# Page 2 Mathematics Study Guide for the ACT

#### 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 “x 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.

#### 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:

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

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

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

And

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: