# Page 3 Mathematics Study Guide for the ACT

#### Elementary Counting Techniques and Simple Probability

Basic counting principle: “If there are A ways to do one thing, and Z ways to do another thing, then there are A x Z ways to do both things.”

Example: How many ways can you mix and match 12 shirts, 5 pairs of pants, and 3 pairs of footwear?

Number of combinations = $12 \cdot 5 \cdot 3 = 180$

Probability: the likelihood that an event will happen is computed by dividing the number of ways an event can happen over the total number of outcomes.

Example: The faces of a die are numbered from 4 to 9. What is the probability that the die will land with the number 5 face shown?

#### Data Collection, Representation, and Interpretation

Charts, various types of graphs, maps and other visual representations of data may be used during the test to measure your comprehension of such resources and your ability to interpret them. The first thing to do is to understand what data is represented. Read the title, the chart headings, the labels of graphs, the value of calibrations or numbers, and the scale in maps. The titles and headings give the main clue. Graphs can be any type: line, bar, pie, pictograph, or scatter plot. It helps to be familiar with the different types. Mastery in dealing with fractions, ratios and percentages help a lot in data interpretation.

#### Simple Descriptive Statistics

Data is summarized using descriptive statistics. We describe the central tendency or central value of a set of data by calculating the mean (average), the median (middle value), or the mode (the value that appears most frequently). Data can be numerical (e.g., scores, height, population), or nominal (e.g., eye color, gender). Here is how we determine the mean, median and mode:

Mean: Add all the given data. Divide the sum by the number of data points added.

Median: Sort the data in either ascending or descending order. Identify the middle number – that is the median. In cases where there are two middle data points (which is what happens when the total number of items is an even number), add the middle pair and divide the sum by two.

Mode: Inspect the list of data and the item that occurs most often is the mode. A set of data can be bimodal (two items occur equally often) or multimodal (involving more than two items).

The spread or dispersion of data from the central value is measured by calculating the range, variance and the standard deviation.

### Elementary Algebra

#### Properties of Exponents and Square Roots

Here are some properties of exponents commonly used in algebra:

$a^0 = 1$ as long as $a \neq 0$

Properties of square roots and nth roots:

Familiarity with these properties will come in handy when simplifying radicals.

Example:

#### Using Substitution to Evaluate Algebraic Expressions

Algebraic expressions make use of variables or letters to represent an unknown values. To evaluate, we substitute numerical values for their corresponding variables or letters.

Example: Evaluate $b^2 + 3b$ when $b = 5$

#### Expressing Functional Relationships using Variables

Functions are special relationships where only one output results for every input. On a graph, a vertical line will never cross a function more than once - this is also called the vertical line test. Variables are used to express functional relationships.

In $f(x) = x+3$, the function f takes the input x to produce the output x + 3.

The domain of a function is the set of input values. The range of a function is the set of output values.