# Page 1-Mathematics Study Guide for the ACT

## General Information

The ACT® test is designed to monitor students’ learning as it ought to be at the end of the 11th grade. The test consists of 60 multiple-choice questions and must be completed within 60 minutes. Though this may seem a short time, this gives you a minute per question—which should be more than enough time. To maximize your efficiency, look over the test once, complete the questions that are easy to understand and solve, and then move on to the more difficult ones. This will allow you to deal with the easy items first, leaving more time for the questions you may struggle with later.

On the ACT® test score report, you will receive one overall Math score and eight subscores. These scores will be derived from your performance on different questions that involve three areas of mathematics. The first area, shown below, is also divided into five categories of math (each of which you have had instruction on in high school or before), providing the five additional subscores.

These are the eight subscores you will receive in Math and the approximate percentage of questions from which each score will be derived. Note that a single question can contribute to more than one subscore.

### Preparing for Higher Math (57% to 60%)

• Number and Quantity (7% to 10%)
• Algebra (12% to 15%)
• Functions (12% to 15%)
• Geometry (12% to 15%)
• Statistics and Probability (20% to 25%)

### Integrating Essential Skills (40% to 43%)

(This pertains to questions that require you to apply skills you learned in math before the eighth grade to problems that also require higher level skills.)

### Modeling (more than 25%)

(This score will be derived from your performance on any question that makes use of a model of some sort, whether you study a model, evaluate it, or devise how to produce it.)

Within each of these sub-areas, these are the concepts with which you should be familiar. If there are any that give you trouble, seek extra practice on them before you take the ACT® test.

## Pre-Algebra

### Basic Operations with:

#### Whole Numbers

When a number has no decimal or fractional part, it is a whole number. Whole numbers are used for counting and representing quantity. They can be added, subtracted, multiplied, and divided.

To add or subtract whole numbers, simply align the digits according to their place value and proceed with addition or subtraction as usual. In addition, the numbers to be added are called addends, and the result of the operation is the sum. In subtraction, the first number is the minuend and the number to be subtracted is called the subtrahend. The result of the operation is called the difference.

Multiplication is simply repeated addition. When multiplying two multiple-digit numbers, multiply the first number with each digit of the second number. The first number is the multiplicand, while the second is the multiplier. Both numbers to be multiplied are called factors, and the result of multiplication is the product.

Division is repeated subtraction or the inverse of multiplication. In $$45 \div 9 = 5$$, we call $$45$$ the dividend, $$9$$ the divisor, and $$5$$ the quotient.

#### Fractions

A fraction represents the value of a thing as a part of a whole. If we say $$\frac{1}{12}$$ of a dozen eggs, we are referring to $$1$$ egg being part of $$12$$ eggs.

#### Decimals

A decimal is another way of representing a part of a whole. The fraction $$\frac{1}{5}$$, for instance, can be written as $$0.2$$ in the decimal form. It is easy to convert a fraction to its decimal form: simply divide the numerator by the denominator.

On the number line, decimals are the numbers located between whole numbers. The number $$0.2$$ is located $$\frac{1}{5}$$ of the way between $$0$$ and $$1$$.

#### Integers

Integers are all of the whole numbers on the number line, including negative whole numbers, zero, and positive whole numbers. Here’s how operations work for integers:

• Addition: When the addends have like signs, add the integers as usual and copy their common sign. When the signs are different, subtract the integers and copy the sign of the bigger integer.

• Subtraction: Change the sign of the subtrahend and proceed to algebraic addition.

• Multiplication and Division: Multiplication and division of integers with like sign will result in a positive value. Multiplication and division of integers with unlike signs will result in a negative value.

### Place Value

Numbers are made of digits, which can be any of the numbers from $$0$$ to $$9$$. These digits each have different values, depending on their location in the number. The digit $$4$$ in the number $$432$$ has a different value as the same digit $$4$$ in the number $$102.45$$. In $$432$$, its value is $$400$$ (or $$4$$ hundreds), while in $$102.45$$, its value is $$0.4$$ (or $$4$$ tenths).

To the left of the decimal point, the place value increases, e.g.: ones, tens, hundreds, thousands, ten thousands, and so on. To the right of the decimal point, the place value decreases, e.g.: tenths, hundredths, thousandths, ten thousandths, and so on.

### Square Roots and Approximations

The square root of a number is that number raised to the exponent of $$\frac{1}{2}$$. Thus, the square root of $$25$$ can be written as $$\sqrt{25}$$ or $$25^{\frac{1}{2}}$$.

Another way to understand square roots is by understanding exponents. Five squared, or $$5$$ raised to the exponent of $$2$$ ($$5^2$$), means multiply $$5$$ by itself or $$5\cdot5 = 25$$.
The inverse of this is $$\sqrt{25} = \pm 5$$.

To approximate the square root of a number is to find a smaller number, which when you multiply it by itself (called squaring a number), will give you the larger number.
A test question may ask for $$\sqrt{625}$$. Try all choices given by squaring them and finding which one equals $$625$$.

### Exponents

Exponents are also called powers or indices. They can be positive, negative, or even fractions.

A positive exponent indicates how many times a number or base is multiplied by itself.

$9^3 = 9 \cdot 9 \cdot 9 = 729$

A negative exponent indicates how many times $$1$$ is divided by the number:

$9^{-3}=\dfrac{1}{9^3}=\frac{1}{9 \cdot 9 \cdot 9} = \frac{1}{729}$

A fractional exponent, $$\frac{1}{n}$$, indicates taking the $$n$$th root of that number.

$27^{\frac{1}{3}} = \sqrt [3]{27} = 3$

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