Complete Guide to Integers on SAT Math (Advanced)

Complete Guide to Integers on SAT Math (Advanced) SAT / ACT Prep Online Guides and Tips Integer questions are some of the most common on the SAT, so understanding what integers are and how they operate will be crucial for solving many SAT math questions. Knowing your integers can make the difference between a score you’re proud of and one that needs improvement. In our basic guide to integers on the SAT (which you should review before you continue with this one), we covered what integers are and how they are manipulated to get even or odd, positive or negative results. In this guide, we will cover the more advanced integer concepts you’ll need to know for the SAT. This will be your complete guide to advanced SAT integers, including consecutive numbers, primes, absolute values, remainders, exponents, and roots- what they mean, as well as how to handle the more difficult integer questions the SAT can throw at you. Typical Integer Questions on the SAT Because integer questions cover so many different kinds of topics, there is no â€Å"typical† integer question. We have, however, provided you with several real SAT math examples to show you some of the many different kinds of integer questions the SAT may throw at you. Over all, you will be able to tell that a question requires knowledge and understanding of integers when: #1: The question specifically mentions integers (or consecutive integers). Now this may be a word problem or even a geometry problem, but you will know that your answer must be in whole numbers (integers) when the question asks for one or more integers. If $j$, $k$, and $n$ are consecutive integers such that $0jkn$ and the units (ones) digit of the product $jn$ is 9, what is the units digit of $k$? A. 0B. 1C. 2D. 3E. 4 (We will go through the process of solving this question later in the guide) #2: The question deals with prime numbers. A prime number is a specific kind of integer, which we will discuss in a minute. For now, know that any mention of prime numbers means it is an integer question. What is the product of the smallest prime number that is greater than 50 and the greatest prime number that is less than 50? (We will go through the process of solving this question later in the guide) #3: The question involves an absolute value equation (with integers) Anything that is an absolute value will be bracketed with absolute value signs which look like this:| | For example: $|-210|$ or $|x + 2|$ $|10 - k| = 3$ $|k - 5| = 8$ What is a value for k that fulfills both equations above? (We will go through how to solve this problem in the section on absolute values below) Note: there are several different kinds of absolute value problems. About half of the absolute value questions you come across will involve the use of inequalities (represented by $$ or $$). If you are unfamiliar with inequalities, check out our guide to inequalities. The other types of absolute value problems on the SAT will either involve a number line or a written equation. The absolute value questions involving number lines almost always use fraction or decimal values. For information on fractions and decimals, look to our guide to SAT fractions. We will be covering only written absolute value equations (with integers) in this guide. #4: The question uses perfect squares or asks you to reduce a root value A root question will always involve the root sign: $√$ $√81$, $^3√8$ You may be asked to reduce a root, or to find the square root of a perfect square (a number that is the square of an integer). You may also need to multiply two or more roots together. We will go through these definitions as well as how all of these processes are done in the section on roots. (Note: A root question with perfect squares may involve fractions. For more information on this concept, look to our guide on fractions and ratios.) #5: The question involves multiplying or dividing bases and exponents Exponents will always be a number that is positioned higher than the main (base) number: $2^7$, $(x^2)^4$ You may be asked to find the values of exponents or find the new expression once you have multiplied or divided terms with exponents. We will go through all of these questions and topics throughout this guide in the order of greatest prevalence on the SAT. We promise that integers are a whole lot less mysterious than...whatever these things are. Exponents Exponent questions will appear on every single SAT, and you will likely see an exponent question at least twice per test. An exponent indicates how many times a number (called a â€Å"base†) must be multiplied by itself. So $4^2$ is the same thing as saying $4 * 4$. And $4^5$ is the same thing as saying $4 * 4 * 4 * 4 * 4$. Here, 4 is the base and 2 and 5 are the exponents. A number (base) to a negative exponent is the same thing as saying 1 divided by the base to the positive exponent. For example, $2^{-3}$ becomes $1/2^3$ = $1/8$ If $x^{-1}h=1$, what does $h$ equal in terms of $x$? A. $-x$B. $1/x$C. $1/{x^2}$D. $x$E. $x^2$ Because $x^{-1}$ is a base taken to a negative exponent, we know we must re-write this as 1 divided by the base to the positive exponent. $x^{-1}$ = $1/{x^1}$ Now we have: $1/{x^1} * h$ Which is the same thing as saying: ${1h}/x^1$ = $h/x$ And we know that this equation is set equal to 1. So: $h/x = 1$ If you are familiar with fractions, then you will know that any number over itself equals 1. Therefore, $h$ and $x$ must be equal. So our final answer is D, $h = x$ But negative exponents are just the first step to understanding the many different types of SAT exponents. You will also need to know several other ways in which exponents behave with one another. Below are the main exponent rules that will be helpful for you to know for the SAT. Exponent Formulas: Multiplying Numbers with Exponents: $x^a * x^b = x^[a + b]$ (Note: the bases must be the same for this rule to apply) Why is this true? Think about it using real numbers. If you have $2^4 * 2^6$, you have: $(2 * 2 * 2 * 2) * (2 * 2 * 2 * 2 * 2 * 2)$ If you count them, this give you 2 multiplied by itself 10 times, or $2^10$. So $2^4 * 2^6$ = $2^[4 + 6]$ = $2^10$. If $7^n*7^3=7^12$, what is the value of $n$? A. 2B. 4C. 9D. 15E. 36 We know that multiplying numbers with the same base and exponents means that we must add those exponents. So our equation would look like: $7^n * 7^3 = 7^12$ $n + 3 = 12$ $n = 9$ So our final answer is C, 9. $x^a * y^a = (xy)^a$ (Note: the exponents must be the same for this rule to apply) Why is this true? Think about it using real numbers. If you have $2^4 * 3^4$, you have: $(2 * 2 * 2 * 2) * (3 * 3 * 3 * 3)$ = $(2 * 3) * (2 * 3) * (2 * 3) * (2 * 3)$ So you have $(2 * 3)^4$, or $6^4$ Dividing Exponents: ${x^a}/{x^b} = x^[a-b]$ (Note: the bases must be the same for this rule to apply) Why is this true? Think about it using real numbers. ${2^6}/{2^2}$ can also be written as: ${(2 * 2 * 2 * 2 * 2 * 2)}/{(2 * 2)}$ If you cancel out your bottom 2s, you’re left with $(2 * 2 * 2 * 2)$, or $2^4$ So ${2^6}/{2^2}$ = $2^[6-2]$ = $2^4$ If $x$ and $y$ are positive integers, which of the following is equivalent to $(2x)^{3y}-(2x)^y$? A. $(2x)^{2y}$B. $2^y(x^3-x^y)$C. $(2x)^y[(2x)^{2y}-1]$D. $(2x)^y(4x^y-1)$E. $(2x)^y[(2x)^3-1]$ In this problem, you must distribute out a common element- the $(2x)^y$- by dividing it from both pieces of the expression. This means that you must divide both $(2x)^{3y}$ and $(2x)^y$ by $(2x)^y$. Let's start with the first: ${(2x)^{3y}}/{(2x)^y}$ Because this is a division problem that involves exponents with the same base, we say: ${(2x)^{3y}}/{(2x)^y} = (2x)^[3y - y]$ So we are left with: $(2x)^{2y}$ Now, for the second part of our equation, we have: ${(2x)^y}/{(2x)^y}$ Again, we are dividing exponents that have the same base. So by the same process, we would say: ${(2x)^y}/{(2x)^y} = (2x)^[y - y] = (2x)^0 = 1$ (Why 1? Because, as you'll see below, anything raised to the power of 0 = 1) So our final answer looks like: ${(2x)^y}{((2x)^{2y} - 1)}$ Which means our final answer is C. Taking Exponents to Exponents: $(x^a)^b = x^[a * b]$ Why is this true? Think about it using real numbers. $(2^3)^4$ can also be written as: $(2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2)$ If you count them, 2 is being multiplied by itself 12 times. So $(2^3)^4 = 2^[3 * 4] = 2^12$ $(x^y)^6 = x^12$, what is the value of $y$? A. 2B. 4C. 6D. 10E. 12 Because exponents taken to exponents are multiplied together, our problem would look like: $y * 6 = 12$ $y = 2$ So our final answer is A, 2. Distributing Exponents: $(x/y)^a = {x^a}/{y^a}$ Why is this true? Think about it using real numbers. $(2/4)^3$ can be written as: $(2/4) * (2/4) * (2/4)$ $8/64 = 1/8$ You could also say $2^3/4^3$ = $8/64$ = $1/8$ $(xy)^z = x^z * y^z$ If you are taking a modified base to the power of an exponent, you must distribute that exponent across both the modifier and the base. $(3x)^3$ = $3^3 * x^3$ (Note on distributing exponents: you may only distribute exponents with multiplication or division- exponents do not distribute over addition or subtraction. $(x + y)^a$ is NOT $x^a + y^a$, for example) Special Exponents: For the SAT you should know what happens when you have an exponent of 0: $x^0=1$ where $x$ is any number except 0 (Why any number but 0? Well 0 to any power other than 0 is 0, because $0x = 0$. And any other number to the power of 0 is 1. This makes $0^0$ undefined, as it could be both 0 and 1 according to these guidelines.) Solving an Exponent Question: Always remember that you can test out exponent rules with real numbers in the same way that we did above. If you are presented with $(x^2)^3$ and don’t know whether you are supposed to add or multiply your exponents, replace your x with a real number! $(2^2)^3 = (4)^3 = 64$ Now check if you are supposed to add or multiply your exponents. $2^[2+3] = 2^5 = 32$ $2^[2 * 3] = 2^6 = 64$ So you know you’re supposed to multiply when exponents are taken to another exponent. This also works if you are given something enormous, like $(x^23)^4$. You don’t have to test it out with $2^23$! Just use smaller numbers like we did above to figure out the rules of exponents. Then, apply your newfound knowledge to the larger problem. And the philosophical debate continues. Roots Root questions are common on the SAT, and you should expect to see at least one during your test. Roots are technically fractional exponents. You are likely most familiar with square roots, however, so you may have never heard a root expressed in terms of exponents before. A square root asks the question: "What number needs to be multiplied by itself one time in order to equal the number under the root sign?" So $√36 = 6$ because 6 must be multiplied by itself one time to equal 36. In other words, $6^2 = 36$ Another way to write $√36$ is to say $^2√36$. The 2 at the top of the root sign indicates how many numbers (2 numbers, both the same) are being multiplied together to become 36. (Note: you do not expressly need the 2 at the top of the root sign- a root without an indicator is automatically a square root.) So $^3√27 = 3$ because three numbers, all of which are the same ($3 * 3 * 3$), multiplied together equals 27. Or $3^3 = 27$. Fractional Exponents If you have a number to a fractional exponent, it is just another way of asking you for a root. So $16^{1/2} = ^2√16$ To turn a fractional exponent into a root, the denominator becomes the value to which you take the root. But what if you have a number other than 1 in the numerator? $16^{2/3} = ^3√16^2$ The denominator becomes the value to which you take the root, and the numerator becomes the exponent to which you take the number under the root sign. Distributing Roots $√xy = √x * √y$ Just like with exponents, roots can be separated out. So $√20$ = $√2 * √10$ or $√4 * √5$ $√x * √y = √xy$ Because they can be separated, roots can also come together. So $√2 * √10$ = $√20$ Reducing Roots It is common to encounter a problem with a mixed root, where you have an integer multiplied by a root (like $3√2$). Here, $3√2$ is reduced to its simplest form, but let's say you had something like this instead: $2√12$ Now $2√12$ is NOT as reduced as it can be. In order to reduce it, we must find out if there are any perfect squares that factor into 12. If there are, then we can take them out from under the root sign. (Note: if there is more than one perfect square that can factor into your number under the root sign, use the largest one.) 12 has several factor pairs. These are: $1 * 12$ $2 * 6$ $3 * 4$ Well 4 is a perfect square because $2 * 2 = 4$. That means that $√4 = 2$. This means that we can take 4 out from under the root sign. Why? Because we know that $√xy = √x * √y$. So $√12 = √4 * √3$. And $√4 = 2$. So 4 can come out from under the root sign and be replaced by 2 instead. $√3$ is as reduced as we can make it, since it is a prime number. We are left with $2√3$ as the most reduced form of $√12$ (Note: you can test to see if this is true on most calculators. $√12 = 3.4641$ and $2 *√3 = 2 * 1.732 = 3.4641$. The two expressions are identical.) Now to finish the problem, we must multiply our reduced form of $√12$ by 2. Why? Because our original expression was $2√12$. $2 * 2√3 = 4√3$ So $2√12$ in its most reduced form is $4√3$ Remainders Questions involving remainders generally show up at least once or twice on any given SAT. A remainder is the amount left over when two numbers do not divide evenly. If you divide 12 by 4, you will not have any remainder (your remainder will be zero). But if you divide 13 by 4, you will have a remainder of 1, because there is 1 left over. You can think of the division as $13/4 = 3{1/4}$. That extra 1 is left over. Most of you probably haven’t worked with integer remainders since elementary school, as most higher level math classes and questions use decimals to express the remaining amount after a division (for the above example, $13/4 = 3 \remainder 1$ or $3.25$). But for some situations, decimals simply do not apply. Joanne’s hens laid a total of 33 eggs. She puts them into cartons that fit 6 eggs each. How many eggs will she have left that do NOT make a full carton of eggs? $33/6 = 5 \remainder 3$. So Joanne can make 5 full baskets with 3 eggs left over. Some remainder questions may seem incredibly obscure, but they are all quite basic once you understand what is being asked of you. Which of the following answers could be the remainders, in order, when five positive consecutive integers are divided by 4? A. 0, 1, 2, 3, 4B. 2, 3, 0, 1, 2C. 0, 1, 2, 0, 1D. 2, 3, 0, 3, 2E. 2, 3, 4, 3, 2 This question may seem complicated at first, so let’s break it down into pieces. The question is asking us to find the list of remainders when positive consecutive integers are divided by 4. This means we are NOT looking for the answer plus remainders- we are just trying to find the remainders by themselves. We will discuss consecutive integers below in the guide, but for now understand that "positive consecutive integers" means positive integers in a row along a number line. So positive consecutive integers increase by 1 continuously. , 12, 13, 14, 15, etc. are an example of positive consecutive integers. We also know that any number divided by 4 can have a maximum remainder of 3. Why? Because if any number could be divided by 4 with a remainder of 4 left over, it means it could be divided by 4 one more time! For example, $16/4 = 4 \remainder 0$ because 4 goes into 16 exactly 4 times. (It is NOT $3 \remainder 4$.) So that automatically lets us get rid of answer choices A and E, as those options both include a 4 for a remainder. Now we also know that, when positive consecutive integers are divided by any number, the remainders increase by 1 until they hit their highest remainder possible. When that happens, the next integer remainder resets to 0. This is because our smaller number has gone into the larger number an even number of times (which means there is no remainder). For example, $10/4 = 2 \remainder 2$, $/4 = 2 \remainder 3$, $12/4 = 3 \remainder 0$, and $13/4 = 3 \remainder 1$ Once the highest remainder value is achieved (n - 1, which in this case is 3), the next remainder resets to 0 and then the pattern repeats again from 1. So we’re looking for a pattern where the remainders go up by 1, reset to 0 after the remainder = 3, and then repeat again from 1. This means the answer is B, 2, 3, 0, 1, 2 Luckily, Joanne's remaining eggs did not go unloved for long. Prime numbers The SAT loves to test students on prime numbers, so you should expect to see one question per test on prime numbers. Be sure to understand what they are and how to find them. A prime number is a number that is only divisible by two numbers- itself and 1. For example, is a prime number because $1 * $ is its only factor. ( is not evenly divisible by 2, 3, 4, 5, 6, 7, 8, 9, or 10). 12 is NOT a prime number, because its factors are 1, 2, 3, 4, 6, and 12. It has more factors than just itself and 1. 1 is NOT a prime number, because its only factor is 1. The only even prime number is 2. Questions about primes come up fairly often on the SAT and understanding that 2 (and only 2!) is a prime number will be invaluable for solving many of these. A prime number $x$ is squared and then added to a different prime number, $y$. Which of the following could be the final result? An even number An odd number A positive number A. I onlyB. II onlyC. III onlyD. I and III onlyE. I, II, and III Now this question relies on your knowledge of both number relationships and primes. You know that any number squared (the number times itself) will be an even number if the original number was even, and an odd number if the original number was odd. Why? Because an even * an even = an even, and an odd * an odd = an odd ($6 * 6 = 36$ $7 * 7 = 49$). Next, we are adding that square to another prime number. You’ll also remember that an even number + an odd number is odd, an odd number + an odd number is even, and an even number + an even number is even. Knowing that 2 is a prime number, let’s replace x with 2. $2^2 = 4$. Now if y is a different prime number (as stipulated in the question), it must be odd, because the only even prime number is 2. So let’s say $y = 3$. $4 + 3 = 7$. So the end result is odd. This means II is correct. But what if both x and y were odd prime numbers? So let’s say that $x = 3$ and $y = 5$. So $3^2 = 9$. $9 + 5 = 14$. So the end result is even. This means I is correct. Now, for option number III, our results show that it is possible to get a positive number result, since both our results were positive. This means the final answer is E, I, II, and III If you forgot that 2 was a prime number, you would have picked D, I and III only, because there would have been no possible way to get an odd number. Remembering that 2 is a prime number is the key to solving this question. Another typical prime number question on the SAT will ask you to identify how many prime numbers fall in a certain range of numbers. How many prime numbers are between 30 and 50, inclusive? A. TwoB. ThreeC. FourD. FiveE. Six This might seem intimidating or time-consuming, but I promise you do NOT need to memorize a list of prime numbers. First, eliminate all even numbers from the list, as you know the only even prime number is 2. Next, eliminate all numbers that end in 5. Any number that ends is 5 or 0 is divisible by 5. Now your list looks like this: 31, 33, 37, 39, 41, 43, 47, 49 This is much easier to work with, but we need to narrow it down further. (You could start using your calculator here, or you can do this by hand.) A way to see if a number is divisible by 3 is to add the digits together. If that number is 3 or divisible by 3, then the final result is divisible by 3. For example, the number 31 is NOT divisible by 3 because $3 + 1 = 4$, which is not divisible by 3. However 33 is divisible by 3 because $3 + 3 = 6$, which is divisible by 3. So we can now eliminate 33 ($3 + 3 = 6$) and 39 ($3 + 9 = 12$) from the list. We are left with 31, 37, 41, 43, 47, 49. Now, to make sure you try every necessary potential factor, take the square root of the number you are trying to determine is prime. Any integer equal to or less than the square root will be a potential factor, but you do not have to try any numbers higher. Why? Well let’s take 36 as an example. Its factors are: 1, 2, 3, 4, 6, 9, 12, 18, and 36. But now look at the factor pairings. 1 36 2 18 3 12 4 9 6 6 (9 4) (12 3) (18 2) (36 1) After you get past 6, the numbers repeat. If you test out 4, you will know that 9 goes evenly into your larger number- no need to actually test 9 just to get 4 again! So all numbers less than or equal to a potential prime’s square root are the only potential factors you need to test. Going back to our list, we have 31, 37, 41, 43, 47, 49. Well the closest square root to 31 and 37 is 6. We already know that neither 2 nor 3 nor 5 factor evenly into 31 and 37. Neither do 4, or 6. You’re done. Both 31 and 37 must be prime. As for 41, 43, 47, and 49, the closest square root of these is 7. We already know that neither 2 nor 3 nor 5 factor evenly into 41, 43, 47, or 49. 7 is the exact square root of 49, so we know 49 is NOT a prime. As for 41, 43, and 47, neither 4 nor 6 nor 7 go into them evenly, so they are all prime. You are left with 31, 37, 41, 43, and 47. So your answer is D, there are five prime numbers (31, 37, 41, 43, and 47) between 30 and 50. Prime numbers, Prime Directive, either way I'm sure we'll live long and prosper. Absolute Values Absolute values are a concept that the SAT loves to use, as it is all too easy for students to make mistakes with absolute values. Expect to see one question on absolute values per test (though very rarely more than one). An absolute value is a representation of distance along a number line, forward or backwards. This means that an absolute value equation will always have two solutions. It also means that whatever is in the absolute value sign will be positive, as it represents distance along a number line and there is no such thing as a negative distance. An equation $|x + 3| = 14$, has two solutions: $x = $ $x = -17$ Why -17? Well $-17 + 3 = -14$ and, because it is an absolute value (and therefore a distance), the final answer becomes positive. So $|-14| = 14$ When you are presented with an absolute value, instead of doing the math in your head to find the negative and positive solution, rewrite the equation into two different equations. When presented with the above equation $|x + 3| = 14$, take away the absolute value sign and transform it into two equations- one with a positive solution and one with a negative solution. So $|x + 3| = 14$ becomes: $x + 3 = 14$ AND $x + 3 = -14$ Solve for $x$ $x = $ and $x = -17$ $|10 - k| = 3$ $|k - 5| = 8$. What is a value for $k$ that fulfills both equations above? We know that any given absolute value expression will have two solutions, so we must find the solution that each of these equations shares in common. For our first absolute value equation, we are trying to find the numbers for $k$ that, when subtracted from 10 will give us 3 and -3. That means our $k$ values will be 7 and 13. Why? Because $10 - 7 = 3$ and $10 - 13 = -3$ Now let's look at our second equation. We know that the two numbers for $k$ for $k - 5$ must give us both 8 and -8. This means our $k$ values will be 13 and -3. Why? Because $13 - 5 = 8$ and $-3 - 5 = -8$. 13 shows up as a solution for both problems, which means it is our answer. So our final answer is 13, this is the number for $k$ that can solve both equations. Consecutive Numbers Questions about consecutive numbers may or may not show up on your SAT. If they appear, it will be for a maximum of one question. Regardless, they are still an important concept for you to understand. Consecutive numbers are numbers that go continuously along the number line with a set distance between each number. So an example of positive, consecutive numbers would be: 4, 5, 6, 7, 8 An example of negative, consecutive numbers would be: -8, -7, -6, -5, -4 (Notice how the negative integers go from greatest to least- if you remember the basic guide to integers, this is because of how they lie on the number line in relation to 0) You can write unknown consecutive numbers out algebraically by assigning the first in the series a variable, $x$, and then continuing the sequence of adding 1 to each additional number. The sum of four positive, consecutive integers is 54. What is the first of these integers? If x is our first, unknown, integer in the sequence, so you can write all four numbers as: $x + (x + 1) + (x + 2) + (x + 3) = 54$ $4x + 6 = 54$ $4x = 48$ $x = 12$ So, because x is our first number in the sequence and $x= 12$, the first number in our sequence is 12. You may also be asked to find consecutive even or consecutive odd integers. This is the same as consecutive integers, only they are going up every other number instead of every number. This means there is a difference of two units between each number in the sequence instead of 1. An example of positive, consecutive even integers: 8, 10, 12, 14, 16 An example of positive, consecutive odd integers: 15, 17, 19, 21, 23 Both consecutive even or consecutive odd integers can be written out in sequence as: $x, x + 2, x + 4, x + 6$, etc. No matter if the beginning number is even or odd, the numbers in the sequence will always be two units apart. What is the median number in the sequence of five positive, consecutive odd integers whose sum is 185? $x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = 185$ $5x + 20 = 185$ $5x = 165$ $x = 33$ So the first number in the sequence is 33. This means the full sequence is: 33, 35, 37, 39, 41 The median number in the sequence is 37. Bonus history lesson- Grover Cleveland is the only US president to have ever served two non-consecutive terms. Steps to Solving an SAT Integer Question Because SAT integer questions are so numerous and varied, there is no set way to approach them that is entirely separate from approaching other kinds of SAT math questions. But there are a few techniques that will help you approach your SAT integer questions (and by extension, most questions on SAT math). #1: Make sure the question requires an integer. If the question does NOT specify that you are looking for an integer, then any number- including decimals and fractions- are fair game. Always read the question carefully to make sure you are on the right track. #2: Use real numbers if you forget your integer rules. Forget whether positive, even consecutive integers should be written as $x + (x + 1)$ or $x + (x + 2)$? Test it out with real numbers! 14, 16, 18 are consecutive even integers. If $x = 14$, $16 = x + 2$, and $18 = x + 4$. This works for most all of your integer rules. Forget your exponent rules? Plug in real numbers! Forget whether an even * an even makes an even or an odd? Plug in real numbers! #3: Keep your work organized. Like with most SAT math questions, integer questions can seem more complex than they are, or will be presented to you in strange ways. Keep your work well organized and keep track of your values to make sure your answer is exactly what the question is asking for. Santa is magic and has to double-check his list. So make sure you double-check your work too! Test Your Knowledge 1. If $a^x * a^6 = a^24$ and $(a^3)^y = a^15$, what is the value of $x + y$? A. 9B. 12C. 23D. 30E. 36 2. If $48√48 = a√b$ where $a$ and $b$ are positive integers and $a b$, which of the following could be a value of $ab$? A. 48B. 96C. 192D. 576E. 768 3. What is the product of the smallest prime number that is greater than 50 and the greatest prime number that is less than 50? 4.If $j, k$, and $n$ are consecutive integers such that $0jkn$ and the units (ones) digit of the product $jn$ is 9, what is the units digit of $k$? A. 0B. 1C. 2D. 3E. 4 Answers: C, D, 2491, A Answer Explanations: 1. In this question, we are being asked both to multiply bases with exponents as well as take a base with an exponent to another exponent. Essentially, the question is testing us on whether or not we know our exponent rules. If we remember our exponent rules, then we know that we must add exponents when we are multiplying two of the same base together. So $a^x * a^6 = a^24$ = $a^{x + 6} = a^24$ $x + 6 = 24$ $x = 18$ We have our value for $x$. Now we must find our $y$. We also know that, when taking a base and exponent to another exponent, we must multiply the exponents. So $(a^3)^y = a^15$ = $a^{3 * y} = a^15$ $3 * y = 15$ $y = 5$ In the final step, we must add our $x$ and $y$ values together: $18 + 5 = 23$ So our final answer is C, 23. 2. We are starting with $48√48$ and we know we must reduce it. Why? Because we are told that our first $48 = a$ and our second $48 = b$ AND that $a b$. Right now our $a$ and $b$ are equal, but, by reducing the expression, we will be able to find an $a$ value that is greater than our $b$ So let's find all the factors of 48 to see if there are any perfect squares. 48 $1 * 48$ $2 * 24$ $3 * 16$ $4 * 12$ $6 * 8$ Two of these pairings have perfect squares. 16 is our largest perfect square, which means that it will be the number we must use to reduce $48√48$ down to its most reduced form. Though we are not explicitly asked to find the most reduced form of $48√48$, we can start there for now. So $48√48 = 48 * √16 * √3$ $48 * 4 *√3$ $192√3$ This means that our $a = 192$ and our $b = 3$, then: $ab = 192 * 3 = 576$ So our final answer is D, 576. (Special note: you'll notice how we are told to find one possible value for $ab$, not necessarily $ab$ when $48√48$ is at its most reduced. So if our above answer hadn't matched one of our answer options, we would have had to reduce $48√48$ only part way. $48√48 = 48 * √4 * √12$ $48 * 2 * √12$ $96√12$ This would make our $a = 96$ and our $b = 12$, meaning that our final answer for $ab$ would be $96 * 12 = 52$.) 3. This question requires us to be able to figure out which numbers are prime. Let us use the same methods we used during the above section on prime numbers. All prime numbers other than 2 will be odd and there is no prime number that ends in 5. So let's list the odd numbers (excluding ones that end in 5's) above and below 50. 41, 43, 47, 49, 51, 53, 57, 59 We are trying to find the ones closest to 50 on either side, so let's first test the highest number in the 40's. 49 is the perfect square of 7, which means it is divisible by more than just itself and 1. We can cross 49 off the list. 47 is not divisible by 3 because $7 + 4 = $ and is not divisible by 3. It is also not divisible by any even number (because an even * an even = an even), by 5, or by 7. This means it must be prime. (Why did we stop here? Remember that we only have to test potential factors up until the closest square root of the potential prime. $√47$ is between $6^2 = 36$ and $7^2 = 49$, so we tested 7 just to be safe. Once we saw that 7 could not go into 47, we proved that 47 is a prime.) 47 is our largest prime less than 50. Now let's test the smallest number greater than 50. 51 is odd, but $5 + 1 = 6$, which is divisible by 3. That means that 51 is also divisible by 3 and thus cannot be prime. 53 is not divisible by 3 because $5 + 3 = 8$, which is not divisible by 3. It is also not divisible by 5 or 7. Therefore it is prime. (Again, we stopped here because the closest square root to 53 is between 7 and 8. And 8 cannot be a prime factor because all of its multiples are even). This means our smallest prime less than 50 is 47 and our largest is 53. Now we just need to find the product of those two numbers. $47 * 53 = 2491$ Our final answer is 2491. 4. We are told that $j$, $k$, and $n$ are consecutive integers. We also know they are positive (because they are greater than 0) and that they go in ascending order, $j$ to $k$ to $n$. We are also told that $jn$ equals a number with a units digit of 9. So let's find all the ways to get a product of 9 with two numbers. $1 * 9$ $3 * 3$ The only way to get any number that ends in 9 (units digit 9) from the product of two numbers is in one of two ways: #1: Both the original numbers have a units digit of 3 #2: The two original numbers have units digits of 1 and 9, respectively. Now let's visualize positive consecutive integers. Positive consecutive integers must go up in order with a difference of 1 between each variable. So $j, k, n$ could look like any collection of three numbers along a consistent number line. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, , 12, 13, 14, 15, 16, etc. There is no possible way that the units digits of the first and last of three consecutive numbers could both be 3. Why? Because if one had a units digit of 3, the other would have to end in either 1 or 5. Take 13 as an example. If $j$ were 13, then $n$ would have to be 15. And if $n$ were 13, then $j$ would have to be . So we know that neither $j$ nor $n$ has a units digit of 3. Now let's see if there is a way that we can give $j$ and $n$ units digits of 1 and 9 (or 9 and 1). If $j$ were given a units digit of 1, $n$ would have a units digit of 3. Why? Picture $j$ as . $n$ would have to be 13, and $ * 13 = 143$, which means the units digit of their product is not 9. But what if $n$ was a number with a units digit of 1? $j$ would have a units digit of 9. Why? Picture $n$ as now. $j$ would be 9. $9 * = 99$. The units digit is 9. And if the last digit of $j$ is 9 and the numbers $j, k, \and n$ are consecutive, then $k$ has to end in 0. So our final answer is A, 0. The Take-Aways Integers and integer questions can be tricky for some students, as they often involve concepts not tested in high school level math classes (when’s the last time you dealt with integer remainders, for example?). But most integer questions are much simpler than they appear. If you know your definitions- integers, consecutive integers, absolute values, etc.- and you know how to pay attention to what the question is asking you to find, you’ll be able to solve most any integer question that comes your way. What’s Next? Whew! You’ve done your paces on integers, both basic and advanced. Now that you’ve tackled these foundational topics of the SAT math, make sure you’ve got a solid grasp of all the math topics covered by the SAT math section, so that you can take on the SAT with confidence. Find yourself running out of time on SAT math? Check out our article on how to buy yourself time and complete your SAT math problems before time’s up. Feeling overwhelmed? Start by figuring out your ideal score and check out how to improve a low SAT math score. Already have pretty good scores and looking to get a perfect 800 on SAT Math? Check out our article on how to get a perfect score written by a full SAT scorer. Want to improve your SAT score by 160 points? Check out our best-in-class online SAT prep program. We guarantee your money back if you don't improve your SAT score by 160 points or more. Our program is entirely online, and it customizes what you study to your strengths and weaknesses. If you liked this Math strategy guide, you'll love our program. Along with more detailed lessons, you'll get thousands of practice problems organized by individual skills so you learn most effectively. We'll also give you a step-by-step program to follow so you'll never be confused about what to study next. Check out our 5-day free trial:

Quantitative, Qualitative, or Mixed Methods research Paper

Quantitative, Qualitative, or Mixed Methods - Research Paper Example The research report under analysis examines the depth and breadth of reverse logistical practices in Ghana’s pharmaceutical manufacturing industry. Apparently, the research used a mixed methods approach. In essence, the research exercise involved a proportional application of both the quantitative methods and qualitative methods. The first evidence of mixed methods approach is observable within the data collection context of the research. With respect to the data and methodology section of the article, it emerges that data and information were gathered through the use of questionnaires, specifically both open ended and closed ended questionnaires. Questions contained within the questionnaires were either open or close ended in nature. In this case, use of closed ended questions is an iconic characteristic of quantitative research methodology. On the other hand, the use of open ended questions is synonymous to qualitative research methodology. In addition, administration of per sonal interviews to managers and professionals within the pharmaceutical industry qualifies as an incident of qualitative research approach. Based on the nature of questions used during the questionnaire data collection exercise, it is admissible that a mixed research methodology was employed.Besides the proportional blend of qualitative and quantitative approaches in obtaining data and information, mixed methods approach is also evidenced by techniques used in the description and analysis of data and information.

Ethics & Social Responsability in Strategic Management Essay

Ethics & Social Responsability in Strategic Management - Essay Example could result in delay in supply of any one of them and consequently delay the launch. Â   The delay in projects, especially when the dates are announced, may corrode the reputation because the businesses build their credibility on the efficient and timely delivery of their projected target objectives. Mutual trust between business partners is also an important part of business processes and withholding of crucial information corrodes their credibility in the market that may result in loss of business in future. Â   The companies had not been able to launch successfully a commercial satellite primarily because it not only is high cost venture with costly software and hardware, the amount of contract is also dependent on the size of the anticipated market, the number of competitors and the whether the imageries would be used by the military and local agencies. Secondly in the case of the delays in launch, the preferences of the market may change, resulting in huge financial and business loss. Satellite programs are dependent on the venture capitalists for their finance and in case, they are not able to acquire advance contracts or their rivals are able to gain leverage, the venture capitalists are likely to withdraw from the business that may force the companies to stop production, leading to bankruptcy. The venture capitalists look for companies that could acquire advance contracts so as to ensure success for their investment. The current problem mainly cropped up because there was great chances of delay in the officially stated launch date of the satellite due to the fact that the satellite camera sub contractor was having problems with the development of thermal stabilizers for the instruments. The development of the thermal stabilizers could take up more time, thus delaying the launch by twelve to eighteen months. If the launch date is officially delayed and advanced by another one year, the company is

Improving Management Essay Example for Free

Improving Management Essay It has been established that Company Q is a small grocery chain that has recently decided to close two of their stores in heavily populated areas due to profit loss and high crime statistics. The necessity to close two stores is a significant economic failure for both the community and the stake holders involved in the franchise. Company Q has demonstrated that improvements need to be made to restore their reputation within the community and gain the confidence it needs to succeed among the share holders of the company. Since Company Q has closed these two stores, the loss of employment by members of the community has been recognized along with the need for citizens to travel further outside of the community for groceries. This takes tax revenue away from the city and has also resulted in two vacant buildings. In a community already plagued with crime, this adds to the potential of additional crime. In order to salvage Company Q’s reputation with the community and restore trust with the shareholders, significant changes will need to be made. Based on the closure of two stores, the social responsibility that Company Q exudes is very poor and concerted effort will need to be displayed in order to prove they are a socially responsible company. Part B For several years, customers of Company Q have requested that Company Q offer a more health conscience and organic variety of foods. Due to the recent closure of two stores, it is noted that there are now funds available to provide this request at the remaining open franchises. Company Q recognizes that in order to regain the customer’s trust, they need to be accommodating to the desires and needs of their customers. The first recommendation is to continue to provide the healthier, organic food varieties being requested. This action recognizes the customer’s desires and validates that Company Q is listening and wants to be accommodating.  Money that is no longer being utilized to manage the two stores that are closed can now be allocated to the high margin items at the stores that remain open. Company Q has been asked by the local food bank to donate day-old products instead of throwing it away. Company Q has denied this request citing worries about fraud and possibilities of employees stealing and stating that they were actually donating it. Our first recommendation is to rescind this policy and work with your local food bank to be able to obtain this otherwise wasted food. The reputation with the community is already fragile after the closure of two nearby stores. In order to build trust with the community and the shareholders of Company Q, simply providing the requested healthier, organic food will not be enough. Extra effort being demonstrated by Company Q will need to be recognized. The next recommendation would be to create an Action Committee within each of Company Q’s open stores. This Action Committee will work together to create a system that  monitors the amount of high margin items being purchased from the distributor and then being sold to the customer. The products that can be donated to the local food bank will be recorded. These records will be maintained by the Action Committee and preferably a member of the managerial staff. The donated product can be considered a tax write off at year end. Since there is a committee monitoring what is coming in and what is being donated out, it will help minimize the potential of fraud or possible theft by employees. This Action Committee can then determine which products are being sold on a consistent basis and which products are being donated. The purchasing of high margin items from the distributors can be adjusted so that more popular items are being sold and fewer items are being donated. This action will demonstrate to the community that Company Q cares about the citizens and they want to be an active part of the community. It also can instill trust with Company Q’s employees that they are not the reason the business wasn’t donating to the local food bank, and that they believe their employees are trusting and have integrity. A final recommendation would be for Company Q to establish their own volunteer program consisting of members of management. If additional employees of Company Q desire to be part of the volunteer committee, it is completely on a volunteer basis. This volunteer team will periodically represent Company Q at local events. They can volunteer time at the food bank, local homeless shelters, charitable events and so forth. The team can organize events to paint over graffiti in the part of town where they previously closed two of their stores. The presence of Company Q in the community can help restore trust that was lost when two stores were closed and the community suffered. The stockholders in Company Q will also see a significant change in the community’s perspective of their company. Based on these three recommendations, we believe that Company Q will exemplify the social responsibility expected by a company this size and that has such a presence in this populated community. Upon request, additional recommendations can be made to ensure Company Q sustains and improves upon its reputation with the public.

Gender Roles in The Yellow Wallpaper -- Yellow Wallpaper essays

Gender Roles in The Yellow Wallpaper  Ã‚        Ã‚  Ã‚   In Charlotte Perkins Gilman's short story "The Yellow Wallpaper," the reader is treated to an intimate portrait of developing insanity. At the same time, the story's first person narrator provides insight into the social attitudes of the story's late Victorian time period. The story sets up a sense of gradually increasing distrust between the narrator and her husband, John, a doctor, which suggests that gender roles were strictly defined; however, as the story is just one representation of the time period, the examination of other sources is necessary to better understand the nature of American attitudes in the late 1800s. Specifically, this essay will analyze the representation of women's roles in "The Yellow Wallpaper" alongside two other texts produced during this time period, in the effort to discover whether Gilman's depiction of women accurately reflects the society that produced it.    "The Yellow Wallpaper" features an unnamed female narrator who serves to exemplify the expectations placed upon women of the time period. As we are told early on, she is suffering from a "nervous condition" (Gilman 1). While we are not told the specific nature of this condition, we do discover that the cure prescribed by John, the narrator's husband and doctor, entails taking "phosphates or phosphites--whichever it is, and tonics, and journeys, and air, and exercise" while intellectual "work" is "absolutely forbidden Ãâ€" until [she is] well again"   (Gilman 1). This poses a particular problem for the narrator, due to her desire to write, which she continues to do "in spite of them," and causes her to hide her writing to avoid facing "heavy opposition" (Gilman 1). The treatment to which t... ...Mitchell, seems all the more plausible. After all, her socially-defined role as the dutiful wife and mother was being constrained by her inability to withstand the treatment foisted upon her by a man trained to disregard his patients' feelings. As a woman, she had no socially sanctioned way to respond to the problems she faced. Rather than wonder, as John does throughout the story, why his wife is becoming increasingly deranged, readers of this story should only wonder why, given the mores of the time period, there weren't far more stories like it.    Works Cited Gilman, Charlotte Perkins. "The Yellow Wallpaper." English 101 Homepage. August 1999 . Mitchell, S. Weir. The Evolution of the Rest Treatment. English 101 Course Packet. Chico: Mr Kopy, 1999. Power, Susan. The Ugly-Girl Papers. English 101 Course Packet. Chico: Mr Kopy, 1999.      

The Matrix Linked Christianity

Matrix movie is full of violence but there are some powerful concepts in it that will help us to understand what really the real world is, that may help us to be different from usual people, leaving the secular society and do the things that are needed to be done. When Neo took the red pill and woke up in the pod or vat, it was just like when we turn to Jesus and woke up from the Matrix that we live in called material society.Like Neo when he woke up in the pod, he realized that humans are slaves to an empire of man-made, intelligent machines and the first year of most of us who turned back after with Christ was extremely confusing (The Christian Science Monitor, 2003). Things we had believed for years no longer made any sense whatsoever. Neo said, â€Å"how come my eyes hurt so much†, and Morpheus replied, â€Å"It's because you never used them before†. Neither had most of us used our spiritual eyes, and when the light of Jesus hit us, we were empowered to see things t hrough our spiritual eyes, but it also hurt like crazy cause no one understood us.Lots of people had never yet wake up from Matrix, and these people are the most person who will always turned us down, people who are our love ones. Analysis of Matrix from Christian View The Matrix: our civilization The Resistance: members of the Christian family including Angels in heaven (people not plugged into the Matrix) and Christians on earth (people plugged into the Matrix. ) The only job of the Resistance is to wake people up from the Matrix and get them to join the Resistance.Sentient programs: angels of the Devil roaming the earth (Matrix) looking for members of the Resistance (Christians) who they can kill or interfere with. Morpheus: John the Baptist Cypher: Judas Neo: Jesus Christ Life in the Matrix is not real life and there is a supreme negative intelligence running the Matrix. It is smarter than any human being. It outsmarts people into thinking they are living life and getting ahead (successful in the Matrix) when they are really asleep, dying and lying dormant in the pod (or vat) with the life being sucked out of them by the source of the Matrix (the Machines- the Devil).In the movie this â€Å"life† is human electromagnetic energy – in real world that â€Å"life† is our spiritual energy, the soul that the devil sucks out of us, as we are lying spiritually unconscious. In the world of The Matrix, men are â€Å"born into bondage, born into a prison that you cannot smell or taste or touch. † Then comes â€Å"the One,† the promised deliverer who will overcome mankind’s enemy and liberate the human race from bondage. Morpheus has been foretold that he will find this figure of prophecy; and, like John the Baptist heralding Jesus as the Lamb of God, Morpheus recognizes Neo to be the One.According to Morpheus, the Matrix is â€Å"the world that has been pulled over your eyes to blind you from the truth† (Wikipedia, 200 6). The Matrix is symbolic of sin that keeps the human race blinded about reality. â€Å"It is the world that has been pulled over your eyes to blind you from the truth. † Each person is â€Å"born into bondage†¦ a prison, for your mind. † The illusion of the matrix is so we do not wake up to the realization of our slavery. It would seem that the machines then are Satan and his demons trying to deceive mankind concerning his true state of bondage.Neo's real name is Thomas Anderson, perhaps referring to â€Å"doubting Thomas† of the Bible (since it took him so long to believe he was the One). The name Anderson means â€Å"son of man,† a title used by Jesus in reference to himself. He mimics the true Savior by having both a death and resurrection. After his resurrection, the matrix (sin) no longer has dominion over him. This is in contrast to the reality of the Savior who died in history. Death and sin had no power over Christ. As Christ died for manki nd, Neo dies to save the world.At the end, Neo determines to show the world the truth, just as Christ said He is the truth. Neo destroys Agent Smith and Christ destroyed evil. The world is set free from the bondage of the matrix, and Christ frees those who believe in Him from the bondage of sin. A character Choi, says to him â€Å"Hallelujah. You're my savior, man. My own personal Jesus Christ. † A plate in Morpheus' ship Nebuchadnezzar says â€Å"Mark III No. 11,† a probable allusion to the Bible: Mark 3:11 reads, â€Å"Whenever the unclean spirits saw him, they fell down before him and shouted, ‘You are the Son of God!‘† (About, 2006). Anderson's hacker alias Neo is an anagram for the One. He is The One who is prophesied to liberate humanity from the chains that imprison them in their computer-generated illusion. First, however, he has to die – and he is killed in room 303. But, after 72 seconds (analogous to 3 days), Neo rises again. Soon t hereafter, he also ascends up into the heavens. There is a battle between the machines (an allegory for the devil and his angels) and the Resistance (Angels and saints.) The members of the Resistance who are not presently plugged into the Matrix are kind of like angels because they can help people in the Matrix (by hacking into it) but are not able to directly interfere in human choice. This battle outside the Matrix is like the battle in the heavens between the angels of the devil and God's angels and saints of God. Members of the Resistance, who are plugged into the Matrix and have an identity there, are able to do things that the people in the Matrix can't do. They are more intuitive, have greater power to fight evil etc.But they are reliant on the members of the Resistance who are back at the ship (Angels). Life as a member of the Resistance is exciting. However, the sentient programs are very interested in destroying these members of the Resistance. The sentient programs have n o interest in the other people in the Matrix because they know they have them already. The machines that run the Matrix see the Resistance as a threat and therefore try to kill them or interfere with their activities (an allegory for Persecution of Christians in the Media etc. )Some people who belong to the Resistance have â€Å"real† jobs in the Matrix like those who are asleep plugged into it. But because the Resistance has been woken up to the fact that the Matrix is unreal, they no longer see things in the Matrix in the same way as other people, even though they are doing regular jobs. They see it as an illusion. These regular jobs are just positions like a spy in the Second World War who would work in a German shoe repair or other trivial job. But they are simply put there as a posting to achieve the greater end.The decision to take the red pill is a completely free choice with no one forcing it. Taking the red pill is an allegory for the choice to completely surrender t o Jesus. It is the only hope for the future. Otherwise they stay asleep. The job of the Matrix is to keep people â€Å"engaged† in the day to day trivial things of life in the Matrix so they stay asleep and die while hooked up to the pod (or vat). And then they are ejected into the sewer mote (hell). When Neo woke up he was ejected into the mote (hell) but was rescued by the Resistance (Angels). This is similar to Jesus in the desert.Angels waited upon Jesus and nursed him back to health after he was 40 days in the desert the same way the members of the Resistance nursed Neo back to health when he woke up from the Matrix. This was the start of Neo's participation in the Resistance. Jesus' ministry started after the 40 days in the desert. Of course there’s also a Judas figure (in one scene he and Neo drink from the same cup, as Jesus and Judas dipped in the same dish). Cypher is a Judas figure. Which is pretty obvious. Cypher knows that the Matrix is not real and that a ny pleasures he experiences there are illusory (What is the Matrix, 1999).When Smith (the sentient program) outsmarts Neo, near the end of the movie, he knew what Neo was thinking and so was one step ahead of him waiting in the room where Neo went to make the phone call. That is when Smith shoots Neo as he went to make the phone call in room 303. When we don't completely surrender to Christ, our best plans will not succeed in getting us out of the Matrix (material world) because the devil can read our actions and behavior. So he is always one step ahead of us like Smith was when he was waiting for Neo in the room where Neo was going to make the phone call.Neo dies and is raised to life again. When Neo came back to life and realized he was the one. He was no longer thinking about what he would do next. He was plugged into his Divine power. Neo was no longer planning ahead. He was responding to his spiritual connectedness in the moment. And so Smith could no longer figure out what Neo was going to do next and was no longer able to be one step ahead of Neo. Smith was no longer able to read what Neo would do next because Neo didn't even know, Neo was simply responding to his spiritual connectedness, which, in fact put Neo one step ahead of Smith.If we trust God, we don't know what he is going to do next with us, but neither does the devil. In that way God can outsmart the devil in our lives. It's when we wrestle to keep control of our lives that we are subject to interference from the devil because he will always be a step ahead of us. Life as a Christian is completely different from when we planned our lives. We don't know what God will do next and we just trust and listen and watch God outsmart evil in our lives. Then the devil can no longer be a step ahead of us (i. e. , waiting at the door to shoot us like Smith did to Neo when Neo was thinking for himself).I understand the fear of surrender but this is a good surrender, it is plugging into a source of power t hat outsmarts evil in my life. I think it is the only source. Jesus is â€Å"The One†. The reality of this world and man’s destiny is clearly revealed in God’s Word. Man is not his own savior, but he is a condemned sinner in bondage to sin; as a result, man is spiritually dead. Jesus Christ is the savior who did not leave it up to people to see how they felt about Him in order to believe. Rather, He gave many signs proving, in fact, that He is God and Savior. His death and resurrection is a historic fact.The man who thinks he will save himself is grievously deceived. It is only in trusting in Jesus Christ’s death and resurrection to atone for man’s sin that a repentant sinner can be released from the bondage of sin and experience eternal fellowship with his Creator. There are also other problematic implications to the film. From a Christian perspective, to begin with, the whole premise of the unknowing enslavement of all of humanity by machines is a staggeringly apocalyptic event that raises serious eschatological questions: Would God allows all of humanity to be subjected to so immense a deception?Descartes argued not. Consider especially the implications of generations of humans living and dying without real physical contact with one another. While it’s possible to imagine Christian faith existing in such a world (and indeed Morpheus mentions people going to church in the Matrix), the Church itself, and in particular apostolic succession and the papacy, cannot be perpetuated under these conditions, since there is no physical laying on of hands.(This problem is mitigated, though, by the fact that the film does establish that not all of mankind is in the Matrix — there is one surviving human community, Zion, where it’s possible to imagine the Church having survived. On the other hand, what we see of Zion in the sequel, The Matrix Reloaded, offers no indication whatsoever of any Christian presence. ) In fa ct, God and religion seem to be basically irrelevant to the characters in the film. Morality, too, tends generally to be a non-issue.Of course there’s the glaring disregard for life seen especially in the lobby massacre. Beyond this, Neo himself has a shady background, and although he is in many ways transformed during the course of the film, this doesn’t include any kind of moral transformation. Likewise, Morpheus sets people free from the Matrix, but there’s no indication that they’re any freer from sin or evil. Consider a scene in which a character named Mouse invites Neo to have virtual sex with a digital woman of his creation.The other crewmembers may needle Mouse as a â€Å"digital pimp,† but there’s no real moral backbone to their criticism. (â€Å"Pay no attention to these hypocrites,† Mouse tells Neo. â€Å"To deny our impulses is to deny the very thing that makes us human. â€Å") Of course there’s no rule that says that characters fighting against a great evil must be depicted as paragons of virtue. On the other hand, the film’s overall lack of moral perspective does make it hard to see it as meaningfully â€Å"Christian. † References The Holy Bible. 2005. December 4, 2006. The Matrix. Wikipedia: The Free Encyclopedia. 4 December 2006. December 4, 2006. . The Matrix and Christianity. About. 2006. December 2, 2006. The Gospel according to Neo. The Christian Science Monitor. May 09, 2003. December 2, 2006. Wake Up!. What is the Matrix. 1999. December 2, 2006. .

Employees Motivation Essay - 1035 Words

Employees Motivation A business seeks profit by provided customers with goods and services (Schoell, et al 15). There are various types of businesses that differ according to their ownership. The three basic forms of private ownership businesses are the sole proprietorship (i.e. sole trader), partnership, and corporation (Schoell, et al 132). The type of ownership that a business organization would apply is dependent on the owners financial status and objectives. Apart from the different types of ownerships, there are various styles of management and leadership. The organizations management and leadership style has a great effect on the working environment and the employees motivation. The development of an optimal†¦show more content†¦The autocratic style of leadership would be considered the most task-oriented type, in which the manager carry out all the decision-making process without any consultations from subordinates. Communication is one-way, where the work p rovided by the manager is to be done without any modifications by the subordinates. In this style of leadership, the delegation of authority is centralized, meaning that the decision-making is only performed by top-line managers and is rarely delegated to subordinates along the chain of command (Bovee, et al 476-8). The democratic leadership style is more of a participative type of leadership (Schoell, et al 286). It is a two-way communication leadership, in which employees are allowed to contribute in the decision-making; however, the manager makes the final decision. The delegation of authority in this style of leadership is less centralized and more decentralized than in the autocratic style. Employees feel more flexible, since they are allowed to modify in the methods of accomplishing the tasks and to contribute in the decision-making. This changes the working environment; making it an environment with a lot of team working, where each employee feels that he plays an ess ential role in the well being of the business (Bovee, et al 476-8). The free-reign style of leadership is the most lenient style, in which the employees have the complete freedom byShow MoreRelatedMotivation And Motivation For Employees755 Words   |  4 PagesMost organizations seek employees that are highly motivated. This helps employees be higher producers within the agency. Organizations expect their employees to do their best work to enhance company. My organization also focuses on motivating employees to be more productive. Managers have to be ready to motivation different types of people. However, most managers should be very diverse when it comes to motivating employees. Intrinsic and extrinsic rewards play a huge part in the success ofRead MoreMotivation And Motivation For Employees2112 Words   |  9 Pages Motivation in the work place has always been a problem for employers. It is a key element in keeping the employees driven to do good work or finish tasks on time. There are several methods of motivation that help employees stay with an organization. Motivation varies from person to person. Different methods of motivation drive people differently. Webster’s dictionary says motivation is something inside people that drives them to action. In other words it is the willingness to work at a levelRead MoreEffect of Motivation on Employees9326 Words   |  38 Pagesas the most critical firm asset, and the ability to attract motivate and retain capable employees is essential in organization’s innovation and quality improvement (Frye, 2004). These sentiments are supported by Jung and Hartog, (2007) who suggest that, one way for organizations to become more innovative is to capitalize on their employees’ ability to innovate. Jung and Hartog, continue to argue that employees can help to improve business performance through their ability to generate ideas and useRead MoreMotivation Of Employees At The Workplace2053 W ords   |  9 PagesMotivation of Employees at the Workplace Anna Mbamalu University of Texas at Arlington Biology 3320 December 7th, 2014 Motivation of Employees at the Workplace Introduction The term â€Å"motivation† is derived from a theoretical construct. Many times, psychologists have used this term to refer to something responsible for our action. The concept of motivation can be classified into categories: intrinsic and extrinsic. Extrinsic motivation is the one related to external factors that influenceRead MoreManagers And Employees Motivation And Its Importance1312 Words   |  6 PagesManagers’ and Employees’ Motivation and Its Importance Joechelle Gemino Indian River State College Abstract Is it very important that managers understand their employees’ motivation. There are different components that need to be bonded together in order to motivate employees. Workplace environment and communication influence managers’ and employees’ motivation. More motivation means more productivity. The relationship between managers’ motivation and their employees is analyzed in this paper. Read MoreMotivation Factors And Indicators Of The Employees2242 Words   |  9 PagesIntroduction: Motivation is the culmination of set of forces that is the reason for people to get engaged in any particular behaviour instead of many alternative behaviour. (Griffin Moorhead, 2012) There are different instances for motivation and its indicators. For instance, a salesperson may be willing to work overtime over the weekend and may demonstrate his motivated behaviours. Motivation is a continuous process. This is a challenging task for an organisation and the managers to find outRead MoreEffect of Motivation on Employees Productivity1775 Words   |  8 Pagescontinue to keep employees beyond a certain maximum number. Research reports has shown that employees motivation is essential in an organization as it is a key to a successful organization need for maintaining continuity and survival. Motivating the staff leads to broaden their skills to meet the organizational demands. The need for achievement always results in a desire for employees to do extra effort to have something done better and have the desire for success. Motivation creates a productiveRead MoreMotivation of Employees in the Public and Private Sector647 Words   |  3 Pagesâ€Å"Motivation is an individuals degree of willingness to exert and maintain an effort towards organizational goals Dieleman, M. et al. (2003). It is true that everyone work for a good pay package and are motivated upon by same so as to meet their basic needs and those of their dependents. However, decision makers often rely on financial incentive as a means of motivation which in itself is fine thanks to the New Public Management scheme of pay-for-performance. Nevertheless, it’s as well importantRead MoreImpact of Employees Motivation on Work Performance1788 Words   |  8 Pages ISSN: 2249-7382 EFFECTS OF EMPLOYEES MOTIVATION ON ORGANIZATIONAL PERFORMANCE – A CASE STUDY Pankaj Chaudhary* ABSTRACT Motivational factors play an important role in increasing employee job satisfaction. This will result in improving organizational performance. High productivity is a long term benefits of employee motivation. Motivated employee is a valuable asset who creates value for an organization in strengthening the business and revenue growth. Motivation is going to work if the rightRead MoreImpact of Employees Motivation on Organizational Effectiveness2341 Words   |  10 Pagesorg Impact of Employees Motivation on Organizational Effectiveness Quratul-Ain Manzoor Department of Management Sciences, The Islamia University of Bahawalpur, Bahawalpur Abstract Purpose- The purpose of this paper is to identify the factors that effects employee motivation and examining the relationship between organizational effectiveness and employee motivation. Approach- A model was designed based on the literature, linking factors of employee motivation with employee motivation and organizational