbs$).

Capacity measurements involve liters ($1\,L$), comparable to a large water bottle. A milliliter ($1\,mL$) is a very small amount, like a medication drop. A kiloliter ($1\,kL$) represents a massive volume, such as the capacity of a swimming pool. Realistic estimates for common objects include $30\,cm$ for a kitchen knife, $1,500\,kg$ for a car, and $150\,L$ for a filled bathtub.

Translating Linguistic Phrases into Algebraic Expressions

Mastering algebraic word problems requires translating specific phrases into mathematical symbols. The phrase "four times a number" becomes $4x$. "Four more than a number" is $x + 4$. Caution is required with "four less than a number," which must be written as $x - 4$, despite the order of words. "Four times the difference of two numbers" translates to $4(x - y)$. Words like "sum" and "and" correspond to addition ($x + y$), "difference" suggests subtraction ($x - y$), and "product" indicates multiplication ($x \times y$). The words "is" or "equals" denote the equal sign ($=$).

For example, if "six more than a number is $14$," the equation is $x + 6 = 14$, leading to $x = 8$. If "five times the difference of two numbers is $35$" and one number is $10$, the equation is $5(x - 10) = 35$. Distributing the $5$ yields $5x - 50 = 35$. Adding $50$ results in $5x = 85$, and dividing by $5$ gives $x = 17$.

Ratios, Rates, and Percent Proportions

Percent proportions use the formula: $\frac{\text{is}}{\text{of}} = \frac{\text{percent}}{100}$. This matches the relationship of a part to its whole. If asking "$18$ is what percent of $42$", the setup is $\frac{18}{42} = \frac{x}{100}$. Cross-multiplying gives $42x = 1800$, resulting in $x = 42.86\%$. Ratios and proportions are solved similarly. If Carmen reads $4$ books in $10$ hours, to find the time for $50$ books, set up $\frac{4\text{ books}}{50\text{ books}} = \frac{10\text{ hours}}{x\text{ hours}}$. Cross-multiplying gives $4x = 500$, so $x = 125$ hours.

Rates describe the connection between two quantities with different units, such as miles and days. A unit rate is a specific form where the denominator is $1$, such as "dollars per hour". If an individual drives $36\text{ miles}$ in $4\text{ days}$, the rate is $\frac{36}{4}$, which simplifies to a unit rate of $9\text{ miles per day}$. Likewise, $96\text{ dollars}$ in $8\text{ hours}$ equates to $\frac{96}{8}$, or $12\text{ dollars per hour}$.

Mathematical Inequalities and Profit Thresholds

Inequalities describe relationships between values that are not equal, establishing ranges or limits. Symbols include greater than ($>$), less than ($<$), greater than or equal to ($\ge$), and less than or equal to ($\le$). If a hospital elevator has a maximum capacity of $2,000\,lbs$ and already holds $460\,lbs$, the inequality is $w + 460 \le 2000$, where $w$ is the additional weight ($w \le 1540\,lbs$). At least phrasing requires $\ge$, such as a score of at least $85\%$ being $x \ge 85\%$.

Profit calculations utilize inequalities where revenue must be greater than costs ($R > C$). If a factory has fixed costs of $3,600\,dollars$ plus $1.40\,dollars$ per product, and each product sells for $4.20\,dollars$, the inequality is $4.2x > 3600 + 1.4x$. Subtracting $1.4x$ yields $2.8x > 3600$. Dividing by $2.8$ gives $x > 1285.71$, meaning the factory must sell at least $1,286$ products to be profitable.

Direct and Inverse Proportionality Patterns

Direct proportion occurs when one quantity increases and the other increases at a consistent rate ($y = kx$, where $k$ is the constant of proportionality). This forms a linear graph passing through the origin. If removing one marble from a bag yields two candies, the proportionality constant is $2$. Conversely, inverse proportionality describes a relationship where as one variable increases, the other decreases ($y = \frac{k}{x}$). A real-world example is assigning more workers to a task to decrease the time needed to complete it.

Non-directly proportional relationships do not follow a linear change and are often slope-based ($y = mx + b$). For these to be direct, the constant $b$ must equal zero. In an inverse relationship graph, the line is a curve that approaches the axes but never touches them. As the independent variable $x$ increases, the dependent variable $y$ approaches zero but never reaches it.

Descriptive Statistics: Mean, Median, Mode, and Range

Mean represents the average of a dataset, calculated by summing all numbers and dividing by the total count. Median is the middle number when a dataset is arranged in numerical order. In sets with an even number of data points, the median is the average of the two middle numbers ($\frac{x + y}{2}$). Mode is the most frequent number; datasets can have multiple modes (bimodal) or no mode if all frequencies are equal. Range is the difference between the highest and lowest values in a dataset ($\text{Max} - \text{Min}$).

Consider a student's quiz scores: $76, 82, 85, 88, 91$. The range is $91 - 76 = 15$. For the cookie sales dataset: $11, 12, 12, 12, 14, 15, 16$, the sum is $91$. Dividing by the count ($7$) gives a mean of $13$. The middle number is $12$, which is the median, and the most frequent number is $12$, which is the mode. In pain score tracking for $10$ patients ($3, 3, 3, 4, 4, 4, 5, 5, 6, 7$), the modes are both $3$ and $4$ because they appear with equal frequency.

Distribution Shapes and Skewness

Distribution symmetry occurs when a vertical line through the center results in mirroring halves, meaning the mean equals the median. A unimodal symmetric distribution (mound shape) has one peak where mean, median, and mode align. A bimodal distribution has two distinct peaks. Uniform distributions feature equal or nearly equal frequencies across all classes, resulting in no distinct peaks and a flat graph profile.

Skewness describes datasets that trail off in one direction. In a left-skewed distribution, the tail extends to the left, and the bulk of the data is on the right; the mean is typically to the left of the median. In a right-skewed distribution, the tail extends to the right, and bulk data clusters at the lower end. A mnemonic for skewness is to look at your feet: a right-skewed graph trails off like the toes on your right foot, with the "big toe" mode on the left and the smaller digits pulling the mean to the right.

Probability and Likelihood Scales

Probability is calculated using the formula: $\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$. The probability scale ranges from $0$ (impossible) to $1$ (certainty). A probability of $0.5$ indicates an equal chance of an event occurring or not. Flipping a coin for heads yields a probability of $\frac{1}{2}$ ($50\%$). Rolling a standard six-sided die for a specific number is $\frac{1}{6}$ ($16.7\%$). The cumulative probability of all possible outcomes for any event must always sum to $1$ or $100\%$.

When calculating the probability of multiple events occurring together ("and" scenarios), the individual probabilities are multiplied. For example, drawing two red marbles from a bag of $10$ marbles (containing $5$ red) without replacement: the first draw is $\frac{5}{10}$, and the second is $\frac{4}{9}$ (since one red marble and one total marble are gone). Multiplying these yields $\frac{20}{90}$, simplified to $\frac{2}{9}$. If the scenario involves an "or" relationship, the probabilities are added.

Graphical Displays of Data

Five primary graphical displays are used to represent numerical information. Cartesian coordinate charts use perpendicular X (horizontal) and Y (vertical) axes to plot data points from an origin ($0,0$). Scatter plots examine relationships between two variables, where the independent variable is manipulated on the X-axis and the resulting dependent variable is observed on the Y-axis. Line graphs are ideal for observing changes over time, such as monthly sales. Pie or circle charts illustrate numerical proportions of a whole. Bar graphs compare average values across multiple distinct groups or categories.

Graphs must be labeled with clear titles and units of measure to avoid ambiguity. Scales on axes should be uniform and consistent. A line of best fit in a scatter plot represents the average trend and should feature roughly half the data points above and half below it. This line should not extend past the range of existing data. Bad graphs often feature deceptive scaling, mislabeled axes (swapping independent and dependent variables), or a lack of descriptive titles.

Trends: Linear, Exponential, and Quadratic

Linear trends are characterized by a constant rate of increase or decrease ($y = mx + b$), appearing as a straight line. Exponential trends occur when a quantity increases by a fixed proportion or factor annually (e.g., doubling), appearing as a curve that steadily steepens ($y = ab^x$). Given enough time, exponential growth will always surpass linear and quadratic growth. Quadratic trends follow the square of a variable ($y = ax^2 + bx + c$), resulting in a distinctive symmetrical U-shaped curve known as a parabola.

Directions of trends are analyzed from left to right. Increasing trends rise, while decreasing trends fall. A "no change" trend results in a flat horizontal line. Outliers are data points that deviate significantly from the general cluster, often caused by measurement errors or unique, genuine deviations from the norm.

Geometry: Perimeter, Area, and Volume

Perimeter is a one-dimensional measure of the total distance around a shape's border, expressed in linear units (e.g., $cm$, $ft$). For polygons, it is the sum of all side lengths. Circumference is the perimeter of a circle calculated as $C = \pi d$, where $d$ is the diameter. Area is a two-dimensional measure of the space inside a shape, expressed in square units (e.g., $cm^2$). Rectangle area is $L \times W$, and triangle area is $\frac{1}{2} \times \text{base} \times \text{height}$. Circle area is $A = \pi r^2$, where $r$ is the radius (half the diameter).

Volume measures three-dimensional space in cubic units (e.g., $cm^3$). Prisms are calculated as $\text{Area of base} \times \text{height}$. Cylinder volume is $V = \pi r^2 h$. Cones and rectangular pyramids are exactly $\frac{1}{3}$ the volume of their corresponding cylinder or prism counterparts: $\frac{\pi r^2 h}{3}$ and $\frac{LWh}{3}$ respectively. The volume of a sphere is calculated using the formula $V = \frac{4}{3} \pi r^3$.

Conversion between Standard and Metric Systems

The United States customary system uses inches, feet ($1\,ft = 12\,in$), yards ($1\,yd = 3\,ft$), and miles ($1\,mi = 5,280\,ft$). Weight is measured in ounces ($oz$) and pounds ($1\,lb = 16\,oz$). Liquid volume capacity follows a hierarchy: one gallon contains $4$ quarts, $8$ pints, or $16$ cups ($1\,tbsp = 3\,tsp$). The metric system is a decimal-based system ($10$x intervals) using meters, grams, and liters. The mnemonic "King Henry Doesn't Usually Drink Cold Milk" helps remember the prefixes: Kilo, Hecto, Deca, Unit, Deci, Centi, Milli.

Converting between these systems is essential in healthcare. Key conversions include: $1\,kg = 2.2\,lbs$, $1\,inch = 2.54\,cm$, $1\,mile = 1.6\,km$, and $1\,meter = 3.28\,feet$. For weight, one ounce is approximately $28.35\,grams$. When converting from a larger unit to a smaller unit, multiply; when returning from small to large, divide.

Questions & Discussion

A student asked: How do you identify the shape of a distribution if the data has two distinct peaks? When a dataset shows two distinct peaks, it is referred to as a bimodal distribution. This implies there are two modes or most frequent values within the set.

If a researcher notices test scores are evenly distributed across all values with no clear peaks, what is the label? An even distribution across all positive values is described as a uniform distribution. It lacks the humps or tails seen in skewed or mound-shaped graphs.

How is a distribution described if it has a long tail to the right? This is a right-skewed distribution. The bulk of the data points cluster at the lower end (left side), and the outliers pull the tail toward the right.

What is the best graph to compare study hours and test scores to see the effect of study time? A scatter plot is the most appropriate. It allows for the identification of a correlation between the independent variable (hours studied) and the dependent variable (test scores), often visualized with a line of best fit.

Which graph is best for comparing the average test scores of four different classes? A bar graph is the most effective tool for comparing average values across different categorical groups. Ensuring the bars are labeled (e.g., Algebra, Biology) is necessary to avoid ambiguity.

What is the relationship if the cost of apples increases by a constant $2 for each additional apple? This is a linear relationship. The constant rate of change ($2\,dollars/apple$) creates a straight line graph.

If a bacterial count doubles every hour, what graph type represents this? This represents an exponential relationship. The growth rate becomes progressively more rapid because it increases by a fixed proportion (factor of $2$) rather than a fixed amount.

In a study where medication dosage is varied to see the impact on blood pressure, which is which? The variable being manipulated (medication dosage) is the independent variable (X-axis). The variable being measured in response (blood pressure) is the dependent variable (Y-axis).