PCAT Exam: Quantitative Reasoning - A Simple Guide To Basic Math

We shall start with a review of a few basic terms for describing numbers.
Recall that integers, or whole numbers, are numbers that are used to count things; the integers also include negative numbers and zero. This means such numbers as -9, -2, 0, 4, and 14 are integers. However, the integers do not include fractions or numbers that have nonzero digits after the decimal point.
One integer is a factor of another integer if it divides it evenly into the second integer. For example, 4 is a factor of 8 because 4 divides evenly into 8. Integers for which 2 is a factor are called even; otherwise, they are called odd.
A prime number is an integer that is greater than one whose only factors are itself and one. The list of the first few prime numbers includes 2, 3, 5, 7, 11, 13, 17, and 19.
A composite number is any integer greater than one that is not a prime number.

Fractions
The basic form of a fraction is , where x and y are integers.

By the definition of a fraction, .

In the expression ,  x is called the numerator, and y is called the denominator. The denominator can be any value except zero.

When working with fractions, the numerator and denominator can be multiplied or divided by the same number (other than zero) without changing the value of the fraction.

This means , as long as a and y are not zero; for example, .

If x and y are integers that do not share any common factors, then the fraction is said to be simplified.

In the example, is simplified, but is not.
With many fraction problems, the fractions may need to be rewritten so they all share the same denominator.

This process is called finding a common denominator for the fractions.

Given two fractions and , multiply the numerator and the denominator of the left fraction by d, and multiply the numerator and the denominator of the right fraction by b.

This operation results in the two fractions and , which share the common denominator .
The reciprocal or multiplicative inverse of the fraction is the fraction .

As with integers, there are 4 basic arithmetic operations that can be performed with fractions: addition, subtraction, multiplication, and division.

To add fractions together, first rewrite them so they share a common denominator.

Next, add the numerators together to get a new numerator.

The denominator is the least common denominator.
Example: .

In the last step, the fraction has been simplified by dividing the numerator and denominator by 3.

To subtract fractions, as with addition, first rewrite the fractions so they share a common denominator. Next, subtract the numerators to get a new numerator.

The denominator remains the common denominator that was found.
Example: .

To multiply fractions, there is no need to find a common denominator.

Instead, simply multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator.

Example: .

In the last step, the fraction has been simplified by dividing the numerator and denominator by 2.

When multiplying fractions, it is sometimes possible to make the problem simpler by doing the following operation first: cancel factors that appear in the numerator of one fraction and in the denominator of the other fraction, and then multiply.
Example: .
To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
Example: .
In the last step, the fraction has been simplified by dividing the numerator and denominator by 3.

A proper fraction has a smaller numerator than denominator. A mixed number has a larger numerator than denominator.

It is possible to rewrite such mixed numbers as a combination of integers and proper fractions, if an answer in that format is desired.

For example, , and can be written as .

Another operation closely related to multiplication is the exponential.

This operation is written as .

In this expression, x is called the base, and n is called the exponent.

When n is a positive integer, this indicates that x is multiplied by itself n times. For example, .

If the exponent is a negative integer, this means that the reciprocal of x is multiplied by itself n times.

For example, .

When the exponent is zero, the result is always equal to one, even if the base is zero.
A perfect square is a whole number that is the square of another whole number. For example, 16 and 36 are perfect squares because 16 is the square of 4 and 36 is the square of 6.

When the exponent is 2, the operation is called squaring the base.

When the exponent is 3, the operation is called cubing the base.
An exponent of for a positive integer n is also called an nth root, and can be written as .

The symbol on the right is also called a radical, and if the n is left blank next to the radical, it is assumed to be 2: .

By definition, the nth root of x is the number that, when raised to the n-th power, results in x.

That is, .

Here are the basic rules for working with exponents.
For any numbers :

- .
-
- .
- .
-
- .
- .
- .

Using these rules, it is possible to determine the value of an exponential expression with any fractional exponent: .

Note that for even roots of negative numbers, the result is an imaginary number.

The order of operations is a rule concerning the order in which to perform each operation in a mathematical expression.

The rule is to do the operations in the following order.
1. Parentheses
2. Exponents
3. Multiplication
4. Division
5. Addition
6. Subtraction

Parentheses take top priority. The operations inside the parentheses are performed first.
To help remember this order, many students like to use the mnemonic PEMDAS. Some students associate this mnemonic with a phrase to help them, such as “Pirates Eat Many Donuts at Sea.”
When working with radicals, it is often desirable to avoid having radicals in the denominator of the final answer. This operation is called rationalizing the denominator.

To do this, multiply the numbers on the top and bottom by enough of the same radicals so the product in the denominator eliminates the radicals.

This operation is best illustrated by an example.

Consider .

To remove the radical from the denominator, multiply by , as follows:


One additional operation that is very useful when discussing certain probabilities is the factorial.

The factorial is defined for non-negative integers in the following manner: , otherwise .

Percentages
A percentage can be thought as a fraction with a denominator of 100. Thus, .

The word “percent” comes from a Latin phrase that means “by the hundred.”
To convert a fraction to a percentage, the fraction is rewritten so that the denominator is 100.

Example: .

To convert a percentage to a fraction, the percentage is written over a denominator of 100, and then the result is simplified. For example, .

Decimals
A decimal number is a way of writing a number that uses a decimal point to show the part of the number that is a proper fraction.

For example, 11.4 means 11 plus .

The decimal point is indicated with a period in the United States (in some countries, a comma is used instead).
The decimal place of a digit in a decimal number describes the location of the digit relative to the decimal point, that is, how far to the right of the decimal point a digit appears. The first spot to the right of the decimal point is the tenths spot and indicates how many tenths are in the number. The second spot is the hundredths spot and indicates how many hundredths are in the number, and so on.

It is possible to convert a decimal to a percentage by multiplying the given percentage by 100, which just moves the decimal point to the right two places: .

To perform the opposite procedure and convert from a percentage to a decimal, divide the number by 100, which moves the decimal point to the left by two places: .
When working with decimals, it is sometimes easier to write a number by multiplying by a power of 10.

A power of 10 can be written as , where n is an integer.

The process of multiplying a decimal by shifts the decimal point n places to the right if n is positive, and n places to the left if n is negative.

If necessary, extra zeros are used as placeholders.

For example, .

When the non-decimal part of the first number is a single digit, this is known as scientific notation.

Often, the number is rounded to only include a certain number of nonzero digits in scientific notation.

For example, 2,503 might be rounded to .

A decimal number that is equal to some fraction is called a rational number.

Rational numbers always have decimal expressions.

Either the expressions terminate, meaning that beyond some point all their digits are zero, or the expressions repeat, meaning that their digits eventually repeat a pattern as they continue to the right.

For example, has a decimal expression of 1.3333… where the 3 continues repeating infinitely.

This number can be written by putting a bar over the digits that repeat: indicates the 3 repeats infinitely.

The number indicates that the 27 repeats infinitely, 1.127272727….

Decimal numbers that are not equal to any fraction are called irrational.
The real numbers are all the rational and irrational numbers.

Unit Conversions
Units express measured quantities. The idea is to compare a quantity to some fixed, standard quantity. For example, when measuring a distance, one can measure the distance in a multiple of a foot, which is a fixed, standard distance. One could also measure the distance as a multiple of a meter, which is another fixed, standard distance. Such standards exist for all kinds of physical quantities. When working with physical quantities, it is essential to keep track of which units are involved and to indicate the units in the answer
Converting between two different units requires knowledge of the ratio between the fixed standards. Given a quantity x in units A, to express the quantity in units of B, multiply x by the number of units of type B per unit of type A.

For example, the ratio between centimeters (cm) and inches (in) is 2.54 cm/in, meaning there are 2.54 centimeters in every inch.

To convert a distance of 4 inches to centimeters, the equation is .

Therefore, the distance of 4 inches can also be expressed as 10.16 centimeters.
One way to help keep track of this is to think of the units as cancelling one another.

If the units are written alongside the quantities, then one writes the previous example as .

The “inches” have cancelled one another.

This quantity can also be written as .
When converting between areas or volumes, if the ratio of the lengths is already known, one can use that ratio to find the area or volume.

For example, there are 100 centimeters in every meter (m); that is, the conversion between lengths is 100 cm/m.

If a problem requires converting 4 cubic meters, or 4 , into centimeters, then, to cancel the 3 factors of the meter unit and get 3 factors of the centimeter unit, multiply by 100 cm/m 3 times: .

As this example shows, keeping track of the units involved is very important.
The conversion ratio from units B to units A is always the reciprocal of the conversion from A to B.

The conversion from inches to centimeters is 2.54 cm/in, and the conversion from centimeters to inches is .

Log Base 10
A logarithm is the reverse operation of exponentiation.

In this guide, only logarithms with a base of 10 are considered.

This operation can be written as , although for base 10, sometimes the number is omitted and the logarithm is simply written as .

The basic definition of the logarithm with a base of 10 of the number x, when used as an exponent for 10, results in x. In other words, .

The obverse is also true: .

The following are the rules for simplifying logarithms with a base of 10.
1.
2.
3.
4.

5.

Ratios
A ratio expresses a proportional relationship between 2 quantities. It is generally written as , which can be read as “a to b.”

A ratio behaves very much like a fraction. For example, one can multiply both sides of a ratio by the same constant without changing the relationship represented by that ratio. For example, 2:3 and 4:6 both express the same ratios.

As with fractions, it is often desirable to express the final answer in a form in which the two sides of the ratio do not share any common factors.

Suppose an office building has 20 offices and 35 employees. The ratio of offices to employees is 20:35. To simplify this ratio, divide both sides by 5, resulting in a ratio of 4:7. This ratio means that in this building, there are 4 offices for every 7 employees.

For another example of applying ratios, suppose that in a school there are 50 girls and 65 boys. The ratio of girls to boys in this school is 50:65. Dividing both sides by 5 results in a ratio of 10:13. This means that in this school, there are 10 girls for every 13 boys.