Complete Guide to Integers on ACT Math (Advanced)

Complete Guide to Integers on ACT Math (Advanced) SAT / ACT Prep Online Guides and Tips Integers, integers, integers (oh, my)! You've already read up on your basic ACT integers and now you're hankering to tackle the heavy hitters of the integer world. Want to know how to (quickly) find a list of prime numbers? Want to know how to manipulate and solve exponent problems? Root problems? Well look no further! This will be your complete guide to advanced ACT integers, including prime numbers, exponents, absolute values, consecutive numbers, and roots- what they mean, as well as how to solve the more difficult integer questions that may show up on the ACT. Typical Integer Questions on the ACT First thing's first- there is, unfortunately, no â€Å"typical† integer question on the ACT. Integers cover such a wide variety of topics that the questions will be numerous and varied. And as such, there can be no clear template for a standard integer question. However, this guide will walk you through several real ACT math examples on each integer topic in order to show you some of the many different kinds of integer questions the ACT may throw at you. As a rule of thumb, you can tell when an ACT question requires you to use your integer techniques and skills when: #1: The question specifically mentions integers (or consecutive integers) It could 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. (We will go through the process of solving this question later in the guide) #2: The question involves prime numbers A prime number is a specific kind of integer, which we will discuss later in the guide. For now, know that any mention of prime numbers means it is an integer question. A prime number a is squared and then added to a different prime number, b. Which of the following could be the final result? An even number An odd number A positive number I only II only III only I and III only I, II, and III (We'll go through the process of solving this question later in the guide) #3: The question involves multiplying or dividing bases and exponents Exponents will always be a number that is positioned higher than the main (base) number: $4^3$, $(y^5)^2$ 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 the process of solving this question later in the guide) #4: The question uses perfect squares or asks you to reduce a root value A root question will always involve the root sign: √ $√36$, $^3√8$ The ACT may ask you to reduce a root, or to find the square root of a perfect square (a number that is equal to an integer squared). 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. (We will go through the process of solving this question later in the guide) (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 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: $|-43|$ or $|z + 4|$ (We will go through how to solve this problem later in the guide) Note: there are generally two different kinds of absolute value problems on the ACT- equations and inequalities. About a quarter 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 ACT inequalities (coming soon!). The majority of absolute value questions on the ACT will involve a written equation, either using integers or variables. These should be fairly straightforward to solve once you learn the ins and outs of absolute values (and keep track of your negative signs!), all of which we will cover below. We will, however, only be covering written absolute value equations in this guide. Absolute value questions with inequalities are covered in our guide to ACT inequalities. We will go through all of these questions and topics throughout this guide in the order of greatest prevalence on the ACT. We promise that your path to advanced integers will not take you a decade or more to get through (looking at you, Odysseus). Exponents Exponent questions will appear on every single ACT, and you'll likely see an exponent question at least twice per test. Whether you're being asked to multiply exponents, divide them, or take one exponent to another, you'll need to know your exponent rules and definitions. An exponent indicates how many times a number (called a â€Å"base†) must be multiplied by itself. So $3^2$ is the same thing as saying 3*3. And $3^4$ is the same thing as saying 3*3*3*3. Here, 3 is the base and 2 and 4 are the exponents. You may also have a base to a negative exponent. This is the same thing as saying: 1 divided by the base to the positive exponent. For example, 4-3 becomes $1/{4^3}$ = $1/64$ But how do you multiply or divide bases and exponents? Never fear! Below are the main exponent rules that will be helpful for you to know for the ACT. 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 $3^2 * 3^4$, you have: (3*3)*(3*3*3*3) If you count them, this give you 3 multiplied by itself 6 times, or $3^6$. So $3^2 * 3^4$ = $3^[2 + 4]$ = $3^6$. $x^a*y^a=(xy)^a$ (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 $3^5*2^5$, you have: (3*3*3*3*3)*(2*2*2*2*2) = (3*2)*(3*2)*(3*2)*(3*2)*(3*2) So you have $(3*2)^5$, or $6^5$ If $3^x*4^y=12^x$, what is y in terms of x? ${1/2}x$ x 2x x+2 4x We can see here that the base of the final answer is 12 and $3 *4= 12$. We can also see that the final result, $12^x$, is taken to one of the original exponent values in the equation (x). This means that the exponents must be equal, as only then can you multiply the bases and keep the exponent intact. So our final answer is B, $y = x$ If you were uncertain about your answer, then plug in your own numbers for the variables. Let's say that $x = 2$ $32 * 4y = 122$ $9 * 4y = 144$ $4y = 16$ $y = 2$ Since we said that $x = 2$ and we discovered that $y = 2$, then $x = y$. So again, our answer is B, y = x 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. ${3^6}/{3^4}$ can also be written as: ${(3 * 3 * 3 * 3 * 3 * 3)}/{(3 * 3 * 3 * 3)}$ If you cancel out your bottom 3s, you’re left with (3 * 3), or $3^2$ So ${3^6}/{3^4}$ = $3^[6 - 4]$ = $3^2$ The above $(x * 10^y)$ is called "scientific notation" and is a method of writing either very large numbers or very small ones. You don't need to understand how it works in order to solve this problem, however. Just think of these as any other bases with exponents. We have a certain number of hydrogen molecules and the dimensions of a box. We are looking for the number of molecules per one cubic centimeter, which means we must divide our hydrogen molecules by our volume. So: $${8*10^12}/{4*10^4}$$ Take each component separately. $8/4=2$, so we know our answer is either G or H. Now to complete it, we would say: $10^12/10^4=10^[12−4]=10^8$ Now put the pieces together: $2x10^8$ So our full and final answer is H, there are $2x10^8$ hydrogen molecules per cubic centimeter in the box. Taking Exponents to Exponents: $(x^a)^b=x^[a*b]$ Why is this true? Think about it using real numbers. $(3^2)^4$ can also be written as: (3*3)*(3*3)*(3*3)*(3*3) If you count them, 3 is being multiplied by itself 8 times. So $(3^2)^4$=$3^[2*4]$=$3^8$ $(x^y)3=x^9$, what is the value of y? 2 3 6 10 12 Because exponents taken to exponents are multiplied together, our problem would look like: $y*3=9$ $y=3$ So our final answer is B, 3. Distributing Exponents: $(x/y)^a = x^a/y^a$ Why is this true? Think about it using real numbers. $(3/4)^3$ can be written as $(3/4)(3/4)(3/4)=9/64$ You could also say $3^3/4^3= 9/64$ $(xy)^z=x^z*y^z$ 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. $(2x)^3$=$2^3*x^3$ In this case, we are distributing our outer exponent across both pieces of the inner term. So: $3^3=27$ And we can see that this is an exponent taken to an exponent problem, so we must multiply our exponents together. $x^[3*3]=x^9$ This means our final answer is E, $27x^9$ And if you're uncertain whether you have found the right answer, you can always test it out using real numbers. Instead of using a variable, x, let us replace it with 2. $(3x^3)^3$ $(3*2^3)^3$ $(3*8)^3$ $24^3$ 13,824 Now test which answer matches 13,824. We'll save ourselves some time by testing E first. $27x^9$ $27*2^9$ $27*512$ 13,824 We have found the same answer, so we know for certain that E must be correct. (Note: when distributing exponents, you may do so 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: It is common for the ACT to ask you 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 equals 0, because $0^x=0$. And any other number to the power of 0 = 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 in our examples above. If you are presented with $(x^3)^2$ and don’t know whether you are supposed to add or multiply your exponents, replace your x with a real number! $(2^3)^2=(8)^2=64$ Now check if you are supposed to add or multiply your exponents. $2^[2+3]=2^5=32$ $2^[3*2]=2^6=64$ 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 $(x^19)^3$. You don’t have to test it out with $2^19$! 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 exponents are down for the count. Instant KO! Roots Root questions are fairly common on the ACT, and they go hand-in-hand with exponents. Why are roots related to exponents? Well, technically, roots are 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 $√81=9$ because 9 must be multiplied by itself one time to equal 81. In other words, $9^2=81$ Another way to write $√{81}$ is to say $^2√{81}$. The 2 at the top of the root sign indicates how many numbers (two numbers, both the same) are being multiplied together to become 81. (Special note: you do not need the 2 on the root sign to indicate that the root is a square root. But you DO need the indicator for anything that is NOT a square root, like cube roots, etc.) This means that $^3√27=3$ because three numbers, all of which are the same (3*3*3), are multiplied together to equal 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 $4^{1/2}= √4$ 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? $4^{2/3}$=$^3√{4^2}$ 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 $√30$ = $√2*√15$, $√3*√10$, or $√5*√6$ $√x*2√13=2√39$. What is the value of x? 1 3 9 13 26 We know that we must multiply the numbers under the root signs when root expressions are multiplied together. So: $x*13=39$ $x=3$ This means that our final answer is B, $x=3$ to get our final expression $2√39$ $√x*√y=√xy$ Because they can be separated, roots can also come together. So $√5*√6$ = $√30$ Reducing Roots It is common to encounter a problem with a mixed root, where you have an integer multiplied by a root (for example, $4√3$). Here, $4√3$ is reduced to its simplest form because the number under the root sign, 3, is prime (and therefore has no perfect squares). But let's say you had something like $3√18$ instead. Now, $3√18$ 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 18. 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.) 18 has several factor pairs. These are: $1*18$ $2*9$ $3*6$ Well, 9 is a perfect square because $3*3=9$. That means that $√9=3$. This means that we can take 9 out from under the root sign. Why? Because we know that $√{xy}=√x*√y$. So $√{18}=√2*√9$. And $√9=3$. So 9 can come out from under the root sign and be replaced by 3 instead. $√2$ is as reduced as we can make it, since it is a prime number. We are left with $3√2$ as the most reduced form of $√18$ (Note: you can test to see if this is true on most calculators. $√18=4.2426$ and $3*√2=3*1.4142=4.2426$. The two expressions are identical.) We are still not done, however. We wanted to originally change $3√18$ to its most reduced form. So far we have found the most reduced expression of $√18$, so now we must multiply them together. $3√18=3*3√2$ $9√2$ So our final answer is $9√2$, this is the most reduced form of $3√{18}$. You've rooted out your answers, you've gotten to the root of the problem, you've touched up those roots.... Absolute Values Absolute values are quite common on the ACT. You should expect to see at least one question on absolute values per test. 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 $|x+4|=12$, has two solutions: $x=8$ $x=−16$ Why -16? Well $−16+4=−12$ and, because it is an absolute value (and therefore a distance), the final answer becomes positive. So $|−12|=12$ When you are presented with an absolute value, instead of doing the math in your head to find the negative and positive solution, you can instead rewrite the equation into two different equations. When presented with the above equation $|x+4|=12$, take away the absolute value sign and transform it into two equations- one with a positive solution and one with a negative solution. So $|x+4|=12$ becomes: $x+4=12$ AND $x+4=−12$ Solve for x $x=8$ and $x=−16$ Now let's look at our absolute value problem from earlier: As you can see, this absolute value problem is fairly straightforward. Its only potential pitfalls are its parentheses and negatives, so we need to be sure to be careful with them. Solve the problem inside the absolute value sign first and then use the absolute value signs to make our final answer positive. (By process of elimination, we can already get rid of answer choices A and B, as we know that an absolute value cannot be negative.) $|7(−3)+2(4)|$ $|−21+8|$ $|−13|$ We have solved our problem. But we know that −13 is inside an absolute value sign, which means it must be positive. So our final answer is C, 13. Absolutely fabulous absolute values are absolutely solvable. I promise this absolutely. Consecutive Numbers Questions about consecutive numbers may or may not show up on your ACT. 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: 5, 6, 7, 8, 9 An example of negative, consecutive numbers would be: -9, -8, -7, -6, -5 (Notice how the negative integers go from greatest to least- if you remember the basic guide to ACT 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 five positive, consecutive integers is 5. What is the first of these integers? 21 22 23 24 25 If x is our first, unknown, integer in the sequence, so you can write all four numbers as: $x+(x+1)+(x+2)+(x+3)+(x+4)=5$ $5x+10=5$ $5x=105$ $x=21$ So x is our first number in the sequence and $x=21$: This means our final answer is A, the first number in our sequence is 21. (Note: always pay attention to what number they want you to find! If they had asked for the median number in the sequence, you would have had to continue the problem with $x=21$, $x+2=$median, $23=$median.) 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: 10, 12, 14, 16, 18 An example of positive, consecutive odd integers: 17, 19, 21, 23, 25 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 largest number in the sequence of four positive, consecutive odd integers whose sum is 160? 37 39 41 43 45 $x+(x+2)+(x+4)+(x+6)=160$ $4x+12=160$ $4x=148$ $x=37$ So the first number in the sequence is 37. This means the full sequence is: 37, 39, 41, 43 Our final answer is D, the largest number in the sequence is 43 (x+6). When consecutive numbers make all the difference. Remainders Questions involving remainders are rare on the ACT, but they still show up often enough that you should be aware of them. A remainder is the amount left over when two numbers do not divide evenly. If you divide 18 by 6, you will not have any remainder (your remainder will be zero). But if you divide 19 by 6, you will have a remainder of 1, because there is 1 left over. You can think of the division as $19/6 = 3{1/6}$. 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, $19/6 = 3$ remainder 1 or 3.167). But you may still come across the occasional remainder question on the ACT. How many integers between 10 and 40, inclusive, can be divided by 3 with a remainder of zero? 9 10 12 15 18 Now, we know that when a division problem results in a remainder of zero, that means the numbers divide evenly. $9/3 =3$ remainder 0, for example. So we are looking for all the numbers between 10 and 40 that are evenly divisible by 3. There are two ways we can do this- by listing the numbers out by hand or by taking the difference of 40 and 10 and dividing that difference by 3. That quotient (answer to a division problem) rounded to the nearest integer will be the number of integers divisible by 3. Let's try the first technique first and list out all the numbers divisible by 3 between 10 and 40, inclusive. The first integer after 10 to be evenly divisible by 3 is 12. After that, we can just add 3 to every number until we either hit 40 or go beyond 40. 12, 15, 18, 21, 24, 27, 30, 33, 36, 39 If we count all the numbers more than 10 and less than 40 in our list, we wind up with 10 integers that can be divided by 3 with a remainder of zero. This means our final answer is B, 10. Alternatively, we could use our second technique. $40−10=30$ $30/3$ $=10$ Again, our answer is B, 10. (Note: if the difference of the two numbers had NOT be divisible by 3, we would have taken the nearest rounded integer. For example, if we had been asked to find all the numbers between 10 and 50 that were evenly divisible by 3, we would have said: $50−10=40$ $40/3$ =13.333 $13.333$, rounded = 13 So our final answer would have been 13. And you can always test this by hand if you do not feel confident with your answer.) Prime Numbers Prime numbers are relatively rare on the ACT, but that is not to say that they never show up at all. So 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, 13 is a prime number because $1*13$ is its only factor. (13 is not evenly divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10, , or 12). 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. Standardized tests love to include the fact that 2 is a prime number as a way to subtly trick students who go too quickly through the test. If you assume that all prime numbers must be odd, then you may get a question on primes wrong. 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 I only II only III only I and III only 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 ($2*2=4$ $3*3=9$). 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 $y=5$. $4+5=$. 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 $3^2=9$ and 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 prime number question you may see on the ACT will ask you to identify how many prime numbers fall in a certain range of numbers. How many prime numbers are between 20 and 40, inclusive? Three Four Five Six Seven 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: 21, 23, 27, 29, 31, 33, 37, 39 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 23 is NOT divisible by 3 because $2+3=5$, which is not divisible by 3. However 21 is divisible by 3 because $2+1=3$, which is divisible by 3. So we can now eliminate 21 $(2+1=3)$, 27 $(2+7=9)$, 33 $(3+3=6)$, and 39 $(3+9=12)$ from the list. We are left with 23, 29, 31, 37. 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 a number's square root could 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. And, since we are dealing with potential primes, we only need to test odd integers equal to or less than the square root. Why? Because all multiples of even numbers will be even, and 2 is the only even prime number. Going back to our list, we have 23, 29, 31, 37. Well the closest square root to 23 and 29 is 5. We already know that neither 2 nor 3 nor 5 factor evenly into 23 or 29. You’re done. Both 23 and 29 must be prime. (Why didn't we test 4? Because all multiples of 4 are even, as an even * an even = an even.) As for 31 and 37, the closest square root of these is 6. But because 6 is even, we don't need to test it. So we need only to test odd numbers less than six. And we already know that neither 2 nor 3 nor 5 factor evenly into 31 or 37. So we are done. We have found all of our prime numbers. So your final answer is B, there are four prime numbers (23, 29, 31, 37) between 20 and 40. A different kind of Prime. Steps to Solving an ACT Integer Question Because ACT integer questions are so numerous and varied, there is no set way to approach them that is entirely separate from approaching other kinds of ACT math questions. But there are a few techniques that will help you approach your ACT integer questions (and by extension, most questions on ACT 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! 6, 8, 10 are consecutive even integers. If x=6, 8=x+2, and 10=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 ACT 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. Got your list in order? Than let's get cracking! Test Your Knowledge 1. 2. 3. 4. 5. Answers: C, D, B, F, H Answer Explanations: 1. We are tasked here with finding the smallest integer greater than $√58$. There are two ways to approach this- using a calculator or using our knowledge of perfect squares. Each will take about the same amount of time, so it's a matter of preference (and calculator ability). If you plug $√58$ into your calculator, you'll wind up with 7.615. This means that 8 is the smallest integer greater than this (because 7.616 is not an integer). Thus your final answer is C, 8. Alternatively, you could use your knowledge of perfect squares. $7^2=49$ and $8^2=64$ $√58$ is between these and larger than $√49$, so your closest integer larger than $√58$ would be 8. Again, our answer is C, 8. 2. Here, we must find possible values for a and b such that $|a+b|=|a−b|$. It'll be fastest for us to look to the answers in order to test which ones are true. (For more information on how to plug in answers, check out our article on plugging in answers) Answer choice A says this equation is "always" true, but we can see this is incorrect by plugging in some values for a and b. If $a=2$ and $b=4$, then $|a+b|=6$ and $|a−b|=|−2|=2$ 6≠ 2, so answer choice A is wrong. We can also see that answer choice B is wrong. Why? Because when a and b are equal, $|a−b|$ will equal 0, but $|a+b|$ will not. If $a=2$ and $b=2$ then $|a+b|=4$ and $|a−b|=0$ $4≠ 0$ Now let's look at answer choice C. It's true that when $a=0$ and $b=0$ that $|a+b|=|a−b|$ because $0=0$. But is this the only time that the equation works? We're not sure yet, so let's not eliminate this answer for now. So now let's try D. If $a=0$, but b=any other integer, does the equation work? Let's say that $b=2$, so $|a+b|=|0+2|=2$ and $|a−b|=|0−2|=|−2|=2$ $2=2$ We can also see that the same would work when $b=0$ $a=2$ and $b=0$, so $|a+b|=|2+0|=2$ and $|a−b|=|2−0|=2$ $2=2$ So our final answer is D, the equation is true when either $a=0$, $b=0$, or both a and b equal 0. 3. We are told that we have two, unknown, consecutive integers. And the smaller integer plus triple the larger integer equals 79. So let's find our two integers by writing the proper equation. If we call our smaller integer x, then our larger integer will be $x+1$. So: $x+3(x+1)=79$ $x+3x+3=79$ $4x=76$ $x=19$ Because we isolated the x, and the x stood in place of our smaller integer, this means our smaller integer is 19. Our larger integer must therefore be 20. (We can even test this by plugging these answers back into the original problem: $19+3(20)=19+60=79$) This means our final answer is B, 19 and 20. 4. We are being asked to find the smallest value of a number from several options. All of these options rely on our knowledge of roots, so let's examine them. Option F is $√x$. This will be the square root of x (in other words, a number*itself=x.) Option G says $√2x$. Well this will always be more than $√x$. Why? Because, the greater the number under the root sign, the greater the square root. Think of it in terms of real numbers. $√9=3$ and $√16=4$. The larger the number under the root sign, the larger the square root. This means that G will be larger than F, so we can cross G off the list. Similarly, we can cross off H. Why? Because $√x*x$ will be even bigger than $2x$ and will thus have a larger number under the root sign and a larger square root than $√x$. Option J will also be larger than option F because $√x$ will always be less than $√x$*another number larger than 1 (and the question specifically said that x1.) Remember it using real numbers. $√16$ (answer=4) will be less than $16√16$ (answer=64). And finally, K will be more than $√x$ as well. Why? Because K is the square of x (in other words, $x*x=x^2$) and the square of a number will always be larger than that number's square root. This means that our final answer is F, $√x$ is the least of all these terms. 5. Here, we are multiplying bases and exponents. We have ($2x^4y$) and we want to multiply it by ($3x^5y^8$). So let's multiply them piece by piece. First, multiply your integers. $2*3=6$ Next, multiply your x bases and their exponents. We know that we must add the exponents when multiplying two of the same base together. $x^4*x^5=x^[4+5]=x^9$ Next, multiply your y bases and their exponents. $y*y^8=y^[1+8]=y^9$ (Why is this $y^9$? Because y without an exponent is the same thing as saying $y^1$, so we needed to add that single exponent to the 8 from $y^8$.) Put the pieces together and you have: $6x^9y^9$ So our final answer is H, 6x9y9 Now celebrate because you rocked those integers! 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 (have you had reason to use remainders much outside of elementary school?). But most integer questions are much simpler than they appear. If you know your way around exponents and you remember your definitions- integers, consecutive integers, absolute values, etc.- you’ll be able to solve most any ACT integer question that comes your way. What’s Next? You've taken on integers, both basic and advanced, and emerged victorious. Now that you’ve mastered these foundational topics of the ACT math, make sure you’ve got a solid grasp of all the math topics covered by the ACT math section, so that you can take on the ACT with confidence. Find yourself running out of time on ACT math? Check out our article on how to keep from running out of time on the ACT math section before it's pencil's down. Feeling overwhelmed? Start by figuring out your ideal score and work to improve little by little from there. Already have pretty good scores and looking to get a perfect 36? Check out our article on how to get a perfect ACT math score written by a 36 ACT-scorer. Want to improve your ACT score by 4 points? Check out our best-in-class online ACT prep program. We guarantee your money back if you don't improve your ACT score by 4 points or more. Our program is entirely online, and it customizes what you study to your strengths and weaknesses. If you liked this Math lesson, 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:

Repeal of the Affordable Care Act (Obama Care) Research Paper

Repeal of the Affordable Care Act (Obama Care) - Research Paper Example It would contribute in reducing the abuses of insurance industry. According to the government, the bill will give more protection to American nation on health issues. Children will get more health care as compared to past; there will be no limit for the annual health care for people (Healthcare.gov, 2011). Patients will get preventive services without sharing any cost. Government tends to implement the plan in practice from 2014. Despite all the goods that plan seem to bring in the health care of Americans, there are a lot many reservations in the public (Healthcare.gov, 2011). Middle class suffers more as the tax cut from small businesses and insurance policies would help poor or low-income people to buy policies for them. The plan is aimed at reducing the premium of businesspersons running small businesses to help the needy people. Republicans have rejected the law for not having any constitutional authority and have raised voice for repeal (Healthcare.gov, 2011). The Repeal of the Affordable Care Act (Obama Care) is a pending piece of political legislation and its repealing has many reasons behind it. This paper discusses the issue of repeal of the Affordable Care Act in detail. ...ore, Richard Neal, William Pascrell, Gary Peters , Earl Pomeroy , Linda Sanchez, Allyson Schwartz, Ike Skelton, Fortney Stark, John Tanner, Michael Thompson, Dina Titus, Christopher Van Hollen and John Yarmuth (GovTrack.us 2009). The Republicans who sponsored the Affordable Care Act are Virginia Brown-Waite, Walter Jones and Todd Platts (GovTrack.us, 2009). American Nurses Association, American Medical Association, American Academy of Family Physicians, American College of Physicians, Association of American Medical Colleges, National Association of Community Health Centers, American Osteopathic Association, Catholic Health Association, American Public Health Association, Asian and Pacific Islander American Health Forum, Doctors for America and National Hispanic Medical Associatio n consider the Affordable Care Act as a revolutionary act that keeps the capability of attending all the problems associated with health care (Foster, 2011). According to the mentioned health care communities, repeal of Affordable Care Act will be depreciative for health care services and will take health care back to its backward form (Foster, 2011). Ostensible Objectives of the Legislation According to the legislature, ObamaCare Act has many ostensible objectives. They are to make certain relaxed admittance to emergency facilities, to make certain that all people are insured under the law, to make employers responsible for their employees for the provision of health care insurance services, to ensure betterment of insurance policies by increasing insurance coverage, to make sure that the premiums of insurance are kept low and to impose tax penalties for people with no health insurance (ACEP, 2010). Hidden Agendas or Objectives to the Proposed Law Democrats and republicans both hav e some

The Great Depression Research Paper Example | Topics and Well Written Essays - 1750 words

The Great Depression - Research Paper Example The crash of the Stock Market, unemployment and bankruptcy, the Smoot-Hawley Tariff led political changes, emigration and new reforms such as The New Deal took the United States in a sweeping landslide (Sauert 129). The dawn of the Great Depression is usually narrowed down to the crash of the Stock Market which occurred on Tuesday, 29th October, 1929 when the Dow Jones Industrial Average fell to a very low rate, almost as low as 23 percent and the market suffered a gigantic loss which ranged from $ 8 billion to $ 9 billion. However, this was just one of the many losses in a period when severe market volatility was rampant and it exposed the people who had bought stocks on loans (Taylor). This crash of the Wall Street Stock Market completely changed the direction of the events. It marked the arena where the United States was thrown into political chaos and economic instability. The crash of the Stock Market was one of the major reasons that led to the Great Depression. However, it mer ely dealt out a card and there were many more reasons to come. For two months after the crash there was a loss of more than $ 40 billion dollars (Ross 32). Stock holders could not pay back their loans and there was a marked reduction in transactions. People stopped purchasing items which led to a lesser number produced and hence lack of work force. More and more people lost their jobs which simultaneously led to their properties being repossessed. Soon unemployment was rampant in the region (Hembree). The crash of the Wall Street of the Stock Market was thoroughly disastrous for the United States for it completely negated the concept of the American Dream as propagated by President Hoover and his predecessors. For the... The consequences of the Great Depression were staggering for it left thousands and thousands of unemployed people roaming the streets by night and day and trying to find work. The tide of the Depression had calamitous consequences. Not only billions of people lost their homes in one sweeping wave but they also had to migrate to places which were not fit for living. Shanty towns had cropped up in various parts of the United States and they were built out of tents and other sorts of garbage like items such as hulks of old cars. They were known as ‘Hoovervilles’ which was a mocking reference to President Hoover as he had fallen into disgrace for many blamed him for their turmoil and the turn of events. There were also other derogatory terms used in reference to him such as ‘Hoover Blank’ which was an old newspaper used as a blanket, ‘Hoover wagon’ which was an vehicle drawn by a horse since the owner could not afford gas. Unable to do anything, the general public vented out their pent-up misery by using such offensive references for Hoover. The coining of these terms depicted the nationwide view of Hoover in the minds of the public. Women took up men’s work so as to support their families along with their men but the availability of jobs was scarce.

The Role Of Information Systems

The Role Of Information Systems Modern business is surrounded by the information systems which are in place to assess the opportunities and limitations available for the businessmen in order to be productive in their respective field. It is impossible to know the information systems without the proper knowledge of the information technology which is changing at a rapid pace nowadays. The movement and processing of data and information to expedite business operations and decisions is called information systems. (McGraw-Hill, 2000) Role of information system The role of the information system is to forecast the needs and demands of the company on the basis of its current usage and to keep in mind the future changes which are going to take place for instance extension of business operations in the new market so the information system can propose larger database which will easily store the data. It is a decision for the top management and includes huge cost. Types of information systems Information systems may differ in their needs but the most common types of information systems are discusses in detail below; Decision support system (DSS) Management information system (MIS) Transaction processing system (TPS) Executive support system (ESS) Operational-level Systems This system has to support operational managers by keeping track of the elementary activities and transactions of the organization. The principle function of systems at this level is to answer regular questions and track the flow of transactions through the organization. This system covers things such as sales, receipts, cash deposits, payroll, credit decisions and flow of materials. Knowledge-level Systems This system looks after the support knowledge and data workers in an organization. The purpose of these systems is to help the organization discover, sort out and put together new and existing knowledge in to the business, and to help control the flow of paperwork. These systems, especially in the form of collaboration tools, workstations, and office systems, are the fastest growing applications in business today.   Management-level Systems This is designed to serve the monitoring, controlling, decision-making, and administrative activities of middle managers. These typically provide periodic reports rather than instant  on operations. Some of these systems support non-routine decision-making, focusing on less-structured decisions for which requirements are not always clear. This will often require from outside the organization, as well as from normal operational-level data. Strategic-level Systems It helps senior management to handle and address strategic issues and long-term trends, both within the organization and in the environment outside the operations. The principal concern is matching organizational capability to changes, and opportunities, occurring in the medium to long term (i.e. 5 10 years) in the external environment. Typically, an organization might have operational, knowledge, management and strategic level systems for each functional area within the organization. This would be based on the management model adopted by the organization, so, while the most commonly-adopted systems structure would simply follow the standard functional model, structures reflecting bureaucratic, product and matrix models are also possible. Operational-level Systems   Transaction-Processing Systems (TPS) Basic business systems Perform daily routine transactions necessary for business functions At the operational level, tasks, resources and goals are predefined and highly structured Generally, five functional categories are identified, as shown in the diagram. Knowledge-level Systems   Office Automation Systems (OAS) Targeted at meeting the knowledge needs of  data workers  within the organization Data workers tend to process rather than create primarily involved in  use, manipulation or dissemination. Typical OAS handles and manages documents, scheduling and communication. Knowledge Work Systems (KWS) Targeted at meeting the knowledge needs of  knowledge workers  within the organization In general, knowledge workers hold degree-level professional qualifications (e.g. engineers, scientists, lawyers), their jobs consist primarily in creating new knowledge and information for that particular department in order to find out the best suitable candidates to work with the organization. KWS, such as scientific or engineering design workstations, promote the creation of new knowledge, and its dissemination and integration throughout the organization. Management-level Systems   Management information  Systems (MIS) MIS provide managers with reports and, in some cases, on-line access to the organizations current performance and historical records Typically these systems focus entirely on internal events, providing the information for short-term planning and decision making. MIS summarize and report on the basic operations of the organization, dependent on the underlying TPS for their data. Decision-Support Systems (DSS) As MIS, these serve the needs of the management level of the organization Focus on helping managers make decisions that are semi-structured, unique, or rapidly changing, and not easily specified in advance Use internal information from TPS and MIS, but also receive data  from the external sources Greater analytical power than other systems, incorporate modeling tools, aggregation and analysis tools, and support what-if  scenarios They must provide user-friendly, interactive tools Strategic-level Systems   Executive Support Systems (ESS/EIS) Serve the strategic level of the organization ESS/EIS address unstructured decisions and create a generalized computing and communications environment, rather than providing any fixed application or specific capability. Such systems are not designed to solve specific problems, but to tackle a changing array of problems ESS/EIS are designed to incorporate data about external events, such as new tax laws or competitors, and also draw summarized data from internal MIS and DSS These systems filter, compress, and track critical data, emphasizing the reduction of time and effort required to obtain data useful to executive management ESS/EIS employ advanced graphics software to provide highly visual and easy-to-use representations of complex and current trends, but they tend not to provide analytical models which can be helpful in carrying out the regular tasks at the operations level. Conclusion We have come a long way from conventional planning in a development project. The reasons for this change are basically related to four conflicting factors that constitute an over-riding problem with formal planning. Large software systems have long development cycles and require extensive planning to control costs, resources, equipment and priorities, that is why organizations have to take into effect extra measures to cope with such large information systems in order to be more productive and to meet the future needs of the business. Planning is very significant as it will be the very nature of the exercise, which is suppose to seek and undertake future activities in a controlled, reasonable and effective manner. Without the effectiveness of such planning, most of the projects would go into chaos at the early stage of their formations. That is why planning has to be meaningful as the future which depicts on it should be sensible and unchanging. If there is need for changes to occur, then they should be of a limited or anticipated nature and without rapid transitions but the long term duration software projects suffer quite easily from the major, unforeseen and generally rapid changes. These are due as (among others) the development setbacks, migration of personnel, economic down-turns, strategic reversals, significantly modified the technology and systems had to be changed as the expectations which were required earlier were dramatically change due to these unforeseen circumstances. Reference and Bibliography Website: http://navismagazine.com/sample/xxi-cent-warships/degaulle.htm Bell S, Frances prestige warship all at sea, The London Times, 25.2.1999, p.20 Byte Magazine, March 1989 Personal Computer World, June 1989 Yeates D (ed), System Project Management, Pitman, 1986, Chapter 3 Bentley C, Introducing PRINCE, NCC Blackwell, 1992, p.1 Donnelly F, Plan for all seasons, Computing, 4.6.1992, p.32 Kavanagh J, Blind leading the blind into IT fog, Interface, The London Times, 6.8.1997, p.10 Gulton A, Managing the unexpected, Computer Weekly, 4.3.1999, p.30 Belford C, Integrated Business Software Systems: The Cost of Change, Executive Brief, URL Source: www.govcomp.com/executivebrief.html Date of Access 25th February 2010 Ritzman. L, Malhotra. M, 2009. Operations Management, 9th Edition, P. 31 Maylor. H, 2005. Project Management, 3rd Edition, P. 28 Peter. S, Cavanagh. R, 2001. The Six Sigma Way, P. 161

This essay describes how I have worked towards and performed four :: Drama

This essay describes how I have worked towards and performed four pieces of practical work using all three art forms. (Drama, dance, music, and a final piece that is a mixture of all three arts Performance studies This essay describes how I have worked towards and performed four pieces of practical work using all three art forms. (Drama, dance, music, and a final piece that is a mixture of all three arts). Each piece must be three minutes long and we have around two months to improvise, rehearse and perform the four pieces. We were given five ‘key words’ to follow as a sort of guide line for our pieces. There was a different set of guide lines for each art form. These consist of†¦ Music – rhythm, melody, harmony, timbre, texture. Dance – motif, action, relationships, dynamics, space. Drama – dialogue,1 characterisation, physicality, proxemics, tension. These fifteen rules do interconnect between the arts which I will explain during this essay. We started our performance studies classes by learning about and experimenting with improvisation as well as learning about the five rules for each of the arts. We then began to look for the five rules in our dance / musical / drama pieces. We experimented with the five rules in our improvised pieces and all so broke down each of the rules to find out exactly what they can cover. Now we split ourselves in to groups and began to experiment using improvisation for the final performance.2 In dance we sat as a group and planned out linking moves that used all of our five rules, and then put them together by improvising links. In drama we used improvisation to create scenes and improvised the scenes endings, and the drama in them. We also improvised characters for these parts, and once we found something that we liked we would enhance it through rehearsal and write a script. In music we used improvisation to begin to create musical pieces. We created and re-created compositions until we could find a sound that would suit the mood we were aiming3 to achieve. In the mixed piece we had to firstly sit and discuss our options and we decided to base it around a theme. Our theme was on circles and squares, creating a piece about the trapped ness of a single person, using the shapes as representatives of the person’s feelings. We did hit some problems during the creations of these pieces. In dance we had a lot of people dropping out of the coarse so our group ended up as a double act. But still we managed to capture all the five

Describe Popular Culture in Britain at the Beginning of the 1960’s

The 1950's were a conservative period. The country was recovering from the ravishes of war and many people wanted society to return to how it was in the 1930's. America was leading the way forwards however much the older generation disliked it. New music was appearing such as Elvis Presley and Cliff Richard. More consumer goods could be afforded by the middle classes creating a better standard of living. Televisions began to be purchased widely as did refrigerators and washing machines. The standard of living of the average person living in Britain rose during the 1950's. The popular catchphrase used in the 1959 election by Harold Macmillan was ‘you've never had it so good' which in my view sums up the 1950's; and by 1960 change was imminent. The culture of the 1960's reflects Britain at this time. Attitudes of many people in Britain were still very conservative entering the 1960's. However, things had begun to change for many groups in society. Women were still second class citizens but a large proportion of them had begun to work. They were paid less and did most of the manual jobs. Some male attitudes towards women still hadn't changed very much and many saw women still as ‘baby machines'. It was generally unheard of in 1960 for women to wear trousers in public, let alone to work. Many women began to demand equal rights, and by the mid to late 60's much had changed for women. In the later years of the 50's Britain saw a large influx of immigration from the Commonwealth. There were jobs available in Britain and better living conditions. By the 1960's Britain had become a multi-cultural society. The large majority of immigrants arrived from the West Indies and India. They suffered from large amounts of racism and were given the poorly paid jobs. Many did menial work and failed to buy a house for many years. It wasn't until the later 1960's that Britain became a more tolerant society. Crime was at a low in the late 1950's to 1960. Few needed to turn to crime due to the increase in wealth by the large majority of people. Capital punishment was still being used despite an increasingly large number of people turning against it. Crime was in fact at a lower rate than in today's society. Drugs were starting to be used by a certain group of people. It was not until the later 1960's when drugs became more widely used. Times were changing but the society was still very conservative with few people using drugs. Attitudes towards sexual behaviour had not yet begun to change by the early 1960's. It was still a very conservative society although in the later 1960's it became more permissive. The contraceptive pill had been discovered but was not widely used. Many women still believed that sex before marriage was a sin although this did change. Britain was still very much a class society although times were changing. The middle classes were expanding and the upper classes were starting to lose the control. Cars in the 1950's became more widely available. Ownership was rapidly increasing throughout the 50's due to the reduction in price. In 1960 the Mini was brought out alongside cars such as the bubble car made by a German company. They were advertised on television and on the radio as a necessary possession and fun. In 1959 the first motorway to be built in Britain the M1 was opened. Travelling long distances was becoming much easier and accessible for the general public. Public transport began to suffer problems due to the huge increase in cars. The steam trains effectively died out being replaced by the newer and more economical electric and diesel powered engines. Less people were using them as a method of transport and there were large cuts in the workforce. There was a new period beginning in the holiday industry. Some were still very traditional like Butlins, bed and breakfasts and other holiday camps. However with people owning cars they experienced a new found freedom. Places such as the Costa del Sol became popular with British holiday makers. Package holidays grew in popularity as did camping and caravanning holidays. The new found freedoms experience with the new transport was shown in various ways throughout culture in the 1960's. As in any period of time the media always shows how the culture was changing. Until the early 1960's the British television was dominated by the upper classes. There were only two channels, BBC and ITV with BBC 2 starting in 1965. However the television industry underwent a period of change in the early 1960's Programmes began to get shown with people from middle and working classes on. Coronation Street was first broadcast in 1960 and others soon followed. The television started to reflect the society around it and soon became the most popular activity. Dramas began to be shown like ‘Cathy Come Home' which were watched throughout Britain and had great impacts on society. Campaigning programmes also began to be broadcast for example Tonight and other news based programmes. There were more programmes for the teenage and child markets. The 6 -5 Special was first followed by Ready Steady Go. These had major impacts on the youth culture of the 1960's. During the 1950's radio was the most accessible and widely listened to form of entertainment. There were such stations as Radio 2 and 4. These were listened to by adults and children alike. However by 1960 things were changing. The teenage market had developed and they were demanding their own radio station playing their music. Many pirate radio stations set up, such as Radio Caroline. These played popular music of the time and aimed themselves to the teenage market. In the late 1960's Radio 1 was created. This was aimed at the older teenage market. The 1960's were a period of change for the music industry. Newspapers were widely read although there were less tabloids than today. These were aimed towards the adult generation and left most teenagers and young people to find out the news from the television. They were generally more serious which reflects the culture of the early 1960's. Magazines were also much more serious. Women read ‘Women's Own' and nothing else was available. Young children had many different comic books at the time, Dandy, Bunty etc. It was not until the mid to late 60's that more revolutionary magazines like Cosmopolitan were created. There were very few music or hobby orientated magazines either. The traditional British film industry was in decline in the late 1950's. Typical romances were shown along with early action films. They all contained only upper class people. By 1960 new comedies were becoming popular. A change was underway with more people with working class accents getting into the film industry. These were more popular with the general public who enjoyed being able to watch a film that they could relate to. The first person to enter the ‘new' film industry was Michael Cain, who soon became a teenage icon. The media of the early 1960's reflects the change that was occurring in British society and culture of the time. Around 1960 there were 5 million people who now classed themselves as teenagers. This had become a whole new market for companies and advertising in the late 1950's. More of these young people had more money than ever before and could afford many new products. Throughout the 1950's the influence of American culture had been great. British teenagers wanted to be like their American counterparts and have coffee bars and their own fashion and music tastes. The society was changing and the demands of the new market were met. New music in 1950 and started the revolution in youth culture. Elvis Presley was frowned upon by the older generation but the youths were fascinated by the new concept of ‘Rock and Roll'. Cliff Richard became the British teen symbol and began the British music industry. Adults however were still listening to Frank Sinatra and the Joe Loss Orchestra. They couldn't understand the new music with their sexual lyrics and movements. In 1962 the Beatles released their first record. They were still relatively unknown playing in places like the ‘Cavern' and Hamburg. New fashions were starting to become the new way to be revolutionary. Teddy Boys began to make a statement in society as did the Rockers. Many teenagers and young people in the early 1960's though, still wore their parents fashions. Boys wore short trousers and shirts whilst girls wore skirts. Fashion as everything else in the late 1950's to early 60's was still very conservative. Popular culture in the early 1960's was still much the same as it had been throughout the 50's. Although a teenage market had emerged, little had really changed by 1962. The British culture was still very conservative. However things were soon to change. The 1960's were dramatically different to the ways the older generation had grown up. America looked to Britain as the culture capital of the world. New pop groups emerged and life became ‘swinging' for many young people. The older generation still did not approve but were powerless to do anything. The early 1960's can be described best as ‘a grey period' and not as the ‘swinging sixties' which came later. Describe popular culture in Britain at the beginning of the 1960’s To many people at the time, they were the â€Å"swinging sixties†. They were a decade when fashions changed continuously and young people appeared to have more freedom then ever before. It was time that many people look back on with found memories, but which other blame for some of the failings in society. The sixties consisted of stars like Elvis Presley, Cliff Richard and Marlin Monroue which I will be mentioning in this essay as well as other stars. These stars were like the modern David Beckham, Tom Cruise and J-lo. In 1958 the National Service ended which was a huge relief for some. People in England were looking to America for their icons, but America band members or artists were not allowed to enter the country, then so people like Cliff Richard was formed. Moreover in the 1960's Britain had not yet developed its own style therefore was still in America's shadow. The traditional and cultural values in the 1950's were now about to change. These were many due to the development of TV, radio and theatre which I will be exploring in this essay. By the end of the 1950's music was still heavily influenced by America a lot of people were listening to American stars like Elvis Presley. But by the beginning of the 1960's British artists like Billy Fury, Cliff Richard and Adam Faith all s tarted to appear in the charts. Many of these British starts were producing new records but mainly basing them Elvis rock music. The British fashion was changing very rapidly and stars like Jackie Kennedy influenced them a lot. Wearing shifts and A-line dressed which had been designed in during the 1954 and 1955 were not worn until the 1960's. These gave women a new sense of freedom. Moreover the British film was not as popular as it was in America but brand new films featuring pop starts interested a lot of teenagers. There was now new British humour which people had never seen before. Radio was introduced and started showing some distinctive style. Radio shows such as â€Å"Beyond the Fringe† was a mocking comedy that criticized the way the country was run making people think more about politics. Many television programmes were brought so people watch television at home but there were only two channels BBC and ITV which could only run for about five to six hours a day. In 1960 the first episode of â€Å"Coronation Street† appeared in black and white in TV sets all over the country. Finally there was theatre which consisted of plays developed to mirror real life situations. As well as this it also helped to break down class barriers. Youth culture began to expand and grow by the late 60's rival groups such as â€Å"Mods† who drove Italian scooters instead of motorbikes were formed also girls began to wear more mini skirts whereas they were very strict. Prices generally went up as wages rose by 34% in the 1960's. In some respects it started popular music, fashion and a rise economy. Youth culture changed Britain's youth forever.In conclusion much of Britain's popular culture was a mix of American and 1950's Britain but changes were beginning to develop which would lead to a new British culture and a new era.

In William Blake’s Songs of Innocence and Songs of...

In William Blake’s Songs of Innocence and Songs of Experience, many of the poems correlate in numerous aspects. For example, The Chimney Sweeper is a key poem in both collections that portrays the soul of a child The Chimney Sweeper in Innocence vs. The Chimney Sweeper in Experience In William Blake’s Songs of Innocence and Songs of Experience, many of the poems correlate in numerous aspects. For example, The Chimney Sweeper is a key poem in both collections that portrays the soul of a child with both a naà ¯ve and experienced persona. Blake uses the aspects of religion, light versus dark imagery, and the usage of the chimney sweeper itself to convey the similarities and differences of the figure in both poems. The†¦show more content†¦After Tom awakes from this dream, he was â€Å"happy and warm† with the knowledge that with God, there was no need to fear death. However, in Songs of Experience, the outlook on life and death is not so joyful. The religious imagery is not so much as in Songs of Innocence, possibly because people tend to believe more religiously when innocence dominates terrible experiences. In the latter poem, however, the â€Å"little black thing† has been â€Å"clothed in the clothes of death† by his parents forcing him to become a chimney sweeper. His parents have â€Å"gone to praise God and his Priest and King, who make up a heaven of our misery† and the boy cannot understand this as he â€Å"sings the notes of woe† and not happiness. This chimney sweeper does not have the innocence and hopefulness of the chimney sweeper in Songs of Innocence. This child possesses experience of hardship and does not hold much faith in God and religion. This version of The Chimney Sweeper lacks the hopefulness and faith found in the former version although it is the same setting, factors, and occupation. William Blake conveys both innocence and experience with the literary technique of light versus dark imagery. In Songs of Innocence, Blake discusses the issue of soot on several instances. In the beginning verse, the young chimney sweeper slept in soot, showing the incorruptibility and despair of the young child. Also, Tom Dacre’s â€Å"white hair† was shaved so that the dark sootShow MoreRelatedWilliam Blake s The Tyger1132 Words   |  5 PagesWilliam Blake’s â€Å"The Tyger† and Tragedies William Blake wrote a set of poems in his collection Songs of Innocence and Songs of Experience. Some of the poems in each collection were meant to be read together to show the difference between innocence and experience. Many people question why Blake wrote a two part series to his poems and what they could actually mean. Two specific poems, â€Å"The Lamb† and â€Å"The Tyger,† were meant to be read together. â€Å"The Lamb† is a part of Blake’s Songs of Innocence andRead MoreWilliam Blake in Contrast of Songs of Innocence and of Experience1452 Words   |  6 PagesEN 222-Intro to British Lit. II April 21, 2012 William Blake in contrast of Songs of Innocence and of Experience William Blake, an engraver, exemplified his passion for children through his many poems. Blake lived in London most of his life and many fellow literati viewed him as eccentric. He claimed to have interactions with angels and prophets, which had a great influence on his outlook of life. Blake believed all prominent entities, those being church, state, and government had become sick withRead MoreSongs of Good and Evil1545 Words   |  7 Pages Simple, limited, and unadventurous all describe William Blake’s life (Greenblatt, Abrams, Lynch, Stillinger). Blake was born November 28, 1757 in London, England and his artistic ability became evident in his early years. Blake had a very simple upbringing and had little education. His formal education was in art and at the age of fourteen he entered an apprenticeship with a well-known engraver who taught Blake his skills in engraving. In Blake’s free time, he began reading writing poetry. At theRead MoreEssay on Biography of William Blake1631 Words   |  7 Pagesyourself with influence and inspire your work and success. William Blake was a famous artist, engraver and poet. However, it was not until 1863 that he became famous when Alexander Gilchrist published his biography(Blake, William, and Geoffrey Keynes).Blake and his poetry have been compared to Shakespeare (Kathleen Raine). As an artist Blake was equated to Michelangelo. Being born during the time of both the American and French Revolution, William Blake was against both the Church and the State. BlakeRead MoreSociological Criticism of William Blake’s Poetry Essay1506 Words   |  7 Pagespolitical and economic ideas of communism and social inequality. William Blake, a Romantic poet, frequently wrote on the topic of class oppression and his opposition to the exploitation of the proletariat by the capitalists. Blake’s ideology and preference towards an equalitarian society quite closely mirror the theories of Karl Marx. Analyzing Blake’s poetry from a Marxist perspective paints a clearer picture of the motives behind Blake’s anger towards social inequality. Poems such as â€Å"The Chimney Sweeper†Read MoreThe Human Abstract Essay1196 Words   |  5 Pagesown. Of the critical interpretations that do exist, many approach the poem by examining its various manifestations in Blakes manuscripts, reading it against A Divine Image, a poem w hich was never finally published by Blake, or comparing it to its Innocence counterpart, The Divine Image. Most critics seem to agree that The Human Abstract represents a philosophical turning point in The Songs of Innocence and of Expe rience, and in Blakes work as a whole. In 1924, Joseph H. Wicksteed observesRead MoreThe Innocence of Lamb in Songs of Innocence by William Blake615 Words   |  3 PagesSongs of Innocence by William Blake collocates the naà ¯ve lives of children and loss of innocence of adults, with m oral Christian values and how religion has the capacity to promote cruelty and prejudice. Blake was born in 1757, up to and after the French Revolution he wrote many works criticizing enlightened rationalism and instead focused on intellectual ideas that avoided institutionalization and propelled ethical and moral order. Blake’s collection of poem exposes and explores the values and limitationsRead MoreWilliam Blake s Songs Of Innocence And Experience1268 Words   |  6 PagesWilliam Blake’s Songs of Innocence and Experience, printed in 1794, â€Å"represents the world as it is envisioned by what he calls ‘two contrary states of the human soul’† (Greenblatt, 1452). This collection of poetry is accompanied by pictures, which create a mutually reliant relationship that allows for complete understanding of Blake’s works. â€Å"To read a Blake poem without the pictures is to miss something important: that relationship is an aspect of the poemâ €™s argument† (1452). Overall, Blake’s worksRead MoreWilliam Blake s The Lamb And The Tyger873 Words   |  4 Pagesbiblical times. William Blake describes the young sheep in similar characteristics in the poem â€Å"The Lamb† and â€Å"The Tyger†. A tiger as we know its characteristics to be is fierce and mysterious. Always lurking around, waiting for the perfect opportunity to strike. In William Blake’s two separate poems he ties each of the poems together with in-depth understanding and symbolism. Both animals play an important part in both of Blake’s poems. The use of the lamb in William Blake’s poem is significantRead MoreThe Romantic Poetry Of Blake And Shelley1494 Words   |  6 Pagessimplistic language, personal experiences, democracy, and liberty, significant in various art forms including poetry. The development of the self and self-awareness became a major theme as the Romantic Period was seen as an unpredictable release of artistic energy, new found confidence, and creative power found in the writings of the Romantic poets Blake, Wordsworth, Coleridge, Keats, and Shelley, who made a substantial impact on the world of poetry. Two of the Romantic poets, William Blake, and Percy Bysshe