GRE Math Review Math Review for the Quantitative Reasoning measure of the GRE® General Test www ets org http //www ets org Overview This Math Review will familiarize you with the mathematical skills a[.]
Integers
The integers are the numbers 1, 2, 3, , together with their negatives, 1, 2, 3, , and 0 Thus, the set of integers is , 3, 2, 1, 0, 1, 2, 3,
Positive integers are greater than zero, negative integers are less than zero, and zero is neither positive nor negative When integers are added, subtracted, or multiplied, the results are always integers; division is discussed separately Familiar elementary number facts include simple operations such as 7 + 8 = 15, 78 - 87 = -9, and 7 - (-18) = 25, along with their products like 7 × 8 = 56 Three important general facts about multiplying integers are summarized, emphasizing the fundamental principles of integer multiplication.
Fact 1: The product of two positive integers is a positive integer
Fact 2: The product of two negative integers is a positive integer
Fact 3: The product of a positive integer and a negative integer is a negative integer
Factors are integers involved in multiplication that contribute to the product For example, in the equation 2 × 3 × 10 = 60, the numbers 2, 3, and 10 are factors of 60 Additionally, integers such as 4, 15, 5, and 12 are also factors of 60, since 4 × 15 = 60 and 5 × 12 = 60 Understanding factors is essential for mastering multiplication and divisibility concepts.
The positive factors of 60 include 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, which are the integers that evenly divide 60 Both positive and negative factors exist, such as -2 and -30, since their product with their negatives also equals 60 These are all the factors of 60, confirming that 60 is divisible by each of its divisors Additionally, 60 is a multiple of each factor, illustrating its divisibility properties, with examples including other multiples and factors that highlight this relationship.
Example 1.1.1: The positive factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100
Example 1.1.2: 25 is a multiple of only six integers: 1, 5, 25, and their negatives
Example 1.1.3: The list of positive multiples of 25 has no end: 25, 50, 75, 100, ; likewise, every nonzero integer has infinitely many multiples
Example 1.1.4: 1 is a factor of every integer; 1 is not a multiple of any integer except 1 and −1
Example 1.1.5: 0 is a multiple of every integer; 0 is not a factor of any integer except 0
The least common multiple of two nonzero integers c and d is the least positive integer that is a multiple of both c and d For example, the least common multiple of 30 and 75 is
150 This is because the positive multiples of 30 are 30, 60, 90, 120, 150, 180, 210,
240, 270, 300, 330, 360, 390, 420, 450, , and the positive multiples of 75 are 75,
150, 225, 300, 375, 450, Thus, the common positive multiples of 30 and 75 are
150, 300, 450, , and the least of these is 150
The greatest common divisor (or greatest common factor) of two nonzero integers is the largest positive integer that divides both numbers without leaving a remainder For example, the greatest common divisor of 30 and 75 is 15 because the positive divisors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30, while the divisors of 75 are 1, 3, 5, 15, 25, and 75 The common positive divisors are 1, 3, 5, and 15, with 15 being the greatest Understanding the concept of the greatest common divisor is essential for simplifying fractions and solving various mathematical problems.
When an integer c is divided by an integer d, where d divides c, the result is always a divisor of c For example, dividing 60 by 6, a divisor of 60, yields 10, which is also a divisor of 60 If d is not a divisor of c, the division result can be interpreted in three ways: as a fraction or decimal, which are useful for viewing the result as a single number; or as a quotient with a remainder, which is beneficial for expressing the division in terms of integers only Each interpretation serves different purposes depending on the context. -**Sponsor**Need help making your content shine and follow SEO rules? It sounds like you're looking for a way to refine your article's key sentences and ensure each paragraph is coherent With [Article Generation](https://pollinations.ai/redirect-nexad/WAxYkiGG?user_id=983577), you can instantly get SEO-optimized content and save time and money compared to hiring a writer Imagine having a tool that helps you rewrite and optimize your articles effortlessly, ensuring they capture the essence of each paragraph while adhering to SEO best practices! It's like having your own content team without the hassle!
When dividing an integer c by a positive integer d that does not evenly divide c, the quotient includes a remainder For example, dividing 19 by 7 results in a quotient greater than 2 but less than 3, since 2 times 7 is less than 19, and 3 times 7 exceeds 19 This illustrates how quotients with remainders are determined when the divisor does not evenly divide the dividend.
When dividing 19 by 7, we observe that 19 is 5 more than 14 (which is 7 multiplied by 2) Therefore, the quotient is 2 with a remainder of 5, often written as 2 remainder 5 Generally, to divide an integer c by a positive integer d, you identify the greatest multiple of d less than or equal to c, expressed as qd, where q is the quotient The remainder is then calculated as the difference between c and this multiple, r = c - qd, and it always satisfies the condition 0 ≤ r < d, ensuring the remainder is non-negative and less than the divisor.
Here are four examples that illustrate a few different cases of division resulting in a quotient and remainder
Example 1.1.6: 100 divided by 45 is 2 remainder 10, since the greatest multiple of 45 that is less than or equal to 100 is ( )( ) 2 45 , or 90, which is 10 less than 100
When dividing 24 by 4, the quotient is 6 with a remainder of 0 because 24 is the greatest multiple of 4 less than or equal to itself In general, the remainder is zero if and only if the dividend is divisible by the divisor, indicating that the number divides evenly without any leftover.
Example 1.1.8: 6 divided by 24 is 0 remainder 6, since the greatest multiple of 24 that is less than or equal to 6 is ( )( )0 24 , or 0, which is 6 less than 6
Example 1.1.9: 32− divided by 3 is 11− remainder 1, since the greatest multiple of 3 that is less than or equal to 32− is (−11 3 ,)( ) or 33− , which is 1 less than 32.−
Here are five more examples
Example 1.1.10: 100 divided by 3 is 33 remainder 1, since 100 = ( )( )33 3 +1.
Example 1.1.11: 100 divided by 25 is 4 remainder 0, since 100 = ( )( ) 4 25 +0.
Example 1.1.12: 80 divided by 100 is 0 remainder 80, since 80 = ( )( 0 100 )+80.
Example 1.1.13: 13− divided by 5 is 3− remainder 2, since −13 = −( )( ) 3 5 + 2.
Example 1.1.14: 73− divided by 10 is 8− remainder 7, since −73 = −( )( ) 8 10 +7.
An integer divisible by 2 is classified as an even integer, while all other integers are considered odd When dividing an odd integer by 2, the remainder is always 1, highlighting its odd nature The set of even integers includes { , -4, -2, 0, 2, 4, 6, }, whereas the set of odd integers comprises { , -3, -1, 1, 3, 5, } Understanding these fundamental classifications helps in various mathematical and programming applications involving integers.
, 5, 3, 1, 1, 3, 5, Here are six useful facts regarding the sum and product of even and odd integers
Fact 1: The sum of two even integers is an even integer
Fact 2: The sum of two odd integers is an even integer
Fact 3: The sum of an even integer and an odd integer is an odd integer
Fact 4: The product of two even integers is an even integer
Fact 5: The product of two odd integers is an odd integer
Fact 6: The product of an even integer and an odd integer is an even integer
A prime number is a natural number greater than 1 that has only two positive divisors: 1 and itself The first ten prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29 Not all numbers are prime; for example, 14 is not a prime number because it has four positive divisors: 1, 2, 7, and 14 Additionally, 1 is not considered a prime number, and unique among primes, 2 is the only even prime number.
Every integer greater than 1 is either a prime number or can be uniquely expressed as a product of prime factors, also known as prime divisors Prime factorization is the process of breaking down a number into its fundamental prime components This unique factorization property underpins many aspects of number theory and is essential for various mathematical applications For example, six different numbers can each be represented through their prime factorization, highlighting the importance of this concept in understanding the structure of integers.
An integer greater than 1 that is not a prime number is called a composite number The first ten composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18.
Fractions
A fraction is a number of the form c, d where c and d are integers and d 0 The integer c is called the numerator of the fraction, and d is called the denominator For example,
A rational number is any number expressed as a fraction, such as 7/5, where 7 minus is the numerator and 5 is the denominator These numbers are known as rational numbers because they can be written as a ratio of two integers It's important to note that every integer, such as n, is also a rational number since it can be represented as a fraction, with the integer as the numerator and 1 as the denominator.
If both the numerator c and the denominator d, where d 0, are multiplied by the same nonzero integer, the resulting fraction will be equivalent to c. d
Example 1.2.1: Multiplying the numerator and denominator of the fraction 7
Multiplying the numerator and denominator of the fraction 7
For all integers c and d, the fractions c, d
If both the numerator and denominator of a fraction have a common factor, then the numerator and denominator can be factored and the fraction can be reduced to an equivalent fraction
To add two fractions with the same denominator, you add the numerators and keep the same denominator
To add two fractions with different denominators, start by finding a common denominator, which is a multiple of both denominators Next, convert each fraction to an equivalent fraction with this common denominator Finally, add the numerators together while keeping the common denominator unchanged to obtain the sum of the fractions.
Example 1.2.5: To add the two fractions 1
5 first note that 15 is a common denominator of the fractions
Then convert the fractions to equivalent fractions with denominator 15 as follows
Therefore, the two fractions can be added as follows
15 The same method applies to subtraction of fractions
To multiply two fractions, multiply the two numerators and multiply the two denominators Here are two examples
To divide one fraction by another, first invert the second fraction (that is, find its reciprocal), then multiply the first fraction by the inverted fraction Here are two examples
48 is called a mixed number It consists of an integer part and a fraction part, where the fraction part has a value between 0 and 1; the mixed number 3
To convert a mixed number to a fraction, convert the integer part to an equivalent fraction with the same denominator as the fraction, and then add it to the fraction part
Example 1.2.10: To convert the mixed number 3
48 to a fraction, first convert the integer 4 to a fraction with denominator 8, as follows
Numbers of the form c, d where either c or d is not an integer and d 0, are called fractional expressions Fractional expressions can be manipulated just like fractions Here are two examples
Solution: Note that 6 is a common denominator of both numbers
2 p is equivalent to the number 3
3 p is equivalent to the number 2
Solution: Note that the numerator of the number is 1
2 and the denominator of the number is 3
5 Note also that the reciprocal of the denominator is 5
which can be simplified to 5
Exponents and Roots
Exponents represent repeated multiplication of a number by itself, such as 3^4, which equals 81, and 5^3, which equals 125 In these expressions, the base is the number being multiplied, and the exponent indicates how many times the base is multiplied by itself For example, in 3^4, 3 is the base and 4 is the exponent, read as “3 to the fourth power.” Understanding exponents is essential for expressing large numbers efficiently and performing exponential calculations.
When the exponent is 2, we call the process squaring Thus, 6 squared is 36; that is, 6 2
= 36 Similarly, 7 squared is 49; that is, 7 2 7 7 = 49
When raising negative numbers to powers, the results can be positive or negative depending on the exponent For example, (-3)^2 equals 9, while (-3)^5 equals -243 A negative number raised to an even power always yields a positive result, whereas raising it to an odd power always results in a negative number.
− = − = − Exponents can also be negative or zero; such exponents are defined as follows
The exponent zero: For all nonzero numbers a, a 0 1 The expression 0 is 0 undefined
Negative exponents: For all nonzero numbers a, 1 1
and so on Note that a a 1 a 1 a 1
A square root of a nonnegative number n is a number r such that r 2 n For example, 4 is a square root of 16 because 4 2 16 Another square root of 16 is −4, since
All positive numbers have two square roots: one positive and one negative The only square root of zero is zero itself The notation with the square root symbol over a nonnegative number represents the nonnegative (or positive) square root of that number For example, √16 equals 4, since 4 squared is 16.
100 10, 100 10, and 0 0 Square roots of negative numbers are not defined in the real number system
Here are four important rules regarding operations with square roots, where a 0 and 0. b
A square root is a root of order 2, representing a number that, when squared, equals the original number n Higher order roots of a positive number n, such as the cube root (3√n) and fourth root (4√n), are defined similarly, where raising these roots to the respective powers (3 or 4) results in n These roots follow rules analogous to those of square roots, but with exponents replaced by 3 or 4, enabling consistent computation and understanding of roots of different orders.
There are some notable differences between odd order roots and even order roots (in the real number system):
For odd order roots, there is exactly one root for every number n, even when n is negative
For even order roots, there are exactly two roots for every positive number n and no roots for any negative number n
For example, 8 has exactly one cube root, 3 8 2, but 8 has two fourth roots, 4 8 and
whereas 8 has exactly one cube root, 3 8 2, but 8 has no fourth root, since it is negative.
Decimals
The decimal number system is based on representing numbers using powers of 10, where each digit's place value corresponds to a specific power of 10 For instance, in the number 7,532.418, each digit has a designated place value that reflects its position relative to the decimal point This system allows for precise representation of both whole numbers and fractional parts by assigning powers of 10 to each digit according to its position Understanding the place value in the decimal system is essential for accurate reading, writing, and calculation with numbers.
That is, the number 7,532.418 can be written as
Alternatively, it can be written as
Converting a decimal with a finite number of digits to an equivalent fraction is a straightforward process, as each decimal can be expressed as an integer divided by a power of 10 Since each place value to the right of the decimal point corresponds to a power of 10, this method is simple and systematic For example, converting 0.75, 0.4, and 0.125 results in the fractions 75/100, 4/10, and 125/1000, respectively.
Every fraction with integers in the numerator and denominator can be converted to an equivalent decimal through division This process involves dividing the numerator by the denominator using long division The resulting decimal will either terminate or repeat, depending on the factors of the denominator Typically, terminating decimals occur when the denominator has only 2 and/or 5 as prime factors Understanding how to convert fractions to decimals helps in better numerical comprehension and simplifies calculations.
25 or repeat without end, as in
To indicate the repeating part of a non-terminating decimal, place a bar over the digits that repeat For example, in decimal conversions of fractions, the repeating sequence is clearly shown using this notation Four common examples demonstrate how fractions translate into decimals with repeating digits This method effectively highlights the recurring portion of long, non-terminating decimals for clearer understanding and proper mathematical notation.
Every rational number can be expressed as either a terminating or repeating decimal Conversely, every decimal that terminates or repeats corresponds to a rational number Fractions with integer numerators and denominators are equivalent to these types of decimals, highlighting the fundamental relationship between rational numbers and their decimal representations.
Not all decimal representations are terminating or repeating For example, the decimal equivalent of the square root of 2 is 1.41421356237 , which neither terminates nor repeats, indicating it is an irrational number Another example is 0.020220222022220222220 , featuring groups of consecutive 2s separated by zeros, with each group increasing in length These non-terminating, non-repeating decimals are classified as irrational numbers.
Real Numbers
The set of real numbers includes all rational and irrational numbers, encompassing integers, fractions, and decimal numbers These numbers are represented visually on the real number line, also known as the number line, which illustrates the continuum of real numbers Arithmetic and the concept of the real number line are fundamental in understanding how different types of numbers relate to each other within the real number system.
Every real number is represented as a point on the number line, establishing a one-to-one correspondence between numbers and points Each point on the number line uniquely corresponds to a real number, creating a seamless visual representation of the number system The placement of numbers on the line reflects their relative values, with all numbers to the left being smaller and those to the right being larger, illustrating the ordered nature of real numbers.
0 are negative and all numbers to the right of 0 are positive As shown in
Arithmetic Figure 2, the negative numbers 0.4, 1, 3
2, 5, and 3 are to the left of 0, and the positive numbers 1
2, 1, 2, 2, 2.6, and 3 are to the right of 0 Only the number 0 is neither negative nor positive
A real number x is less than a real number y when it is located to the left of y on the number line, denoted as x < y Conversely, a real number y is greater than x when it is positioned to the right of x on the number line, represented as y > x For example, on the number line, numbers increase from left to right, making it easy to compare their sizes based on their positions Understanding these inequalities helps in grasping the concept of order among real numbers.
Arithmetic Figure 2 shows the following three relationships
In mathematics, a real number x is considered less than or equal to a real number y (x ≤ y) if x is located to the left of, or at the same point as, y on the number line Conversely, a real number y is greater than or equal to a real number x (y ≥ x) if y is located to the right of, or at the same point as, x on the number line These inequalities visually represent the relative positions of numbers on the number line, aiding in understanding their order and relationships.
A real number x is between 2 and 3 on the number line if it satisfies the inequality 2 < x < 3, meaning x is greater than 2 and less than 3 The set of all such numbers is called an interval, with 2 < x < 3 representing this interval Notably, the endpoints 2 and 3 are not included in the interval Depending on whether the endpoints are included or excluded, there are four types of intervals, which can be represented by different inequalities.
There are four types of intervals with a single endpoint, each including all real numbers either to the right or left of that point, with some intervals including the endpoint itself and others excluding it These different interval types are represented by specific inequalities, which define whether the interval includes or excludes the endpoint and whether it extends infinitely in one direction Understanding these interval classifications is essential for accurately representing solutions to inequalities and understanding the properties of real number sets in mathematics.
The entire real number line is also considered to be an interval
The distance between a number x and 0 on the number line is called the absolute value of x, written as x Therefore, 3 3 and 3 3 because each of the numbers 3 and
3 is a distance of 3 from 0 Note that if x is positive, then x x; if x is negative, then
; x x and lastly, 0 0 It follows that the absolute value of any nonzero number is positive Here are three examples
Here are twelve general properties of real numbers that are used frequently In each property, r, s, and t are real numbers
Property 5: If rs = 0, then either r = 0 or s = 0 or both
Property 6: Division by 0 is undefined
Property 7 states that when both r and s are positive, their sum (r + s) and their product (rs) are also positive According to Property 8, if both r and s are negative, then their sum (r + s) remains negative, while their product (rs) becomes positive Property 9 explains that if r is positive and s is negative, their product (rs) is negative Understanding these properties helps clarify how the signs of numbers influence their sum and product, which is essential for mastering algebraic concepts.
Property 10: r s r s This is known as the triangle inequality
Example: If r = 5 and s 2, then 5 2 5 2 3 3 and
Property 12: If r 1, then r 2 r If 0 s 1, then s 2 s.
Ratio
The ratio of one quantity to another expresses their relative sizes and is commonly represented as a fraction, with the first quantity as the numerator and the second as the denominator For example, if s and t are positive quantities, their ratio can be written as s/t, or using the notation “s to t,” which also signifies the same relationship To illustrate, if a basket contains 2 apples and 3 oranges, the ratio of apples to oranges is 2:3, highlighting the comparison between the two quantities.
Ratios compare quantities and can be simplified to their lowest terms, similar to fractions For instance, if a basket contains 8 apples and 12 oranges, the ratio of apples to oranges is 2:3 after reduction Likewise, the ratio 9:12 is equivalent to 3:4 when simplified.
When comparing three or more positive quantities, such as r, s, and t, their relative sizes can be expressed as a ratio, for example, "r to s to t." For instance, if a basket contains 5 apples, 30 pears, and 20 oranges, the ratio of apples to pears to oranges is 5 to 30 to 20 This ratio can be simplified to 1 to 6 to 4 by dividing each quantity by their greatest common divisor, which is 5.
A proportion is an equation relating two ratios; for example, 9 3
12 4 To solve a problem involving ratios, you can often write a proportion and solve it by cross multiplication
Example 1.6.1: To find a number x so that the ratio of x to 49 is the same as the ratio of 3 to 21, you can first write the following equation
You can then cross multiply to get 21 x 3 49 , and finally you can solve for x to get
Percent
The term "percent" means "per hundred" or "hundredths," representing ratios used to illustrate parts of a whole as 100 parts Percentages are commonly employed to convey proportions in various contexts, such as discounts, interest rates, and data analysis They can easily be converted into fractions or decimals for easier calculations For example, understanding how to convert percentages into their fractional or decimal forms enhances mathematical comprehension and practical application.
Example 1.7.1: 1 percent means 1 part out of 100 parts The fraction equivalent of 1 percent is 1
100, and the decimal equivalent is 0.01
Example 1.7.2: 32 percent means 32 parts out of 100 parts The fraction equivalent of
100, and the decimal equivalent is 0.32
Example 1.7.3: 50 percent means 50 parts out of 100 parts The fraction equivalent of
100, and the decimal equivalent is 0.50
In a fraction, the part is represented by the numerator and the whole by the denominator, illustrating their relationship Percents are commonly written using the % symbol instead of the word “percent,” making them quick to recognize For example, the percent values can be converted into fractions and decimals; here are five examples of percentages expressed with the % symbol, along with their fractional and decimal equivalents for better understanding and mathematical accuracy.
Be careful not to confuse 0.01 with 0.01% The percent symbol matters For example, 0.01 =1% but 0.01
To calculate a percentage, divide the part by the whole to obtain the decimal form, then multiply the result by 100 The final percentage is expressed as that number followed by the word “percent” or the "%" symbol, providing a clear measure of the part relative to the total.
Example 1.7.9: If the whole is 20 and the part is 13, you can find the percent as follows
Example 1.7.10: What percent of 150 is 12.9 ?
Solution: Here, the whole is 150 and the part is 12.9, so
To determine a specific percentage of a whole, you can simply multiply the total by the decimal form of the percentage Alternatively, setting up a proportion allows you to find the part accurately Both methods are effective for calculating a percentage of a whole efficiently.
Example 1.7.11: To find 30% of 350, you can multiply 350 by the decimal equivalent of 30%, or 0.3, as follows
To find 30% of 350 using a proportion, set up a ratio where 30 parts out of 100 parts equals an unknown number x out of 350 This method involves solving the proportion \( \frac{30}{100} = \frac{x}{350} \), which allows you to determine the exact value of x Using proportions provides a clear and accurate way to calculate percentages of a number by maintaining the relationship between parts and the whole This approach is especially useful for understanding how percentages relate to their original quantities in a straightforward manner.
To calculate the whole from a given percent and part, you can convert the percentage to its decimal form or set up a proportion and solve it Using the decimal equivalent simplifies the process, allowing for quick calculation, while setting up a proportion provides a clear method to find the total when part and percentage are known Both approaches are essential methods for accurate percentage and whole calculations in various situations.
Example 1.7.12: 15 is 60% of what number?
Solution: Use the decimal equivalent of 60% Because 60% of some number z is 15, multiply z by the decimal equivalent of 60%, or 0.6
0.6z Now solve for z by dividing both sides of the equation by 0.6 as follows
15 25 z 0.6 Using a proportion, look for a number z such that
Understanding percentages involves recognizing that a part is compared to a whole, known as the base While percentages often represent a part less than the whole, it's important to note that the part can be equal to or greater than the whole When the numerator exceeds the base, the resulting percentage is greater than 100%, indicating a value that surpasses the total.
Note that the decimal equivalent of 300% is 3.0 and the decimal equivalent of 250% is 2.5
Percent Increase, Percent Decrease, and Percent Change
Percent change is used to measure how a quantity shifts from an initial positive amount to a new positive amount, such as when an employee’s salary is increased It helps calculate the magnitude of the change as a percentage of the original value, providing a clear understanding of the relative growth or decrease.
The initial amount is 600, and the increased amount is 750 The absolute increase is calculated as 750 minus 600, which equals 150 To find the percentage increase, divide the increase (150) by the original amount (600) and multiply by 100, resulting in a 25% increase.
We say the percent increase is 25% Sometimes this computation is written as follows
If a quantity doubles in size, then the percent increase is 100% For example, if a quantity increases from 150 to 300, then the percent increase is calculated as follows amount of increase 300 150 150 100% base 150 150
If a quantity decreases from 500 to 400, calculate the percent decrease as follows amount of decrease 500 400 100 20 20% base 500 500 100
To accurately calculate a percent increase, use the original smaller number as the base Conversely, when determining a percent decrease, the larger initial number serves as the base In both cases, the base refers to the original value before the change occurred, ensuring precise percentage calculations.
Example 1.7.15: An investment in a mutual fund increased by 12% in a single day If the value of the investment before the increase was $1,300, what was the value after the increase?
Solution: The percent increase is 12% Therefore, the value of the increase is 12% of
$1,300, or, using the decimal equivalent, the increase is 0.12 $1,300 $156 Thus, the value of the investment after the change is
The final investment amount is obtained by adding the initial investment of $1,300 to the 12% increase, resulting in a total of 112% of the original amount This can also be calculated by multiplying the initial investment by the decimal equivalent of 112%, which is 1.12.
A quantity can experience multiple successive percent changes, with each change calculated based on the previous result This means that the percentage increase or decrease is compounded over each step, affecting the overall total Understanding how successive percent changes work is essential for accurately analyzing dynamic processes in finance, statistics, and data analysis.
Example 1.7.16: On September 1, 2013, the number of children enrolled in a certain preschool was 8% less than the number of children enrolled at the preschool on
September 1, 2012 On September 1, 2014, the number of children enrolled in the preschool was 6% greater than the number of children enrolled in the preschool on
September 1, 2013 By what percent did the number of students enrolled change from September 1, 2012 to September 1, 2014?
Solution: The initial base is the enrollment on September 1, 2012 The first percent change was the 8% decrease in the enrollment from September 1, 2012, to
September 1, 2013 As a result of this decrease, the enrollment on September 1, 2013, was ( 100 −8 % ) , or 92%, of the enrollment on September 1, 2012 The decimal equivalent of 92% is 0.92
So, if n represents the number of children enrolled on September 1, 2012, then the number of children enrolled on September 1, 2013, is equal to 0.92 n
The new base is the enrollment on September 1, 2013, which is 0.92 n The second percent change was the 6% increase in enrollment from September 1, 2013, to
September 1, 2014 As a result of this increase, the enrollment on September 1, 2014, was
( 100 +6 %, ) or 106%, of the enrollment on September 1, 2013 The decimal equivalent of 106% is 1.06
Thus, the number of children enrolled on September 1, 2014, was ( 1.06 0.92 )( n ) , which is equal to 0.9752 n
The percent equivalent of 0.9752 is 97.52%, which is 2.48% less than 100% Thus, the percent change in the enrollment from September 1, 2012 to September 1, 2014, is a 2.48% decrease
Exercise 3 Which of the integers 312, 98, 112, and 144 are divisible by 8 ?
(a) What is the prime factorization of 372 ?
(b) What are the positive divisors of 372 ?
(a) What are the prime divisors of 100 ?
(b) What are the prime divisors of 144 ?
Exercise 6 Which of the integers 2, 9, 19, 29, 30, 37, 45, 49, 51, 83, 90, and 91 are prime numbers?
Exercise 7 What is the prime factorization of 585 ?
Exercise 8 Which of the following statements are true?
Exercise 10 If a person’s salary increases from $200 per week to $234 per week, what is the percent increase in the person’s salary?
Exercise 11 If an athlete’s weight decreases from 160 pounds to 152 pounds, what is the percent decrease in the athlete’s weight?
A stock originally valued at $40 per share experiences a 20% increase in its value, raising it to $48 Subsequently, the stock's value decreases by 25%, reducing it to $36 per share After these fluctuations, the stock's final value per share is $36.
The kennel contains a total of 20 dogs and cats combined, with a ratio of 3:2 for dogs to cats To find the number of cats, we can set up a proportion based on the ratio By dividing the total into parts representing the ratio, we determine that there are 8 cats at the kennel This problem demonstrates how to use ratios and basic algebra to solve real-world math questions involving total counts and proportional relationships.
Exercise 14 The integer c is even, and the integer d is odd For each of the following integers, indicate whether the integer is even or odd
Exercise 15 When the positive integer n is divided by 3, the remainder is 2, and when n is divided by 5, the remainder is 1 What is the least possible value of n ?
(b) The positive divisors of 372 are 1, 2, 3, 4, 6, 12, 31, 62, 93, 124, 186, and 372.
(a) 100 = ( )( ) 2 2 5 , 2 so the prime divisors are 2 and 5
(b) 144 = ( )( ) 2 4 3 , 2 so the prime divisors are 2 and 3
(c) False; division by 0 is undefined.
The algebra review covers key topics such as algebraic expressions, equations, inequalities, and functions, providing a solid foundation for problem-solving It emphasizes applying these concepts through numerous real-life word problems to enhance understanding The review concludes with an exploration of coordinate geometry and graphing functions, highlighting their importance as essential algebraic tools for analyzing and solving mathematical problems effectively.
Algebraic Expressions
A variable is a symbol, often represented by letters like x or y, that stands for an unknown quantity In algebra, variables are used to construct expressions that incorporate one or more unknown values An algebraic expression can be a single term or a combination of multiple terms, illustrating various relationships between variables Examples of algebraic expressions include different combinations of variables and constants, showcasing their versatility in representing mathematical ideas Understanding variables and algebraic expressions is fundamental to mastering algebra and solving equations effectively.
In the examples above, 2x is a single term, 1 y − 4 has two terms, w z 3 +5z 2 − z 2 + 6 has four terms, and 8 n + p has one term
In the expression w z 3 + 5z 2 − z 2 + 6, the terms 5z 2 and −z 2 are called like terms because they share the same variables with identical exponents A term without any variables is known as a constant term Additionally, the numerical factor multiplied by the variables is called the coefficient of the term.
A polynomial is a mathematical expression consisting of a finite sum of terms, where each term is either a constant or a product of a coefficient and variables with positive integer exponents The degree of a term is determined by summing the exponents of the variables within it, with variables written without exponents considered to have a degree of 1 Constant terms have a degree of 0 The degree of the entire polynomial is determined by the highest degree among its individual terms.
Polynomials of degrees 2 and 3 are known as quadratic and cubic polynomials, respectively
Example 2.1.5: The expression 4x 6 +7x 5 −3x+ 2 is a polynomial in one variable, x The polynomial has four terms
The first term is 4x 6 The coefficient of this term is 4, and its degree is 6
The second term is 7x 5 The coefficient of this term is 7, and its degree is 5
The third term is 3 − x The coefficient of this term is 3,− and its degree is 1
The fourth term is 2 This term is a constant, and its degree is 0
Example 2.1.6: The expression 2x 2 −7xy 3 −5 is a polynomial in two variables, x and y The polynomial has three terms
The first term is 2x 2 The coefficient of this term is 2, and its degree is 2
The second term is −7xy 3 The coefficient of this term is −7; and, since the degree of x is 1 and the degree of y 3 is 3, the degree of the term −7xy 3 is 4
The third term is −5, which is a constant term The degree of this term is 0
In this example, the degrees of the three terms are 2, 4, and 0, so the degree of the polynomial is 4
Example 2.1.7: The expression 4x 3 −12x 2 − +x 36 is a cubic polynomial in one variable
The same rules that govern operations with numbers apply to operations with algebraic expressions
In an algebraic expression, like terms can be combined by simply adding their coefficients, as the following three examples show
A number or variable that is a factor of each term in an algebraic expression can be factored out, as the following three examples show
Example 2.1.13: For values of x where it is defined, the algebraic expression
++ can be simplified as follows
First factor the numerator and the denominator to get ( )
Since (x + 2) appears as a common factor in both the numerator and denominator, canceling it simplifies the expression to an equivalent form This simplification holds for all values of x where the original expression is defined, meaning (x + 2) is not zero Consequently, the simplified expression is equal to 7x for all x in the domain where the original expression is valid.
A fraction is undefined when its denominator equals zero In the original expression, the denominator is 2(x + 2), which equals zero when x = -2 Therefore, the expression is undefined at x = -2, and it is valid for all other values of x except -2.
To multiply two algebraic expressions, each term of the first expression is multiplied by each term of the second expression, and the results are then added together This process is fundamental in algebra and helps simplify complex expressions Understanding how to correctly perform multiplication of algebraic expressions is essential for solving various algebraic problems effectively.
First multiply each term of the expression x +2 by each term of the expression
Then simplify each term to get 3x 2 − 7x +6x −14.
Finally, combine like terms to get 3x 2 − −x 14.
So you can conclude that ( x + 2)(3x −7) = 3x 2 − −x 14
A statement of equality between two algebraic expressions that is true for all possible values of the variables involved is called an identity Here are seven examples of identities
Identities can be used to modify and simplify algebraic expressions, as in the following example
Example 2.1.15: Simplify the algebraic expression
Solution: Use the identity a 2 b 2 ab a b to factor the numerator of the expression, and use the identity ca cb c a b to factor the denominator of the expression to get
When the factor \( x - 3 \) appears in both the numerator and denominator, it can be canceled out provided \( x \neq 3 \), because the expression is undefined when the denominator equals zero Therefore, for all \( x \neq 3 \), the simplified expression is equivalent to 3.
Rules of Exponents
In the algebraic expression x a , where x is raised to the power a, x is called the base and a is called the exponent For all integers a and b and all positive numbers x, except
1, x the following property holds: If x a x b , then a b.
Example: If 2 3 c 1 2 , 10 then 3c 1 10, and therefore, c 3
Here are seven basic rules of exponents In each rule, the bases x and y are nonzero real numbers and the exponents a and b are integers, unless stated otherwise
Example B: ( )−3 0 =1 Note that 0 is not defined 0 Rule 5: ( )( ) x a y a = ( ) xy a
Algebraic identities are essential tools for simplifying expressions efficiently While some algebraic expressions may appear similar and seem to be reducible, they are not always simplifiable in the same way, so caution is necessary To prevent common mistakes when working with exponents, it is important to remember six key cases that govern their manipulation Understanding these specific rules ensures accurate simplification and enhances overall algebraic problem-solving skills.
Instead, ( ) x a b = x ab and x x a b = x a b + ; for example, ( ) 4 2 3 = 4 6 and 4 4 2 3 = 4 5
In particular, note that ( x+ y ) 2 = x 2 + 2xy + y 2 ; that is, the correct expansion contains the additional term 2 xy
Instead, ( )− x 2 = x 2 Note carefully where each negative sign appears
But it is true that x y x y. a a a
Solving Linear Equations
An equation is a statement of equality between two mathematical expressions When an equation involves variables, the solutions are the values that make the equation true Solving an equation involves finding these variable values that satisfy the equation Equations with the same solutions are called equivalent equations; for example, x + 1 = 2 and 2x + 2 = 4 are equivalent because they are true when x = 1 The main approach to solving equations is to transform them into simpler, equivalent forms until the solutions become obvious.
The following three rules are important for producing equivalent equations
Adding or subtracting the same constant from both sides of an equation maintains its equality, making the new equation equivalent to the original This fundamental rule is essential for solving equations, as it allows for simplification without altering the solution Applying this principle helps in isolating variables and progressing toward the solution efficiently Understanding this concept is crucial for mastering algebraic manipulations and solving equations accurately.
Rule 2 states that multiplying or dividing both sides of an equation by the same nonzero constant preserves the equality This mathematical principle ensures that the new equation remains equivalent to the original one Applying this rule is essential for solving equations efficiently and accurately It is a fundamental concept in algebra that helps simplify complex equations while maintaining their integrity.
Rule 3: When an expression that occurs in an equation is replaced by an equivalent expression, the equality is preserved and the new equation is equivalent to the original equation
Example: Since the expression 2( x +1) is equivalent to the expression
2x+ 2,when the expression 2( x +1) occurs in an equation, it can be replaced by the expression 2x+ 2, and the new equation will be equivalent to the original equation
A linear equation is an algebraic expression involving one or more variables, where each term is either a constant or a variable multiplied by a coefficient These equations do not contain variables multiplied together or variables raised to powers greater than one For example, equations like 3x + 2 = 0 or y - 5 = 0 are simple linear equations that demonstrate this form Understanding linear equations is essential for solving various algebraic problems in mathematics and related fields.
2x+ =1 7x and 10x −9y − =z 3 are linear equations, but x + y 2 = 0 and xz = 3 are not
Linear Equations in One Variable
To solve a linear equation in one variable, systematically simplify the equation by combining like terms and applying the rules for creating equivalent equations Continue this process until the equation is reduced to a form where the solution becomes clear and easy to determine.
Example 2.3.1: Solve the equation 11x − −4 8x = 2( x + 4) −2x as follows
Combine like terms on the left side to get 3x− =4 2( x +4) −2 x
Replace 2 ( x + 4 ) by 2x +8 on the right side to get 3x− =4 2x + −8 2 x
Combine like terms on the right side to get 3x− =4 8.
Add 4 to both sides to get 3x
Divide both sides by 3 to get 3 12
You can verify your solution by substituting it back into the original equation If the calculated value on the right-hand side matches the value on the left-hand side, then your solution is correct This quick check ensures the accuracy of your solution and confirms its validity.
Substituting the solution x = 4 into the left-hand side of the equation
Substituting the solution x = 4 into the right-hand side of the equation gives
2 x +4 −2x = 2 4+ 4 −2 4 = 2 8 ( ) − =8 8 Since both substitutions give the same result, 8, the solution x = 4 is correct
Linear equations can sometimes have no solutions; for example, the equation 2x + 3 = 7(1 + x) has no solution because it simplifies to a false statement Conversely, some equations that appear to be linear may actually be identities, true for all values of x, such as 3x - 6 = 2(3 - x) Recognizing these scenarios is important when solving linear equations, as they indicate whether an equation has no solution, a unique solution, or infinitely many solutions.
Linear Equations in Two Variables
A linear equation in two variables, x and y, can be written in the form ax +by = c
GRE Math Review 46 where a, b, and c are real numbers and neither a nor b is equal to 0 For example,
3x+2y = 8 is a linear equation in two variables
A solution to an equation is an ordered pair of numbers (x, y) that satisfy the equation when substituted into it For example, both the ordered pairs (2, 1) and (-2, 3, 5) are solutions of the given equation, demonstrating how specific pairs make the equation true Understanding solutions is essential for solving equations and analyzing relationships between variables in mathematics.
3x+2y = 8, but the ordered pair (1, 2 is not a solution Every linear equation in two ) variables has infinitely many solutions
A system of equations consists of two or more equations involving multiple variables, where the individual equations are known as simultaneous equations Solving a system of equations in two variables, like x and y, involves finding ordered pairs (x, y) that satisfy all given equations Similarly, solving a system with three variables—x, y, and z—requires determining ordered triples (x, y, z) that fulfill all equations simultaneously The concept extends to systems with more than three variables, with solutions defined in a consistent manner across all equations.
Linear systems of equations in two variables typically include two equations involving one or both of the variables, and they often have a unique solution—meaning a single ordered pair satisfies both equations However, some systems may have no solutions at all or infinitely many solutions, depending on their specific relationships.
There are two primary methods for solving systems of linear equations: substitution and elimination The substitution method involves manipulating one equation to express a variable in terms of the other, then substituting this expression into the second equation This approach simplifies the system, making it easier to find the solution Both methods are essential techniques for solving linear systems efficiently.
Example 2.3.2: Use substitution to solve the following system of two equations
Part 1: You can solve for y as follows
Express x in the second equation in terms of y as x = −2 2 y
Substitute 2− 2y for x in the first equation to get 4 2 ( −2 y ) +3 y =13.
Replace 4 2( −2y ) by 8−8y on the left side to get 8−8y +3y
Combine like terms to get 8−5y
Part 2: Now, you can use the fact that y = −1 to solve for x as follows
Substitute 1− for y in the second equation to get x+ 2( )− =1 2.
Thus, the solution of the system is x = 4 and y = −1, or ( x y, ) (= 4, 1 − )
The elimination method aims to make the coefficients of a specific variable identical in both equations, allowing for the elimination of that variable This is achieved by adding or subtracting the equations, simplifying the system and making it easier to solve Using this technique streamlines solving simultaneous equations efficiently.
Example 2.3.3: Use elimination to solve the following system of two equations
+ + (Note that this is the same system of equations that was solved by substitution in
Solution: Multiplying both sides of the second equation by 4 yields 4( x + 2y ) = 4 2 ,( ) or 4x+8y = 8.
Now you have two equations with the same coefficient of x
+ + If you subtract the equation 4x +8y = 8 from the equation 4x+3y , the result is
− = Thus, y = −1, and substituting 1− for y in either of the original equations yields x = 4.
Again, the solution of the system is x = 4 and y = −1, or ( x y , ) (= 4, 1 − )
Solving Quadratic Equations
A quadratic equation in the variable x is an equation that can be written in the form
2 0 ax + bx + =c where a, b, and c are real numbers and a ≠ 0 Quadratic equations have zero, one, or two real solutions
One way to find solutions of a quadratic equation is to use the quadratic formula:
− ± − where the notation ± is shorthand for indicating two solutions − one that uses the plus sign and the other that uses the minus sign
Example 2.4.1: In the quadratic equation 2x 2 − − =x 6 0, we have a = 2, b = −1,and c = −6 Therefore, the quadratic formula yields
When the expression under the square root sign is simplified, we get
( ) 12 2( ) 49 x = − − ± which can be further simplified to
1 49 x = ±4 Finally, since 49 = 7, we get that
Hence this quadratic equation has two real solutions: 1 7
Example 2.4.2: In the quadratic equation x 2 +4x + =4 0, we have a =1, b = 4, and 4. c = Therefore, the quadratic formula yields
When the expression under the square root sign is simplified, we get
The expression under the square root sign is equal to 0 and 0 = 0 Therefore, the expression can be simplified to
( ) 4 2 x = 2 1− = − Thus this quadratic equation has only one solution, x = −2.
Example 2.4.3: In the quadratic equation x 2 + + =x 5 0, we have a =1, b =1, and 5. c = Therefore, the quadratic formula yields
Since the expression under the square root sign equals 19, and square roots of negative numbers are not real, x is not a real number Therefore, there is no real solution to this quadratic equation.
Solving Quadratic Equations by Factoring
Some quadratic equations can be solved more quickly by factoring
Example 2.4.4: The quadratic equation 2x 2 − − =x 6 0 in Example 2.4.1 can be factored as ( 2 x +3 )( x −2 ) = 0 When a product is equal to 0, at least one of the factors must be equal to 0, so either 2x+ =3 0 or x − =2 0.
Example 2.4.5: The quadratic equation 5x 2 +3x− =2 0 can be factored as
Solving Linear Inequalities
A mathematical statement that uses one of the following four inequality signs is called an inequality
≤ less than or equal to
≥ greater than or equal to
Inequalities involve variables and are similar to equations but use inequality signs instead of an equality sign For example, the inequality 4x + 1 ≤ 7 is a linear inequality in one variable, indicating that 4x + 1 is less than or equal to 7 Solving an inequality involves finding all values of the variable that satisfy the inequality, known as the solution set Two inequalities with identical solution sets are called equivalent inequalities.
Solving a linear inequality involves simplifying it by isolating the variable on one side, similar to solving a linear equation This process typically follows two key rules: performing addition or subtraction to move constants and using multiplication or division to solve for the variable, while carefully reversing inequalities when multiplying or dividing by negative numbers Understanding these steps ensures accurate solutions to linear inequalities.
Adding or subtracting the same constant to both sides of an inequality does not change its direction, ensuring the inequality remains equivalent to the original This rule is essential for simplifying and solving inequalities effectively Understanding this principle helps in manipulating inequalities confidently without altering their solutions, which is crucial for various algebraic and mathematical applications.
Rule 2 explains that when multiplying or dividing both sides of an inequality by the same nonzero number, the inequality's direction remains the same if the number is positive However, if the number is negative, the inequality's direction reverses In any case, the resulting inequality remains equivalent to the original, ensuring accurate algebraic manipulations in inequality solving.
Example 2.5.1 : The inequality −3x + ≤5 17 can be solved as follows
Subtract 5 from both sides to get −3x ≤12.
Divide both sides by −3 and reverse the direction of the inequality to get 3 12
Therefore, the solution set of −3x + ≤5 17 consists of all numbers greater than or equal to −4.
11 5 x+ < can be solved as follows
Multiply both sides by 11 to get 4x +