Quantitative Reasoning Study Guide for the DAT

General Information

The Dental Admission Test (DAT) Quantitative Reasoning section is 45 minutes and contains a total of 40 questions. Thirty of the questions present problems in numeracy, algebra, data analysis, and probability and statistics. The other 10 questions require you to apply mathematical skills as you solve word problems. There is no advanced mathematics or calculus on this test and, you will have access to an on-screen calculator during this section of the DAT.

The following information can be used to assess your competence in the most important basic math skills you need to succeed on this section of the DAT.

Numeration and Calculation

To succeed in math on any type of exam, you need to be proficient with numbers and their calculations. This part of the study guide will support your ability to work with numbers, perform calculations accurately, and apply mathematical concepts effectively.

Numbers

Numbers form the foundation of mathematics, and there are various types that serve different purposes.

Types of Numbers

In mathematics, we categorize numbers into these main types:

• natural numbers—the positive integers used for counting (e.g., \(1, \, 2, \, 3\))

• whole numbers—all the natural numbers along with \(0\) (e.g., \(0, \, 1, \, 2, \, 3\))

• integers—the positive natural numbers and their negative counterparts, as well as \(0\) (\(... -2, \, -1, \, 0, \, 1, \, 2, \, …\))

• rational numbers—numbers that can be expressed as fractions (e.g., \(10, \, \frac{1}{7}, \, 14.32)\)

• irrational numbers—numbers that cannot be expressed as a fraction and have non-repeating, non-terminating decimals (e.g., \(\sqrt{2}, \, \pi, \, \sqrt{3}\))

Properties of Numbers

Understanding the properties of numbers is important for performing calculations accurately. These are the vital properties to know for the DAT:

• commutative property of addition and multiplication—Addition and multiplication are commutative operations, meaning changing the order of the numbers does not change the result.

\[a + b = b + a \text{ and } a \times b = b \times a\] \[2 + 3 = 3 + 2 \text{ and } 2 \times 3 = 3 \times 2\]

• associative property of addition and multiplication—Addition and multiplication are associative operations, meaning changing the grouping of the numbers does not change the result.

\[(a + b) + c = a + (b + c) \text{ and } (a \times b) \times c = a \times (b \times c)\] \[(4 + 10) + 2 = 4 + (10 + 2) \quad \text{and} \quad (4 \times 10) \times 2 = 4 \times (10 \times 2)\]

• distributive property—Multiplication distributes over addition and subtraction.

\[a \times (b+c) = (a \times b) + (a \times c)\] \[2 \times (10 + 15) = (2 \times 10) + (2 \times 15)\]

There are also specific properties for \(0\), \(-1\), and \(1\) that are very useful in calculations:

• property of \(0\)—Adding \(0\) to any number does not change its value (\(a + 0 = a\)), subtracting \(0\) from any number does not change its value (\(a - 0 = a\)), and multiplying any number by \(0\) yields \(0\) (\(a \times 0 = 0\)).

• property of \(1\)—Multiplying any number (except \(0\)) by \(1\) yields the number itself (\(a \times 1 = a\)).

• property of \(-1\)—Multiplying any number by \(-1\) changes its sign (\(a \times (-1) = -a\)).

The Number Line

The number line is a visual representation of numbers arranged in order, allowing you to locate integers, fractions, and decimals. It extends infinitely in both positive and negative directions. A typical number line is shown below (with integers only):

The numbers to the right of \(0\) are positive and the numbers to the left of \(0\) are negative. As you move right along the number line, the values of the numbers increase. You can estimate the position of a decimal or fraction on the number line relative to nearby integers that are labeled. For example, the decimal \(1.2\) would be closer to \(1\) than \(2\) on the number line, and the fraction \(2 \frac{4}{5}\) would be closer to \(3\) than \(2\).

Absolute Value

Absolute value helps us understand the size of numbers, regardless of whether they are positive or negative. It tells us how far a number is from zero on the number line. The absolute value of a number is always positive.

For example, even though \(-7\) is negative, its absolute value is positive (\(7\)) because it is seven units away from \(0\). Similarly, for the number \(3\), its absolute value is \(3\) because it is three units away from \(0\).

This is a useful idea that helps us determine the size of numbers without worrying about their positive or negative signs. It helps us solve problems involving equations and inequalities, where we need to consider both positive and negative solutions. In geometry, it’s handy for measuring distances between points, helping us figure out how far apart things are.

Operations

Operations serve as the backbone of quantitative reasoning. Understanding when and how to use each operation is essential for solving mathematical problems accurately and efficiently. The four basic operations are addition, subtraction, multiplication, and division.

Basics

The first basic operation is addition. Here, we combine addends to find a sum. For example, in the problem \(2 + 4 = 6\), \(2\) and \(4\) are addends and the result, \(6\), is the sum.

With subtraction, we find the difference between two numbers. The minuend is the number from which we are taking away another number, the subtrahend, and the result of the problem is the difference. In the problem \(10 - 2 = 8\), \(10\) is the minuend, \(2\) is the subtrahend, and the result, \(8\), is the difference.

Multiplication is a process of repeated addition or grouping of numbers. The multiplicand is multiplied by the multiplier to get a product. In the problem \(2 \times 4 = 8\), \(2\) and \(4\) are the multiplicand and multiplier, respectively, and \(8\) is the product. For ease, we sometimes call both the numbers being multiplied factors.

Division is the process of sharing or distributing a quantity in equal parts. The number being divided is the dividend, the number used to divide is the divisor, and the result is the quotient. In the problem \(200 \div 5 = 40\), \(200\) is the dividend, \(5\) is the divisor, and \(40\) is the quotient.

Laws of Operations

Mathematical operations follow specific rules known as the laws of operations:

• commutative law—Addition and multiplication follow the commutative law, which states that changing the order of numbers does not affect the result.

\[a + b = b + a \text{ and } a \times b = b \times a\] \[10 + 2 = 2 + 10\text{ and }10 \times 2 = 2 \times 10\]

Note: Division and subtraction are not commutative.

• associative law—Addition and multiplication adhere to the associative law, which states that changing the grouping of numbers does not alter the result.

\[(a + b) + c = a + (b+c)\text{ and }(a \times b) \times c = a \times (b \times c)\] \[(4 + 5) + 2 = 4 + (5 + 2)\text{ and }(4 \times 5) \times 2 = 4 \times (5 \times 2)\]

Note: As with the commutative law, subtraction and division are not associative.

• distributive law—Multiplication distributes over addition (and subtraction), allowing expressions to be simplified.

\[a(b+c) = ab + ac\text{ and } a(b-c) = ab - ac\] \[10(2+3) = 10(2) + 10(3) = 20 + 30 = 50 \text{ and } 10(3-2) = 10(3) - 10(2) = 30 - 20 = 10\]

Operations with Integers

Integers, or signed numbers, represent whole numbers that can be positive, negative, or zero. For example, the number \(-4\) is negative and \(4\) is positive. Positive integers represent quantities greater than zero, negative integers represent quantities less than zero, and zero serves as a neutral value.

When performing operations such as addition, subtraction, multiplication, and division with integers, certain rules come into play:

• addition—When adding integers, the process depends on the signs of the numbers involved. If both integers have the same sign (either both positive or both negative), their sum will have the same sign. For instance, \(5 + 3 = 8\) and \((-5) + (-3) = -8\). When adding integers with different signs, subtract the smaller magnitude from the larger magnitude and keep the sign of the integer with the larger magnitude. For example, \(5 + (-3) = 2\) and \((-5) + 3 = -2\).

• subtraction—Subtracting integers can be treated as addition by taking the opposite of the second integer (changing its sign) and then doing the addition process. For example, \(- 7 - 3\) can be written as \(- 7 + (-3)\), which follows the rules of addition (\(- 7 + (-3) = - 10\)).

• multiplication—When multiplying integers, adhere to the following rules:

  • If both integers have the same sign (either both positive or both negative), the product is positive. For example, \(2 \times 6 = 12\) (both positive) or \((-4) \times (-2) = 8\) (both negative).

  • If the integers have different signs, the product is negative. For example, \((-5) \times 2 = -10\) (one negative and one positive).

• division—Division involving integers follows similar rules as multiplication, as you must consider the signs of the divisor and dividend to determine the sign of the quotient:

  • If both integers have the same sign, the quotient is positive. For example, \(10 \div 2 = 5\) (both positive) or \((-12) \div (-3) = 4\) (both negative).

  • If the integers have different signs, the quotient is negative. For example, \(15 \div (-5) = -3\) (one positive and one negative).

The Order of Operations

The order of operations is a set of rules that defines the sequence in which mathematical operations must be performed in an expression (or equation). This sequence is crucial because it ensures accuracy and consistency in interpreting mathematical expressions. Without adhering to a specific order, different individuals will arrive at different results when evaluating the same expression.

The order of operations streamlines mathematical equations and ensures uniformity. One common mnemonic device used to remember the order of operations is PEMDAS, which stands for parentheses, exponents, multiplication and division, addition and subtraction.

To follow PEMDAS:

Do everything in parentheses (P), in order, left to right.
Evaluate any exponents (E), in order, left to right.
Do all multiplication and division (MD), in order, left to right.
Then do all addition and subtraction (AS), in order, from left to right.
A good way to remember is: Please Excuse My Dear Aunt Sally.

Note: We will discuss exponents in greater detail later in this study guide.

Let’s look at an example using PEMDAS.

What is the value of \(4 \times (6 - 2 + 3)^{2}\)?

Solution

Using PEMDAS, we will evaluate the expression inside the parentheses first:

\[4 \times (6-2+3)^{2}\]

Now, we will evaluate the exponent:

\[= 4 \times (7)^{2}\]

Last, we will multiply to find the final answer:

\[= 4 \times 49\] \[= 196\]

Fractions

Fractions represent a part of a whole or a ratio between two numbers. In essence, they allow us to express values that fall between whole numbers. Fractions are used in various aspects of mathematics and everyday life, including measurements, calculations, and comparisons.

Parts of Fractions

A fraction consists of several components:

• numerator—The numerator, which is the top number of a fraction, represents the number of equal parts being considered.

• denominator—The denominator, which is the bottom number of a fraction, indicates the total number of equal parts into which the whole is divided.

• fraction bar—The fraction bar, also known as the vinculum, separates the numerator and denominator.

For example, in the fraction \(\frac{4}{5}\), \(4\) is the numerator and \(5\) is the denominator.

Types of Fractions

There are different types of fractions:

• proper fraction—A proper fraction contains a numerator that is less than the denominator, indicating a value less than \(1\). For example, \(\frac{1}{4}\) and \(\frac{2}{7}\) are both proper fractions.

• improper fraction—An improper fraction contains a numerator that is equal to or greater than the denominator, indicating a value greater than or equal to \(1\). For example, \(\frac{7}{6}\) and \(\frac{11}{2}\) are both improper fractions.

• mixed number—A mixed number consists of a whole number and a proper fraction. The number \(2 \frac{1}{4}\) (read as “two and one fourth”) is a mixed number. Mixed numbers are also known as mixed fractions.

Reducing Fractions

Reducing fractions simplifies them to their smallest possible form. This process is important for clarity and simplicity in mathematical calculations. One common method is the canceling method, where common factors between the numerator and denominator are canceled out.

If we want to reduce \(\frac{6}{10}\), we need to divide both the numerator and the denominator by a common factor. Let’s divide by \(2\):

\[\frac{6}{10} = \frac{6 \div 2}{10 \div 2} = \frac{3}{5}\]

We cannot further reduce the fraction. It’s in its simplest form.

Operations with Fractions

Just like with signed numbers, we can perform the four main operations (addition, subtraction, multiplication, and division) with fractions.

Addition

To add fractions with the same denominator, add the numerators without changing the denominator.

\[\frac{1}{4} + \frac{2}{4} = \frac{1+2}{4} = \frac{3}{4}\]

To add fractions whose denominators are not the same, we’ll need to find a common denominator. This is done by determining the least common multiple (LCM) of the denominators. Once you have the LCM, express both fractions with the common denominator.

Let’s use \(\frac{2}{3}\) and \(\frac{5}{7}\). The common denominator is \(3 \times 7 = 21\). So, the fractions can be written as:

\(\frac{2 \times 7}{3 \times 7} = \frac{14}{21}\) and \(\frac{5 \times 3}{7 \times 3} = \frac{15}{21}\)

Some pairs of denominators are both factors of one number, so you don’t need to multiply them together to find a common denominator. For example, look at the fractions \(\frac{1}{2}\) and \(\frac{1}{6}\). Since \(2\) is actually a multiple of \(6\), we can use \(6\) as the common denominator and keep the fraction \(\frac{1}{6}\) as it is. We just need to convert \(\frac{1}{2}\) to sixths:

\[\frac{1 \times 3}{2 \times 3}=\frac{3}{6}\]

To add \(\frac{1}{4}\) and \(\frac{1}{2}\), we first need to get the same denominator and then follow the adding process shown above:

\(\frac{1}{4}\) stays \(\frac{1}{4}\) and \(\frac{1}{2}\) becomes \(\frac{2}{4}\). Now, we add:

\[\frac{1}{4} + \frac{2}{4} = \frac{1+2}{4} = \frac{3}{4}\]

When dealing with mixed numbers, you must convert them into improper fractions, perform addition, and then convert the result back into a mixed number.

Here is an example:

\[2 \frac{1}{4} + 3 \frac{2}{3}\] \[= \frac{(2 \times 4) + 1}{4} + \frac{(3 \times 3)+2}{3}\] \[= \frac{9}{4} + \frac{11}{3}\] \[= \frac{9(3) + 11(4)}{4 \times 3}\] \[= \frac{71}{12}\] \[= 5 \frac{11}{12}\]

Subtraction

Subtracting fractions follows a process similar to addition. Ensure the denominators are the same, then subtract the numerators.

\[\frac{4}{5} - \frac{1}{5} = \frac{4-1}{5} = \frac{3}{5}\]

Multiplication

To multiply two fractions, simply multiply the numerators together and the denominators together.

\[\frac{1}{5} \times \frac{3}{7} = \frac{1 \times 3}{5 \times 7} = \frac{3}{35}\]

If a mixed number is involved, first change it to an improper fraction and then follow the same process.

\[4 \frac{2}{3} \times \frac{3}{5}\]

First, we change \(4 \frac{2}{3}\) into an improper fraction:

\[4 \frac{2}{3} = \frac{(4 \times 3)+2}{3} = \frac{14}{3}\]

Now, multiply as usual:

\[= \frac{14}{3} \times \frac{3}{5}\] \[= \frac{14 \times 3}{3 \times 5}\] \[= \frac{42}{15}\]

Note: , You may need to reduce the answer to its lowest terms, \(\frac{14}{5}\), or revert to a mixed number, \(2\frac{12}{15}\) or \(2\frac{4}{5}\).

Division

To divide fractions, multiply the first fraction by the reciprocal of the second fraction. A reciprocal is obtained by flipping the fraction so the numerator and denominator change places. To divide \(\frac{4}{5}\) by \(\frac{2}{3}\), we will multiply \(\frac{4}{5}\) by the reciprocal of the divisor, which gives us \(\frac{3}{2}\). So, we have:

\[\frac{4}{5} \div \frac{2}{3}\] \[= \frac{4}{5} \times \frac{3}{2}\] \[= \frac{4 \times 3}{5 \times 2}\] \[= \frac{12}{10}\]

We can reduce the resulting fraction if necessary.

When using mixed numbers, convert them into improper fractions, perform the operation, and then convert the result back into a mixed number.

Comparing Fractions

Comparing fractions is crucial in various mathematical contexts, from basic arithmetic to more complex problem-solving. When comparing fractions, the goal is to determine which fraction represents a larger or smaller portion of a whole. The following methods are useful for this process.

Cross-Multiplication Method

One method of comparing fractions with different denominators is the cross-multiplication method. To do this, multiply the numerator of the first fraction by the denominator of the second fraction and vice versa. The fraction whose numerator results in the bigger number has the greater value.

Let’s compare \(\frac{2}{3}\) and \(\frac{3}{5}\):

\[2 \times 5 = 10\text{ and }3 \times 3 = 9\] \[10>9\]

Therefore:

\[\frac{2}{3} > \frac{3}{5}\]

Finding a Common Denominator

If the fractions have different denominators, find a common denominator by determining the least common multiple (LCM) of the denominators. Then, express both fractions with the common denominator and compare the numerators. Let’s compare \(\frac{2}{3}\) and \(\frac{5}{7}\). The common denominator is \(3 \times 7 = 21\). So, the fractions can be written as:

\[\frac{2 \times 7}{3 \times 7} = \frac{14}{21}\text{ and }\frac{5 \times 3}{7 \times 3} = \frac{15}{21}\] \[15 > 14\]

Therefore:

\[\frac{5}{7} > \frac{2}{3}\]

Visual Representation

Another way to compare fractions is through visual representation, such as using a number line or a bar graph. Plot the fractions on the number line or represent them on a bar graph to visually compare their sizes.

Equivalent Fractions

Keep in mind that fractions can be equivalent even if they contain different numbers. If two fractions represent the same portion of a whole, they are considered equivalent.

\[\frac{2}{3} = \frac{6}{9}\]