Page 2 - Mathematics Study Guide for the ACT

More Pre-Algebra Concepts

Scientific Notation

Scientific notation is a special way of writing very large or very small numbers. It is also called the standard form or standard index form. The rule is to write a number in two parts:

Write the digits first and place the decimal point after the first digit. Add “\(\times 10\) to a certain power”. The power is the number of places that the decimal was moved to its present position. The power is positive if the decimal was moved to the left, and negative if it was moved to the right.

Example:

Convert \(123,456,789\) into a scientific notation.

The correct answer is:

\[1.23456789 \; x \; 10^8\]

Factors

Two numbers are the factors of their product; such as \(3\) and \(5\) are the factors of \(15\). A number can have several pairs of factors. For instance, the factors of \(45\) are \(5, 9, 3,\) and \(15\). It is also correct to say that \(45\) is a multiple of any of its factors.

Ratio, Proportion, and Percent

Ratios are used to compare one value to another. They represent part-to-part comparison. They may also represent a part-to-whole comparison in the same way that fractions represent a part-of-a-whole. When two ratios are equal to each other, they are said to be proportional. The fraction \(\frac{5}{6}\) is in proportion with \(\frac{15}{18}\).

Example:

A box contains oranges and lemons in the ratio of \(1:2\). If there were \(20\) oranges, how many fruits were there all in all?

The given ratio will help you solve that there were \(40\) lemons and that there were \(60\) fruits inside the box.

It can also be shown that the ratio of the oranges to the total was \(1:3\) or \(\frac{1}{3}\) of the total number of fruits.

Linear Equations in One Variable

The terms in a linear equation with one variable have exponents of \(0\) or \(1\). There will be no fractional powers or powers greater than \(1\). To solve an equation of this type involves simply isolating the variable to one side of the equation and evaluating the constant terms.

Absolute Value and Ordering Numbers by Value

The absolute value of a number is always positive, because it is the distance of any number to zero and distance, by convention, is always taken as positive. This property is stated as:

\[\vert a \vert \ge 0\]

The following properties are also useful in solving problems involving absolute values:

\[\vert a \vert = \sqrt{a^2}\] \[\vert a \cdot b \vert = \vert a \vert \cdot \vert b \vert\]

Another property of absolute values commonly used in algebra is given by:

\[\vert u \vert = \pm a\]

Example: Solve for \(y\) in \(\vert y + 17 \vert = 20\)

\[y+17= \pm 20\] \[y = 20-17 = 3\]

And

\[y = -20-17 = -37\]

When absolute values are mixed with inequalities, it gets a little bit trickier. Here’s an example to illustrate:

Find the values of \(x\) if: \(\vert x \vert \lt 10\)

This means that the distance of \(x\) is from \(0\), less than \(10\), and that \(x\) is any value within the interval \((-10, 10)\) but excluding \(-10\) and \(10\).

Ordering the values of \(x\), we write it as:

\[-10 \lt x \lt 10\]

(There is more about this concept under the Intermediate Algebra heading.)

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 \(A \times Z\) 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?

\[Probability = 1 \div 6 = \frac{1}{6}\]

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 helps 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.

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