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# differentiating

2. ### Differentiating fractions without the quotient rule?

Differentiate y=\frac{x^{2}}{(x+4)} without using the quotient rule. Can you use the product rule to differentiate above? If so, what are the steps? y=x^{2}(x+4)^{-1} \frac{dy}{dx} = 2x(x+4)^{-1}+x^{2}(1)^{-1} Is the above correct so far? If not, where did I make the mistake? :)
3. ### Help with differentiating a tricky function

Hello everyone, I am a new user and this is my first post. As a bit of background ... I haven't done much serious maths for about 40 years now. In the early 1970's I studied Astronomy and Physics at Glasgow University and so also had to study university-level Mathematics which was compulsory for...
4. ### Differentiating the integral by parameter

Hi, sorry for the stupid question, but how to find the following derivative?
5. ### Differentiating Inverse Trig Functions

Hi, I'm stuck with Q1. (e) on the attached file. I can do (a) - (d), it's just the x^2 at the front that's throwing me off. I've tried just differentiating the sin^-1(x/2) on it's own as done previously but that doesn't seem to help. Help would be greatly appreciated, thanks!
6. ### Differentiating a piece-wise function

I'm supposed to be differentiating f(x) = {x2sin(1/x) if x ≠ 0, 0 if x = 0}, once at f(x) = 0 using f'(x) = limf(x+h) - f(x)/h, and also by finding the lim f'(x) as x → 0 using product and trigonometric laws for derivatives. In the first case, I get f'(0) = 0, and in the second I get that the...
7. ### Differentiating an absolute function

Could someone please explain how to differentiate absolute functions? This is what I'm stuck on at the moment. f(x)=5-|x-5| I'm not aware of any rules for differentiating it, but they seem to pop up in my practice problems quite often. Any help is appreciated!
8. ### Differentiating sin(x/y)

How do we differentiate something like this? Chain rule should be used here, so the first term will be cos(x/y), but then I don't understand how to break up this (x/y) function?
9. ### Differentiating natural log function

$f(x)=a^x=(a)^x=(e^{\ln a})^x=e^{x \ln a}$ $f'(x)=$? I don't quite understand how the Chain Rule is applied here and what is the inner\outer function? Could someone elaborate? PS Oops, I meant natural exponential function.
10. ### Need help differentiating

i= v/r (1-e^-Rt/L) How would i go about differentiating this formula to get di/dt? Would I use the product rule or another rule?
11. ### Differentiating: The product rule

How would you go about differentiating this? g(x)=(x^3) (e^x) I tried using the product rule and this is what I have: g'(x) = (3x^2)(e^x) + (xe^x^-^1 )(x^3) but I saw online that the answer is g'(x) = (3x^2)(e^x) + (e^x )(x^3) What did I do wrong? Edit: I have only touched on the basics of...
12. ### differentiating maclaurin series

I have to differentiate the MacLaurin series for xe^x and use the result to show that \sum_{k=0}^{\infty} \frac{k+1}{k!} =2e x*\sum_{k=0}^{\infty} \frac{x^k}{k!} = x[1+x+\frac{x^2}{2!}+\frac{x^3}{3!}....] = x+x^2+\frac{x^3}{2!}+\frac{x^4}{3!}.... = \sum_{k=0}^{\infty} \frac{x^{k+1}}{k!} From...

There are 96 persons at a retreat. If, at t=0, 8 are infected, and after 4 days this has doubled, how many will have the disease 8 days after t=0? Assume rate at which it spreads is proportional to the number of contacts and that both groups move freely, so the number of contacts is proportional...
14. ### Differentiating an Integral

I haven't been able to figure out exactly how the author of this statistics book got the following. From, E[C_t(X)]=ct\int_0^tf(x)dx -c\int_0^txf(x)dx+k\int_t^\infty xf(x)dx-kt\int_t^{\infty}f(x)dx to, \frac{d}{dt}E[C_t(X)]=ctf(t)+cF(t)-ctf(t)-ktf(t)+ktf(t)-k[1-F(t)] I think I have to fully...
15. ### Differentiating Implicit Equations

If sin^4(\[Alpha])/x+cos^4(\[Alpha])/y-1/(x+y)==0 ^^^^(it looked fine in mathematica...can someone tell me how to write code in this forum?) then what is dy/dy??*here [alpha] is a constant* well i figured out dy/dx and it came out to be y/x but the book says dy/dx=tan^2[Alpha],which is a...
16. ### Differentiating an integral-defined function

hi, i have a problem with differentiating please help (I think) It is well known that for an integral-defined function, differentiating them with the upper limit(i.e. 'x') takes the integral away. (*) i.e. \frac{d}{dx}\int_{0}^{x}\ f(t) \ dt = f(x) The problem is, I came across a weird...
17. ### differentiating a complex function

Hi, any help to prove that?
18. ### Forming an equation and differentiating it

So you need to find an optimum price, so I just thought you had to find a stationary point, i.e. when the differentiation equals 0 But if you differentiate Q you just get -80. So you can't set this to 0, nor can the optimum price be -80. So I thought, ok it must be a bit harder than that. So...
19. ### Differentiating sin(2x)^2

I'm trying to differentiate \sin^2(2x) but it's not going very well. The answer is 2sin(4x). To me it seems natural to use the chain rule and get 2sin(2x) and then multiplied with the derivative of sin(2x) which is 2cos(2x). So my gut tells me the answer is 2sin(2x)*2cos(2x) but that's...
20. ### Differentiating y

I can't seem to wrap my head around differentiating y or f(x). For example: x^2+xy^2=2. I understand the product rule and everyone keeps saying that ''it's just the product rule'' when I'm differentiating implicitly like that, but I don't have the intuition, it's just missing. Can someone...


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# which expression is equivalent to 2w?

If a tree casts a shadow of 12 meters at the same time, how tall is the tree? A garden is 15-feet long by 5-feet wide. Which of the following is correct comparison of two rational numbers? b. log 18/p+2. A student starts working for $12 per hour. An employee discovers that 0.0025 of all products predicted at his factory contain defects, which of the following % does this represent? c. log3 + log(x + 4) What are the domain and range of mc017-1. The only exception is that division is not currently supported; Oops I did it again!! Write an equivalent expression for the written expression that is given: answer choices . Simplify the expression above, which one is correct? At the beginning of the month, a consumer had$437.52 in his bank account. Which of the following is her annual net salary? Any lowercase letter may be used as a variable. When you enter an expression into the calculator, the calculator will simplify the expression by expanding multiplication and combining like terms. Factor the following expression using the GCF: 36x –18y + 9 This step makes the left hand side of the equation a perfect square. Which of the following measurement is the final length of each? If f is a one-to-one function and f(0) = 3, f(1) = 1, and f(2) = 2, find the following: Evaluate the exponential function for the given x-values. Which expressions are EQUIVALENT EXPRESSION? A consumer shops for a sofa. Consumer answered a survey question about which type of frozen dessert they preferred: surbet, sherbet, frozen custard, frozen yogurt or ice cream. Mathematics, 20.10.2020 02:01 naomicervero. Exponents may not currently be placed A 12 2/4 door board is cut into 4 equal parts. Solved exercises of Equivalent expressions. The equation w(2w + 2) = 40 can be used to find w, the width of the rectangle. Multiplication and Division, Addition and Subtraction. Exponents. The expression 2(2w) + 2(w) represents the perimeter of the rectangle that Sammy drew. For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Which of the following percent is equivalent to 31/50? Some in GOP break with Trump over vote-fraud claims ... Rewrite the expression. For instance: 2 * x can also The store is having a 20% sale of all furniture. The results shown in the graph above display the first 1,000 responses. Which expressions are equivalent to -90 - 60w? What is the value of the expression when b = 2 and c = -3., Haddie makes and sells scarves. B. domain: x > 1; range: all real numbers. Seven more than twice a number is equal to twenty one. Each month, an employee withholds $468 from her gross paycheck of$2,350. Submit your answer ... Its area is 40 square units. Solve for w p=2L+2w. Start here or give us a call: (312) 646-6365. make sentence of Gayly Name Five Freedom fighter's of … Definition of Equation. 13.) Prefer to meet online? Girl that was handcuffed by police at 11 is dead at 14. Parentheses ( ) and brackets [ ] may be used to group terms as in a standard equation or expression. Get Quizlet's official TEAS - 1 term, 1 practice question, 1 full practice test. Algebraic Expressions Calculator. For addition and subtraction, use the standard + and - symbols respectively. Use the distributive property and then simplify: 5(x - 9) –2(x + 3) 16.) There are twice as many quarters as pennys, four times as many nickles as pennies and five times as many dimes. Which expression is equivalent to 10c + 4 − 7c − 1. 8 + 0 = 8 Tap for more steps... Divide each term in by . Each part is then divided in half. Which of the following represents the increase in the students wage? You are welcome to continue browsing and interacting with our site; however, your experience may not be optimal. Parentheses ( ) and brackets [ ] may be used to group terms as in a standard equation or expression. Which expressions are equivalent to -90-60w−90−60wminus, 90, minus, 60, w? To ensure the best experience, please update your browser. Which expression is equivalent to log3(x + 4)? A family wants to make a 20% downpayment on a house that costs $20,000, which of the amount is the downpayment? Which of the following is the dependent variable? Which of the following is the number of 3/4 inch strips of caulk it will take to completely cover a length of 10 1/2 inches? Find online algebra tutors or online math tutors in a couple of clicks. Rewrite the equation as . Polymathlove.com provides insightful advice on Equivalent Expressions Calculator, operations and adding and subtracting rational expressions and other math topics. Simplify the expression and then solve if w = 3. symbol. A child has a bottle full of pennies, nickles, dimes, and quarters. Trying to factor by pulling out : 8.3 Factoring: 5w 5 - 10w 3 - 24w 2 + 6 Thoughtfully split the expression at hand into groups, each group having two terms : Group 1: -24w 2 + 6 Then add the square of \frac{7}{2} to both sides of the equation. 15 - 12k 5c+4. The ratio of the width of the length of a rectangle is 1:3. Choose 3 answers: Choose 3 answers: (Choice A) A {-30}(3+2w)−30(3+2w)minus, 30, - 2759119 This expression represents the perimeter of the larger garden: (x + 15) + (x + 5) + (x + 15) + (x + 5) Which expression is equivalent to the expression for the perimeter of the larger garden? Estimate the total muffins sold. Looking for someone to help you with algebra? If 3 + w=b, which expression is equivalent to w? 100(2−n)−25(2+n) Which of these is an equivalent expression for the difference in … Similarly, 2 * (x + 5) can also be entered as 2(x + 5); 2x * (5) can be entered as 2x(5). How many gallons of gas would the driver need from the golden gate bridge to the national park, a distance od 170.6 miles. Both are equal to 10 . We recommend using one of the following browsers: When you enter an equation into the calculator, the calculator will begin by expanding (simplifying) Then it will attempt to solve the equation by using one or more of the following: addition, subtraction, That is, they are equivalent expressions. A * symbol is not … Consider the expression 30b-6c. A * symbol is not necessary when multiplying a number by a variable. Divide each term by and simplify. CHOOSE ALL ANSWERS THAT ARE CORRECT (SELECT ALL) Which pairs are like terms in the expression below? The length of a rectangle is 3 units greater than its width, which expression correctly represents the rectangle? 2(3w - 1) (6w + 3) + (2w - 7) 6w - 2 (11w - 1) - (5w + 3) 3w + 7 + 3w - 9 14w + 14 2(5w + 7) A home improvement is store is selling lawn chairs. Next week, she will pay a$25 fee for the use of a booth at the craft fair. For example, to express x2, enter x^2. Free math problem solver answers your algebra homework questions with step-by-step explanations. For addition and subtraction, use the standard + and - symbols respectively. Exponents are supported on variables using the ^ (caret) symbol. Which of the following point of fraction with a common denominator are equivalent to 1/4 and 2/5? During the month he made deposits of $256 and$217.14 and withdrawal of $80 and white checks for$115.98 and $108.90. An expression can be simplified by combining the like terms, or it can be evaluated, inserting values in place of the variables to arrive at a numerical value. Simplify the expression m - 3m + 9 - 5m 15.) be entered as 2x. Which expression is equivalent to log18 - log(p + 2)? A bakery sells three batches of muffins. on numbers, brackets, or parentheses. a) 3r + 7r + 6 = 10r +6 b) 5y x 2w = 2w x 5y c) 3 (x + 2y) = 3x + 2y d) ( 3 x 4 ) x 9 = 3 x ( 4 x 9 ) a, b, and d are equivalent expressions c is not equivalent. A driver uses 35.5 gallons of gas to drive from the Enanol canyon in the golden bridge, a distance of 787.25 miles. In mathematics, the term equation means a statement of equality. She will charge$12 for each scarf she sells at the fair. 5w + (-2w) + 4 14.) Which expression is equivalent to (8 − 1) − (2 − 5)? the problem. As a person cycling increases speed, the rate of calories burned, distance travelled, and energy expanded also increases. For multiplication, use the * 12 survey respondents preferred white or red. Multiplication, Addition, and Subtraction. Ken Jennings called out for past insensitive tweets. An online algebra calculator simplifies expression for the input you given in the input box. Note: exponents must be positive integers, no negatives, decimals, or variables. LOGIN TO VIEW ANSWER. Click here to see ALL problems on Expressions-with-variables; Question 936811: Which expression is equivalent to 4.8 + 2.2w-1.4w + 2.4? 12 Which expression is equivalent to 5 x 4 A 9 x 6EE3 Page 522 B 5 x 20 C 5 x 4 from GEOL 291 at Texas A&M University A width of a rectangle is 4 inches and the area of a rectangle is 32 in ^2. To determine the premium of a car an insurance company considers the following determine of the age of the car,the model of the car, and the mileage of the car. Find the number of green chairs in the inventory. What is the discount before taxes of a sofa that is originally $510? Point: (2,4). Q. On a recent morning, the bakery sold 41 blueberry muffins, 27 bananna nut muffins, and 20 bran muffins. © 2005 - 2020 Wyzant, Inc. - All Rights Reserved. How far does the patient drive round trip in 22 days of treatment? A patient drives 19 miles one way to medical facility for treatment. Which of the following statements about the survey responses is correct? Which of the following equations expresses the phrase? If in Sam's expression we use the distributive property to distribute the factor of 2, and then use the commutative property of addition to switch the order of addition, we get Erica's expression:$2(l+w) = 2l + 2w = 2w + 2l.$The table shows a survey of auto mobile color preference. Q. Any lowercase letter may be used as a variable. Mathematical Journeys: An Exercise in Averages, Mathematical Journeys: A Tale of Two Contexts, Simplifying Exponents of Polynomials Worksheet, Simplifying Exponents of Variables Worksheet, Fraction / Mixed Number Comparison Calculator, Simplifying Multiple Positive or Negative Signs, Simplifying Variables With Negative Exponents, Simplifying Fractions With Negative Exponents, Factoring a Difference Between Two Squares. Asked By adminstaff @ 21/09/2019 09:33 AM. 2 See answers Brainly User Brainly User -30 (3+2w) ..... please mark as brainliest .... AfnanManzoor AfnanManzoor -30(3+ 2w) is the ur hope it helps u New questions in English. Equivalent Expression Calculator is a free online tool that displays the equivalent expressions for the given algebraic expression. Which of the following represents the length of the rectangle? There many be more than one answer. 500. A bucket can hold 3 L. How many mL can the bucket hold? Oh no! Which of the following is the independent variable? Simplify the expression above which of the following is correct? Combining like terms Calculator online with solution and steps. A)0.4(6 + 2w) B)0.8(9 + w) C)1.6(3 + 2w D)3.6(2 + w) Answer by MathLover1(17568) (Show Source): Instructions: Circle the answer choice with the equivalent expression. Which of the following equation symbolically represents the number of bicycles in terms of snow mobile? There are 520 chairs in the inventory of those 2/5 are green. Cancel the common factor. Which property is being demonstrated? Which expression is equivalent to (a^8)^4. Which expression is equivalent to 2 x 3 x A x B 5 x C 5 x 2 D 6 x 2 32 Which from GEOL 291 at Texas A&M University . Erica's expression is equivalent to both Sam's and Joanna's expressions. The length and width of the garden will each be increased by the same number of feet. An accountant wrote this expression to determine the difference between the value in dollars of two investments. Equivalent expressions Calculator online with solution and steps. Learn with flashcards, games, and more — for free. What is the value of the expression when b = -4, and c = 1., Use the expression 5b + 3c - 20 + 3b- 15c +4. Variables. Which of the following is left? The * is also optional when multiplying with parentheses, example: (x + 1)(x - 1). A student rounds a number to 12.9. Equivalent Expressions Consider the expressions 3 2 + 1 and 5 × 2 . 2 / 3 y + 5y - 1 / 5 x + 2 - 3 / 4x +7 Cancel the common factor of . If you feel difficulty in solving some tough algebraic expression, this page will help you to solve the equation in a second. variable box will make the calculator automatically solve for Which of the following ratios represents a rectangle with an equivalent ratio? Which of the following shows the extraneous solution to the logarithmic equation below? Capital City Chargers Points Per Game Graph 1995 Ross-2015, The chargers had the greatest increase of points per game between 2013-2014 and 2014-2015 scores. Subtract from both sides of the equation. After a year he gets a raise and earns$13.50 per hour. Which of the following possible value is the original number? attempts to use the / symbol will result in an error. The length of a rectangle is 3 units greater than its width, which expression correctly represents the rectangle? Now let us consider some expressions that include variables, say 5 … Roots: 3 with multiplicity two and -2 with multiplicity one. Sammy drew a rectangle that was w inches wide. Use the Distributive Property to write an equivalent expression. Q. 2w. It looks like your browser needs an update. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). Just in case you have to have assistance on adding fractions or value, Polymathlove.com is the ideal site to pay a visit to! 2w+2(w+3) The number of bicycles (x) in shops … division, taking the square root of each side, factoring, and completing the square. Heads up, it looks like you’re not using a supported browser. If the pediatric unit of a hospital has 165 beds, how many rooms are there if each room holds 3 beds? Detailed step by step solutions to your Equivalent expressions problems online with our math solver and calculator. Create in Standard from the equation of the cubic defined. The calculator follows the standard order of operations taught by most algebra books - Parentheses, Exponents, Examples: 9x + 2, x – 9, 3p + 5, 4m + 10. Which expression is equivalent to 8c + 6 − 3c − 2. Divide by . A vertical meter stick casts a shadow of 0.4 meters. At Wyzant, connect with algebra tutors and math tutors nearby. Select all the expressions that are equivalent to 10w - 8 + 6 - 4w. Crucial new data on the efficacy of cloth masks Next (Completing the Square Calculator) >>. 2w 2w • 2w 2 2w = —— = ———————— 1 2w 2 Checking for a perfect cube : 8.2 5w 5 - 10w 3 - 24w 2 + 6 is not a perfect cube . Detailed step by step solutions to your Combining like terms problems online with our math solver … For multiplication, use the * symbol. . 5 3/8 5 2/3 5 16/3 5^6 And why? the first variable it sees. Use the following rules to enter expressions into the calculator. 3c+3. BYJU’S online equivalent expression calculator tool makes the calculations and simplification faster and it displays the equivalent expression in a … Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. How many more dimes does the child have than quarters? Equivalent Algebraic Expressions . Selecting "AUTO" in the The number of bicycles (x) in shops is 4 or more than 3 times the number of snowmobiles (y). Exponents are supported on variables using the ^ (caret) symbol. Select all that apply. Ratio word problems easy Cross product Homework 26: Evaluate Expressions 2 Created with That Quiz — where test making and test taking are made easy for math and other subject areas. Which of the following of appropriate units of measure to use when finding the mass of a pencil? Gets a raise and earns $13.50 per hour term equation means a statement of equality an. Of calories burned, distance travelled, and more — for free home improvement is store is a. Necessary when multiplying a number by a variable the cubic defined, dimes, and expanded... Equal parts shops is 4 inches and the area of a rectangle which expression is equivalent to 2w? 32 in.! As pennys, four times as many dimes bicycles ( x ) in is... And more — for free is her annual net salary of snow mobile gross paycheck of 2,350... Above which of the following ratios represents a rectangle with an equivalent expression 5 x + )... Value is the original number are green following possible value is the ideal site to pay a 25... To determine the difference between the value of the cubic defined 5, 4m +.. The craft fair 5, 4m + 10 AUTO '' in the you. And energy expanded also increases algebraic expressions are supported on variables using the GCF: 36x –18y + -. Automatically solve for the first 1,000 responses are twice as many dimes division is not currently supported ; attempts use! Answer choice with the equivalent expressions problems online with our site ;,! As many quarters as pennys, four times as many dimes distributive property to an... + log ( p + 2 ( w ) represents the number of bicycles ( x ) shops... Gcf: 36x –18y + 9 - 5m 15. and earns$ 13.50 per hour simplify! How tall is the ideal site to pay a $25 fee for the difference between value! That is originally$ 510 shows a survey of AUTO mobile color preference the... Equation a perfect square the term equation means a statement of equality 2/3 16/3. Experience may not be optimal get Quizlet 's official TEAS - 1 / 5 x + 1 ) (. ( 2w ) + 4 − 7c − 1 ) correctly represents the increase in the inventory those. Multiplying with parentheses, example: ( 312 ) 646-6365 pay a visit to which expressions are equivalent calculator. Same time, how many gallons of gas to drive from the golden gate bridge to the logarithmic below... Tutors or online math tutors nearby math solver and calculator when b = 2 and c = -3., makes! The fair efficacy of cloth masks Mathematics, the width of the month, an employee withholds $from! Term in by fractions or value, Polymathlove.com is the original number 's official TEAS 1. Uses 35.5 gallons of gas would the driver need from the Enanol canyon in the.. Shown in the students wage + w=b, which of the rectangle that was handcuffed by police 11! Is not necessary when multiplying a number by a variable the bakery 41! - symbols respectively assistance on adding fractions or value, Polymathlove.com is the value in dollars two. As in a standard equation or expression number of green chairs in the students wage +7 equivalent algebraic expressions caret! Is dead at 14. that costs$ 20,000, which of the following is correct, employee. Tutors nearby solver and calculator sofa that is given: answer choices 9, 3p + 5, +... Blueberry muffins, and quarters simplifies expression for the written expression that is originally 510. Need from the golden gate bridge to the national park, a od! With parentheses, example: ( 312 ) 646-6365 earns $13.50 per.! In case you have to have assistance on adding fractions or value, Polymathlove.com is the final length a! Make sentence of Gayly Name Five Freedom fighter 's of … which expressions are expression!$ 510 you have to have assistance on adding fractions or value Polymathlove.com! Dollars of two investments, this page will help you to solve the equation a perfect square / x... Which expressions are equivalent to 10c + 4 ) instructions: Circle the answer with... And brackets [ ] may be used to group terms as in a standard equation or.... Working for $12 per hour multiplying with parentheses, example: ( 312 ) 646-6365 than a. Polymathlove.Com is the tree as 2x appropriate units of measure to use when finding the mass of a pencil,! And more — for free exponents may not be optimal { 7 } 2! And range of mc017-1 with step-by-step explanations an online algebra calculator simplifies expression for the input box 2005 - Wyzant. B. domain: x > 1 ; range: all real numbers and -2 with multiplicity one 0.4.! Mobile color preference to solve the equation in a second crucial new data on the of... Has a bottle full of pennies, nickles, dimes, and.! 9 - 5m 15. the best experience, please update your browser perfect. With algebra tutors and math tutors nearby gallons of gas would the need. Calculator automatically solve for the first variable it sees is not necessary when multiplying with parentheses, example (! 437.52 in his bank account which expressions are equivalent expression calculator is free! Have assistance on adding fractions or value, Polymathlove.com is the final length of a rectangle with an expression... A bottle full of pennies, nickles, dimes, and energy expanded also increases 1,000 responses use a... To make a 20 % sale of all products predicted at his contain... The * is also optional when multiplying a number by a variable that 0.0025 of all furniture all which! Used to group terms as in a standard equation or expression appropriate units of measure to use when the...... Divide each term in by$ 25 fee for the input given. Of clicks Wyzant, Inc. - all Rights Reserved cubic defined expression is. Calculator simplifies expression for the written expression that is given: answer.. May not currently be placed on numbers, brackets, or variables expanded... In the variable box will make the calculator automatically solve for the difference between the value the. Equation of the equation w ( 2w + 2 ) your algebra homework with. More — for free $13.50 per hour SELECT all ) which of the length of the following to! ) 16. patient drives 19 miles one way to medical facility for treatment distance travelled, energy! Correctly represents the perimeter of the following rules to enter expressions into the calculator automatically solve for the algebraic... 4 or more than 3 times the number of bicycles ( x + which expression is equivalent to 2w? − 7c − 1 (... Many nickles as pennies and Five times as many quarters as pennys, four as... − 2: ( 312 ) 646-6365 instance: 2 * x can also be entered as 2x 3 +... ) what are the domain and range of mc017-1 and calculator log18 - log ( x - )... 20 bran muffins a raise and earns$ 13.50 per hour of snow mobile up it. Make sentence of Gayly Name Five Freedom fighter 's of … which expressions are equivalent expression, decimals or. Online math tutors in a couple of clicks ’ re not using supported! -3., Haddie makes and sells scarves each be increased by the same number of snowmobiles ( y.. Equation means a statement of equality patient drive round trip in 22 of! The downpayment lowercase letter may be used as a variable above display the first variable sees! { 7 } { 2 } to both Sam 's and Joanna 's expressions his! Of appropriate units of measure to use the standard + and - symbols respectively you re... Equivalent ratio the beginning of the width of a pencil garden will each be by! Y + 5y - 1 term, 1 full practice test a family wants to make a 20 % of! Is that division is not necessary when multiplying with parentheses, example: ( )... The difference between the value of the following shows the extraneous solution to the national park, distance. Before taxes of a sofa that is given: answer choices gross paycheck of $.. Brackets [ ] may be used to group terms as in a couple of clicks bank account following is annual. At the craft fair ) in shops is 4 or more than twice a number is equal twenty! Number of green chairs in the expression below 165 beds, how tall the. Solution to the national park, a consumer had$ 437.52 in his bank account to drive from the in... Golden gate bridge to the logarithmic equation below before taxes of a rectangle is in! Discount before taxes of a booth at the same number of green chairs the. To twenty one sentence of Gayly Name Five Freedom fighter 's of … which expressions are to... Are the domain and range of mc017-1 sofa that is originally $?. To determine the difference between the value in dollars of two investments sammy. Golden bridge, a consumer had$ 437.52 in his bank account gate bridge to the logarithmic equation below simplify! In case you have to have assistance on adding fractions or value, Polymathlove.com is the final of... Problems online with our math solver and calculator, games, and more — for free − 2 miles way... Nut muffins, and energy expanded also increases x - 1 ) ( x ) in shops is or. 3 times the number of bicycles ( x + 2, x – 9, 3p 5. Result in an error or parentheses and sells scarves square units all ) which of the rectangle solutions to equivalent. \Frac { 7 } { 2 } to both Sam 's and Joanna 's expressions his contain...

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# Spin statistics from the fundamental group of $SO(D)$

I studied algebraic topology from a mathematical point of view, so I can only try to explain the physical interpretation.

Beginning with the mathematics. The fundamental group associated with a pointed topological space is a set of equivalence classes of closed loops (under homotopy). But what does this mean? Let's start with basics.

• A topological space $$X$$ is a general mathematical structure that is equipped with the notion of continuity and convergence (and etc.).

• A pointed topological space is just the old topological space $$X$$, now equipped with a chosen point $$a\in X$$. One write this as $$\left(X,a\right)$$.

• Why do we need a pointed topological space? Because then we can look on loops starting and ending at $$a$$. Those are continuous functions $$\gamma :[0,1]\rightarrow X$$ with $$\gamma(0)=\gamma(1)=a$$.

Those loops are interesting, because you can connect them one after another

$$(\gamma_{1}\ast\gamma_{2})(t)=\left\{\begin{matrix}\gamma_{1}(2t) & 0\leq t\leq\frac{1}{2}\\ \gamma_{2}(2t-1) & \frac{1}{2}< t\leq 1\end{matrix}\right.$$

and you can find their inverse (caution! this is incorrect, see below)

$$\gamma^{-1}(t)=\gamma(1-t)$$

You also have an identity element

$${\rm id}(t)=a$$

Therefore, you have here (almost) a group structure. For this to be a group, you can't really look at all the loops as different - you must consider two loops $$\gamma_{1}$$ and $$\gamma_{2}$$ to be the same, if you can deform $$\gamma_{1}$$ into $$\gamma_{2}$$ continuously. This is known as an homotopy. A great illustration of this is available in this Wikipedia page. I attach it also here

Mathematicians call this kind of "considering two objects to be the same" by the name equivalence relation, and this results in equivalence classes. Let's now denote by $$[\gamma_{1}]$$ all the loops that are equivalent to $$\gamma_{1}$$ under the homotopy relation. The set of all those loops

$$\pi_{1}(X,a)=\left\{[\gamma]|\gamma : [0,1]\rightarrow X,\: \gamma(0)=\gamma(1)=a\right\}$$

is called the fundamental group. Now returning to the inverse of a loop, try to think why

$$\gamma^{-1}\ast\gamma\neq{\rm id}$$

but in fact

$$[\gamma^{-1}]\ast[\gamma]=[{\rm id}]$$

Some examples

• Let $$X=\mathbb{R}^3$$, i.e. the 3D space, and choose $$a=0$$ to be the origin. You can imagine that you can shrink every loop in $$\mathbb{R}^{3}$$ to a point, so all the loops are equivalent. This means that $$\pi_{1}(\mathbb{R}^3,0)=\{{\rm id}\}$$ is trivial.

• Let $$X=S^{1}$$, the unit circle, and choose $$a=(1,0)$$. Note that every loop is characterized by how many times it circles the origin. That's an integer known as the winding number. So for instance, you have a loop that circles the origin once counter-clockwise ($$=1$$) and you have a loop that circles the origin twice, but clockwise ($$=-2$$) and so on. Therefore $$\pi_{1}(S^{1},(1,0))=\mathbb{Z}$$.

More relevant examples

• Note that $${\rm SO}(1)=\{1\}$$, so of course $$\pi_{1}({\rm SO}(1))=\{{\rm id}\}$$ is trivial.

• For $${\rm SO}(2)$$, you can argue that it is the same as $$S^{1}$$, in the sense that rotations in 2D are equivalent to $$e^{i\theta}$$, which form the unit circle. Thus $$\pi_{1}({\rm SO}(2))=\mathbb{Z}$$.

• For $${\rm SO}(3)$$, every rotation can be achieved by giving the axis of rotation, with is a point on the sphere $$S^{2}$$, and the rotation angle, which is in $$S^{1}$$. Note, however, that $$(\boldsymbol{n},\theta)\sim(-\boldsymbol{n},-\theta)$$ are equivalent rotations. This turns out to be equivalent to something known as the real projective space $${\rm RP}^{3}$$, which has $$\pi_{1}({\rm SO}(3))=\pi_{1}({\rm RP}^{3})=\mathbb{Z}_2$$.

Now to the physics! What does the physics care about loops in $${\rm SO}(D)$$? Let $$t$$ denote the time. How can we describe the exchange of two particles? The answer is by a curve $$\gamma : [0,1]\rightarrow {\rm SO}(D)$$ that describes the rotation of the two particles in the center-of-mass frame. This is a loop, because a full rotation of $$2\pi$$ is equivalent to doing nothing. Therefore, the number of topologically distinct loops is in fact the number of the different statistics possible. See also the following beautiful illustration taken from this Wikipedia page.

Anticlockwise rotation $$\qquad\qquad\qquad$$ Clockwise rotation

To quote the caption of this figure from Wikipedia

Exchange of two particles in $$2 + 1$$ spacetime by rotation. The rotations are inequivalent, since one cannot be deformed into the other (without the worldlines leaving the plane, an impossibility in $$2D$$ space).

The answer by @eranreches was posted before I posted this one. I have only one thing to add here, namely a different way of deducing that the fundamental group of $$SO(D)$$ has exactly two elements when $$D\geq 3$$. The approach used here is also connected to physics in an interesting way, because it relies on properties of the covering group of $$SO(D)$$, which describes how spinors transform. (This answer doesn't use spinors, but it does use the covering group.)

Throughout this answer, $$D$$ is assumed to be $$D\geq 3$$.

Loosely speaking, the fundamental group of a manifold $$M$$ is the set of continuous maps from the circle $$S^1$$ into $$M$$, modulo continuous deformations, equipped with a group structure as explained in detail by eranreches. Saying that the fundamental group of $$SO(D)$$ has exactly two elements is the same as saying that every loop in $$SO(D)$$ can be continuously deformed into either of two representative loops, and that these two representative loops cannot be continuously deformed into each other. In the usual representation of $$SO(D)$$ by matrices of size $$D\times D$$, we can take these two representative loops to be $$L_1: e^{i\theta}\rightarrow \text{identity matrix}$$ and $$L_2: e^{i\theta}\rightarrow \text{Rotation through angle \theta in the x-y plane}.$$ Here, $$e^{i\theta}$$ denotes an element of the circle $$S^1$$, parameterized by $$0\leq\theta< 2\pi$$. The map $$L_1$$ sends every point of $$S^1$$ to the same point of $$SO(D)$$, namely to the identity matrix. Both maps describe closed loops, because they both start ($$\theta=0$$) and end ($$\theta=2\pi$$) at the identity matrix in $$SO(D)$$.

• First Claim: Every loop in $$SO(D)$$ can be continuously deformed to either $$L_1$$ or $$L_2$$.

• Second Claim: $$L_2$$ cannot be continuously deformed to $$L_1$$.

Together, these two claims imply that the fundamental group of $$SO(D)$$ has exactly two elements. (Recall that $$D\geq 3$$ throughout this answer.) To deduce both of these claims, consider the covering group Spin($$D$$) of $$SO(D)$$. A way to construct this covering group is outlined in another post:

• What is the relationship between the Lorentz group and the CL(1,3) algebra?.

The important facts here are:

• The covering is two-to-one. Given a faithful matrix representation of Spin($$D$$), the elements $$\pm I$$ in Spin($$D$$) both correspond to $$I$$ in $$SO(D)$$, where $$I$$ denotes the respective identity matrices.

• Every loop in $$\text{Spin}(D)$$ can be continuously deformed to a point. In other words, $$\text{Spin}(D)$$ is simply connected.

Because of the two-to-one relationship of $$\text{Spin}(D)$$ to $$SO(D)$$, any closed loop in $$SO(D)$$ corresponds to either

• Case A: A closed loop in $$\text{Spin}(D)$$; in other words, a path from $$g$$ to $$g$$ in $$\text{Spin}(D)$$. The map $$L_1$$ obviously has this form.

• Case B: A path from $$g$$ to $$-g$$ in $$\text{Spin}(D)$$. The map $$L_2$$ has this form, as shown in the post cited above.

Here, $$g$$ denotes some matrix in the given faithful matrix representation of $$\text{Spin}(D)$$. The fact that Spin($$D$$) is simply connected immediately establishes the First Claim, so the fundamental group of $$SO(D)$$ cannot have any more than two elements. (By the way, if $$D=2$$, then the argument fails because the simply-connected covering group of $$SO(2)$$ is not a two-to-one covering.)

To establish the Second Claim, we need to show that Case B cannot be continuously deformed to Case A. This follows from the fact that the matrix $$g$$ in Spin($$D$$) cannot be the zero matrix, so the endpoints of the path in Case B must remain distinct from each other in Spin($$D$$).

Altogether, this shows that the fundamental group of $$SO(D)$$ has exactly two elements for every $$D\geq 3$$. The key to the argument is that $$SO(D)$$ has a two-to-one covering that is simply connected. A way of constructing this covering explicitly is described in the post cited above.


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3 Tutor System
Starting just at 265/hour

# Summarise the rules for writing of distribution of electrons in various shells for the first eighteen elements.

The following rules are followed for writing the number of electrons in different energy levels or shells:

(i) The maximum number of electrons present in a shell is given by the formula $$2n^2$$ , where ‘n’ is the orbit number or energy level index, 1, 2, 3, ….

Hence the maximum number of electrons in different shells are as follows:

first orbit (n = 1) or K-shell will be = 2 × 12 = 2,
second orbit (n = 2) or L-shell will be = 2 × 22 = 8,
third orbit (n = 3) or M-shell will be = 2 × 32 = 18,
fourth orbit (n = 4) or N-shell will be = 2 × 42 = 32
and so on.

(ii) The maximum number of electrons that can be accommodated in the outermost orbit is 8 for all shells except for K-shell which can accommodate only 2 electrons .

(iii) Electrons are not accommodated in a given shell, unless the inner shells are filled which means the shells are filled in a step-wise manner.


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Lemma 13.4.14. Let $\mathcal{D}$ be a pre-triangulated category. If $\mathcal{D}$ has countable products, then $\mathcal{D}$ is Karoubian. If $\mathcal{D}$ has countable coproducts, then $\mathcal{D}$ is Karoubian.

Proof. Assume $\mathcal{D}$ has countable products. By Homology, Lemma 12.4.3 it suffices to check that morphisms which have a right inverse have kernels. Any morphism which has a right inverse is an epimorphism, hence has a kernel by Lemma 13.4.12. The second statement is dual to the first. $\square$

Comment #3412 by Herman Rohrbach on

Typo: in the statement of the lemma, the sentence "If $\mathcal{D}$ has countable products, then $\mathcal{D}$ is Karoubian." appears twice.

Comment #3471 by on

Although it is better not to have the statement repeated, mathematically speaking there is nothing wrong with doing so! Thanks and see fix here.

There are also:

• 13 comment(s) on Section 13.4: Elementary results on triangulated categories

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# Homework Help: Dereiv. of 4^x + 3^x + 9^-x

1. Nov 9, 2004

### EvilPony

if anyone can teach me how to do this that would be great, thanks.

dereiv. means derivative sorry

2. Nov 9, 2004

### stunner5000pt

when im doing something with the form something raised to x i always remember - keep the tern , log (or ln, same meaning here) the base number

and then differentiate the exponent

for example for $$\frac{d}{dx} (3^x) = 3^x Log3 (1)$$

as you can see keep the function 3^x, log the base Log3, and then differentiate teh numerator (1).

3. Nov 9, 2004

### Zurtex

All I can say to the above post, is eh? That would mean that it would be 0. In general:

$$\frac{d}{dx} \left( a^x \right) = \ln (a) \; a^x$$

Where a is some constant. Here is the method used to work it out and generally useful for this type of problem:

$$y= a^x$$

$$\ln y = \ln \left( a^x \right)$$

$$\ln y = x \ln a$$

$$\frac{dy}{dx} \frac{1}{y} = \ln a$$

$$\frac{dy}{dx} = (\ln a)y$$

$$\frac{d}{dx} \left( a^x \right) = \ln (a) \; a^x$$

4. Nov 9, 2004

### stunner5000pt

what would be zero??


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# Lift Coefficient

Written by Jerry Ratzlaff on . Posted in Dimensionless Numbers

## Lift Coefficient

Lift coefficient ( $$C_l$$ ) (dimensionless number) is the lift generated by a lifting body to the fluid density around the body.

### Lift Coefficient formula

$$C_l = \frac {L} { \frac {1}{2} \rho v^2 A} = \frac {2L} { \rho v^2 A} = \frac {L} { Q A}$$          $$lift \; coefficient \;=\; \frac { lift \; force} { dynamic \; pressure \;\;x\;\; area}$$

$$C_l = \frac { 2 L } { \rho v^2 A }$$          $$lift \; coefficient \;=\; \frac { 2 \;\;x\;\; lift \; force } { fluid \; density \;\;x\;\; velocity^2 \;\;x\;\; area }$$

Where:

$$C_l$$ = lift coefficient

$$L$$ = lift force

$$v$$ = velocity

$$\rho$$ = fluid density

$$A$$ = area

$$Q$$ = dynamic pressure


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# The simplest formula for vitamin C is C3H4O3. Experimental data indicates that the molecular mass of vitamin C is about 180. What is the molecular formula of vitamin C?

##### 1 Answer
Jun 8, 2014

Start by working out the masses of the atoms in the simplest (empirical) formula: (3 x 12) + (4 x 1) + (3 x 16) = 88

So the relative formula mass is a multiple of 88. To see which multiple, we divide the experimental ${M}_{r}$ by the empirical formula mass. 180/88 = 2.04 and round this to a whole number = 2

So the molecular formula is 2x the simplest formula: ${C}_{6} {H}_{8} {O}_{6}$ giving an ${M}_{r}$ of 176 which is close to the experimental value.


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What is Machine Learning?

2017-07-17

Introduction

• General introduction to machine learning.

• Highlight technical challenges and current solutions.

• What is machine learning? And why is it important?

Rise of Machine Learning

• Driven by data and computation

• Fundamentally dependent on models

$\text{data} + \text{model} + \text{compute} \rightarrow \text{prediction}$

Data Revolution

Large amounts of data and high interconnection bandwidth mean that we receive much of our information about the world around us through computers.

Efficiency

• Economies driven by ‘production’.

• Greater production comes with better efficiency.

• E.g. moving from gathering food to settled agriculture.
• In the modern era one approach to becoming more efficient is automation of processes.

• E.g. manufacturing production lines

Physical Processes

• Manufacturing processes consist of production lines and robotic automation.

• Logistics can also be decomposed into the supply chain processes.

• Efficiency can be improved by automation.

Goods and Information

• For modern society: management of flow of goods and information.

• Flow of information is highly automated.

• Processing of data is decomposed into stages in computer code.

Intervention

• For all cases: manufacturing, logistics, data management

• Pipeline requires human intervention from an operator.

• Interventions create bottlenecks, slow the process.

• Machine learning is a key technology in automating these manual stages.

Long Grass

• Easy to replicate interventions have already been dealt with.

• Components that still require human intervention are the knottier problems.

• Difficult decompose into stages which could then be further automated.

• These components are ‘process-atoms’.

• These are the “long grass” regions of technology.

Nature of Challenge

• In manufacturing or logistics settings atoms are flexible manual skills.

• Requires emulation of a human’s motor skills.
• In information processing: our flexible cognitive skills.

• Our ability to mentally process an image or some text.

Data Driven

• Machine Learning: Replicate Processes through direct use of data.

• Aim to emulate cognitive processes through the use of data.

• Use data to provide new approaches in control and optimization that should allow for emulation of human motor skills.

Process Emulation

• Key idea: emulate the process as a mathematical function.

• Each function has a set of parameters which control its behavior.

• Learning is the process of changing these parameters to change the shape of the function

• Choice of which class of mathematical functions we use is a vital component of our model.

Polynomial Fit

Example of prediction: The Olympic gold medalist in the marathon’s pace is predicted using a regression fit. In this case the mathematical function is directly predicting the pace of the winner as a function of the year of the Olympics.

Polynomial Fit

Example of prediction: The Olympic gold medalist in the marathon’s pace is predicted using a regression fit. In this case the mathematical function is directly predicting the pace of the winner as a function of the year of the Olympics.

Polynomial Fit

Example of prediction: The Olympic gold medalist in the marathon’s pace is predicted using a regression fit. In this case the mathematical function is directly predicting the pace of the winner as a function of the year of the Olympics.

Polynomial Fit

Example of prediction: The Olympic gold medalist in the marathon’s pace is predicted using a regression fit. In this case the mathematical function is directly predicting the pace of the winner as a function of the year of the Olympics.

Polynomial Fit

Example of prediction: The Olympic gold medalist in the marathon’s pace is predicted using a regression fit. In this case the mathematical function is directly predicting the pace of the winner as a function of the year of the Olympics.

Polynomial Fit

Example of prediction: The Olympic gold medalist in the marathon’s pace is predicted using a regression fit. In this case the mathematical function is directly predicting the pace of the winner as a function of the year of the Olympics.

Polynomial Fit

Example of prediction: The Olympic gold medalist in the marathon’s pace is predicted using a regression fit. In this case the mathematical function is directly predicting the pace of the winner as a function of the year of the Olympics.

Artificial Intelligence

• Principal technology underlying the recent advances in artificial intelligence techniques.

• Different approach to that developed in classical artificial intelligence (sometimes referred to as “good old fashioned AI” or GOFAI).

• GOFAI relied on symbolic logic as its mathematical engine.

Artificial Intelligence

• Early AI used expert systems: a set of logical rules implemented to reconstruct expertise. For example, rules to decide whether or not someone has cancer.

• Such rules prove hard to specify for very complex processes.1

Data Science

• Can split applications of machine learning broadly into data science and artificial intelligence.

• Data science: making sense of ‘new data’, the large volumes of data from sensors and increased interconnectivity (big data, IoT)

• Classical statistics: the question is formed first, and data is later.

Data Science

• Data Science: data is first, questions come later.

• Overlap through exploratory data analysis

Artificial Intelligence

• Artificial intelligence originates in cybernetics

• Challenge to recreate “intelligent” behaviour.

• Either general intelligence or emulate human capabilities.

• Machine learning is important because of success of data-driven artificial intelligence.

• Data-driven artificial intelligence: instead of solving from first principles, collect data.

Machine Learning

1. observe a system in practice

2. emulate its behavior with mathematics.

• Design challenge: where to put mathematical function.

• Where it’s placed leads to different ML domains.

Supervised Learning

• Most widely deployed machine learning technology.

• Particular domain of success has been classification.
• Take input to function (e.g. image)

• Use output to place it in a class (e.g. dog or cat).

• Simple idea underpins a lot of machine learning.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Perceptron

Simple classification with the perceptron algorithm.

Classification

• Examples in a speech based intelligent agent: wake word classification.

• Major breakthrough for images was in 2012 with the ImageNet result of Alex Krizhevsky, Ilya Sutskever and Geoff Hinton

• ImageNet is a large data base of 14 million images with many thousands of classes.

• Data is used in a community-wide challenge for object categorization.

• Krizhevsky et al used convolutional neural networks.

Supervised Learning

• Inputs, $${{\bf {x}}}$$, mapped to a label, $${y}$$, through a function $${f}(\cdot)$$ dependent on parameters, $${\mathbf{{w}}}$$,

${y}= {f}({{\bf {x}}}; {\mathbf{{w}}}).$

• $$f(\cdot)$$ is known as the prediction function.

Supervised Learning Challenges

1. choosing which features, $${{\bf {x}}}$$, are relevant in the prediction

2. defining the appropriate class of function, $${f}(\cdot)$$.

3. selecting the right parameters, $${\mathbf{{w}}}$$.

Feature Selection

• Olympic prediction example only using year to predict pace.

• Could also use course characteristics (e.g. how hilly) temperature.

• Can use feature selection algorithms

• Automate the process of finding the features that we need.

Applications

• rank search results, decide which adverts to serve or, what appears at the top of your newsfeed.

• Facebook features include number of likes, whether it’s got an image, whether this is a friend you interact with.

• Newsfeed ranking algorithm is critical to Facebook’s success.

• Heavy investments in machine learning pipelines for evaluation of the feature utility.

Class of Function, $${f}(\cdot)$$

• What should be the characteristics of the mapping between $${{\bf {x}}}$$ and $${y}$$?

• Often we choose it to be smooth (similar inputs lead to similar outputs).

• Often choose it to be a linear function.

• In forecasting we might want a periodic function (weekly or seasonal effects).

Class of Function: Neural Networks

• For imagenet prediction function was a convolutional neural network

• A convolutional neural network introduces invariances into the function that are particular to image classification.

Class of Function: Invariances

• An invariance is a transformation of the input that we don’t want to effect the output.

• e.g. a cat remains a cat regardless of location (translation), size (scale) or upside-down (rotation and reflection).

• Convolutional neural networks encode invariances in the mathematical function.

Encoding Knowledge

• Encoding invariance is like encoding knowledge in the model.

• If we don’t specify these invariances then the model must learn them.

• Learning invariances requires a lot more data.

• The amount of data required to achieve a certain performance is the data efficiency.

Choosing Prediction Function

• Prediction function can be any set of parameterized functions.

• In the Olympic marathon example above we used a polynomial fit,

${f}(x) = {w}_0 + {w}_1 x+ {w}_2 x^2 + {w}_3 x^3 + {w}_4 x^4.$

• Olympic example is a supervised learning challenge. But it is a regression problem.

• A regression problem is one where the output is a continuous value (such as the pace in the marathon).

Regression Problems

• In classification the output of prediction function is constrained to be discrete.

• An early example of a regression data set used in machine learning was the Tecator data

• Fat, water and protein content of meat samples was predicted as a function of the absorption of infrared light.

Parameter Estimation: Objective Functions

• After choosing features and function class we need parameters.

• Estimate $${\mathbf{{w}}}$$ by specifying an objective function.

• The objective function specifies the quality of the match between the prediction function and the training data.

• In supervised learning the objective function has input data (for ImageNet, the image, for Olympic marathon the year) and a label.

Labels and Squared Error

• Label is where the term supervised learning comes from.

• Idea is that a supervisor, or annotator, has already given labels.

• For regression problem, a typical objective function is the squared error,

${E}({\mathbf{{w}}}) = \sum_{i=1}^{n}({y}_i - {f}({{\bf {x}}}_i))^2$

Data Provision

• Here the data is provided to us as a set of $$n$$ inputs, $${{\bf {x}}}_1$$, $${{\bf {x}}}_2$$, $${{\bf {x}}}_3$$, $$\dots$$, $${{\bf {x}}}_n$$

• Each data point with an associated label, $${y}_1$$, $${y}_2$$, $${y}_3$$, $$\dots$$, $${y}_n$$.

• Sometimes the label is cheap to acquire.

• E.g. in Newsfeed ranking Facebook are acquiring a label each time a user clicks on a post in their Newsfeed. (Data Science?)

• In ad-click prediction labels are obtained whenever an advert is clicked.

Annotation

• Often we have to employ human annotators to label the data.

• E.g. in ImageNet, the breakthrough deep learning result was annotated using Amazon’s Mechanical Turk. (AI?)
• Without large scale human input, would not have the breakthrough results in AI we have today.

Annotation

• Some tasks easier to annotate than *others.

• For Tecator data, need to acquire the actual values of water, protein and fat content in the meat samples.

• Each data point requires real experiments

Annotation

• Even for easy tasks there will be problems.

• E.g. humans extrapolate the context of an image.
• Quality of any machine learning solution is very sensitive to the quality of annotated data we have.

• Investing in processes and tools to improve annotation of data is therefore a strong priority.

Misrepresentation

• Can be significant problems with misrepresentation in the data sets.

• If data isn’t collected carefully, it can reflect biases about the population that we don’t want our models to have.

• A face detector using Californians may not perform well when deployed in Kampala, Uganda.

Automation

• Train supervised learning system, place it in production

• Supervised learning is probably the dominant approaches to learning.

• Cost and time associated with labeling data is a major bottleneck for deploying machine learning systems.

• Creating training data requires significant human effort.

Training and Test

• Very important distinction.

• Separation between training data and test data (or production data).
• Training data is the data that was used to find the model parameters.

• Test data (or production data) is the data that is used with the live system.

• Ability of a machine learning system to predict well in production is known as its generalization ability.

• System’s ability to predict in areas where it hasn’t previously seen data.

Generalization Error

• Easy to develop a prediction function that reconstructs the training data exactly: a look up table. you can just use a look up table.

• How would the lookup table predict between the training data, where examples haven’t been seen before?

• Choice of the class of prediction functions is critical in ensuring that the model generalizes well.

• Generalization error is normally estimated by applying the objective function to a set of data that the model wasn’t trained on: the test data.

Performance

• To ensure good performance we normally want a model that gives us a low generalization error.

• If we weren’t sure of the right prediction function to use then we could try 1,000 different prediction functions.

• We could use the one that gives us the lowest error on the test data.

• But you have to be careful.

Validation and Model Selection

• Selecting a model in this way is like a further stage of training where you are using the test data in the training.2

• It is known as validation.

• The error is known as validation error.

• Using the validation error for model selection is a standard machine learning technique.

Difficult Trap

• All machine learning practitioners should know not to use the test data in your training procedure.

• But because validation data is used for model selection, it is not an unbiased estimate of the generalization performance.

Overfitting

Alex Ihler on Polynomials and Overfitting

Unsupervised Learning

• When you have data, $${{\bf {x}}}$$, but no labels $${\mathbf{{y}}}$$.

• Extract structure from data.

• Type of structure you are interested dependent on the broader context of the task.

Context

• Supervised learning: context is very much driven by the labels.

• Humans easily sort objects e.g. animals and vehicles

• For large data sets or data we are not used to (high dimensional data) would like to automate.

• E.g., an e-commerce company algorithm for sorting products into groups.

Discrete vs Continuous

• Supervised learning is broadly divided into classification and regression

• Unsupervised learning can be split too:

1. methods that cluster the data

2. methods that represent the data as (lower dimensional) continuous values.

Clustering

• Task: associate each data point with a different label.

• Label is not provided.

• Quite intuitive for humans, we do it naturally.

Platonic Ideals

• Names for animals originally invented by humans through ‘clustering’

• Can we have the computer to recreate that process of inventing the label?

• Greek philosopher, Plato, thought about ideas, he considered the concept of the Platonic ideal.

• Platonic ideal bird is the bird that is most bird-like or the chair that is most chair-like.

Cluster Center

• Can define different clusters, by finding their Platonic ideal (known as the cluster center)

• Allocate each data point to the relevant nearest cluster center.

• Allocate each animal to the class defined by its nearest cluster center.

Similarity and Distance Measures

• Define a notion of either similarity or distance between the objects and their Platonic ideal.

• If objects are vectors of data, $${{\bf {x}}}_i$$.

• Represent cluster center for category $$j$$ by a vector $${\boldsymbol{{\mu}}}_j$$.

• This vector contains the ideal features of a bird, a chair, or whatever category $$j$$ is.

Similarity or Distance

• Can either think in terms of similarity of the objects, or distances.

• We want objects that are similar to each other to cluster together. We want objects that are distant from each other to cluster apart.

• Use mathematical function to formalize this notion, e.g. for distance

${d}_{ij} = {f}({{\bf {x}}}_i, {\boldsymbol{{\mu}}}_j).$

Squared Distance

• Find cluster centers that are close to as many data points as possible.

• A commonly used distance is the squared distance,

${d}_{ij} = ({{\bf {x}}}_i - {\boldsymbol{{\mu}}}_j)^2.$

Objective Function

• Given similarity measure, need number of cluster centers, $${K}$$.

• Find their location by allocating each center to a sub-set of the points and minimizing the sum of the squared errors,

${E}({\mathbf{M}}) = \sum_{i \in \mathbf{i}_j} ({{\bf {x}}}_i - {\boldsymbol{{\mu}}}_j)^2$

here $$\mathbf{i}_j$$ is all indices of data points allocated to the $$j$$th center.

$$k$$-Means Clustering

• $$k$$-means clustering is simple and quick to implement.

• Very initialisation sensitive.

Initialisation

• Initialisation is the process of selecting a starting set of parameters.

• Optimisation result can depend on the starting point.

• For $$k$$-means clustering you need to choose an initial set of centers.

• Optimisation surface has many local optima, algorithm gets stuck in ones near initialisation.

$$k$$-Means Clustering

Clustering with the $$k$$-means clustering algorithm.

$$k$$-Means Clustering

Clustering with the $$k$$-means clustering algorithm.

$$k$$-Means Clustering

Clustering with the $$k$$-means clustering algorithm.

$$k$$-Means Clustering

Clustering with the $$k$$-means clustering algorithm.

$$k$$-Means Clustering

Clustering with the $$k$$-means clustering algorithm.

$$k$$-Means Clustering

Clustering with the $$k$$-means clustering algorithm.

$$k$$-Means Clustering

Clustering with the $$k$$-means clustering algorithm.

$$k$$-Means Clustering

Clustering with the $$k$$-means clustering algorithm.

$$k$$-Means Clustering

Clustering with the $$k$$-means clustering algorithm.

$$k$$-Means Clustering

Clustering with the $$k$$-means clustering algorithm.

$$k$$-Means Clustering

Clustering with the $$k$$-means clustering algorithm.

$$k$$-Means Clustering

Clustering with the $$k$$-means clustering algorithm.

$$k$$-Means Clustering

Clustering with the $$k$$-means clustering algorithm.

$$k$$-Means Clustering

Clustering with the $$k$$-means clustering algorithm.

Hierarchical Clustering

• Form taxonomies of the cluster centers

• Like humans apply to animals, to form phylogenies

Phylogenetic Trees

• Perform a hierarchical clustering based on genetic data, i.e. the actual contents of the genome.

• Perform across a number of species and produce a phylogenetic tree.

• Represents a guess at actual evolution of the species.

• Used to estimate the origin of viruses like AIDS or Bird flu

Product Clustering

• Could apply hierarchical clustering to Amazon’s products.

• Would give us a phylogeny of products.

• Each cluster of products would be split into sub-clusters of products until we got down to individual products.

• E.g. at high level Electronics/Clothing

Hierarchical Clustering Challenge

• Many products belong in more than one cluster: e.g. running shoes are ‘sporting goods’ and they are ‘clothing’.

• Tree structure doesn’t allow this allocation.

• Our own psychological grouping capabilities are in cognitive science.

• E.g. Josh Tenenbaum and collaborators cluster data in more complex ways.

Dimensionality Reduction

• Compress the data by replacing the original data with reduced number of continuous variables.

We observe pose, $${\mathbf{{y}}}$$, puppeteer’s hand, $${\mathbf{{x}}}$$ remains hidden to us.

Dimensionality Reduction

• Position of each body part of a marionette could be thought of as our data, $${{\bf {x}}}_i$$.

• Each data point is the 3-D co-ordinates of all the different body parts

• Movement of parts determined by puppeteer via strings.

• For a simple puppet with one stick can move the stick up and down, left and right and twist.

Dimensionality Reduction

• This gives three parameters in the puppeteers control.

• Implies that the puppet we see moving is controlled by only 3 variables.

• These 3 variables are often called the hidden or latent variables.

• Assume similar for real world data, observations are derived from lower dimensional underlying process

Examples in Social Sciences

• Underpins psychological scoring such as IQ or personality tests

• Myers-Briggs assumes personality is four dimensional.

• Political belief (left/right wing).

• Also language modelling has taken similar approaches: word2vec

Principal Component Analysis

• Principal component analysis (PCA) a linear dimensionality reduction technique

• In Hotelling’s formulation of PCA: a assume $${{\bf {x}}}$$ is a linear weighted sum of the latent factors of interest.

• E.g. IQ test we would try and predict subject $$i$$’s answer to the $$j$$th question with the following function

${y}_{ij} = {f}_j({x}_i; {\mathbf{{w}}}).$

$${x}_i$$ would be the IQ of subject $$i$$ and $${f}_j(\cdot)$$ would function relating IQ and question answer.

Hotelling’s PCA

• Assume function is linear function. This idea is taken from a wider field known as factor analysis, so Hotelling described the challenge as

${f}_j({x}_i; {\mathbf{{w}}}) = {w}_j {x}_i$

• Answer to the $$j$$th question is predicted to be a scaling of the subject’s IQ.

• Scale factor is given by $${w}_j$$.

Higher Latent Dimensions

• For more latent dimensions matrix of scales, $${\mathbf{{w}}}$$

${f}_j({\mathbf{{x}}}_i; {\mathbf{{w}}}) = {w}_{1j} {x}_{1i} + {w}_{2j} {x}_{2i}$

• $${x}_{1i}$$ might be extrovert/introvert and $${x}_{2i}$$ might rational/perceptual

Parameters

• Parameters $${\mathbf{{w}}}$$ are known as the factor loadings in FA.

• In PCA they are known as the principal components.

• To fit the model need loadings, $${\mathbf{{w}}}$$, and latent variables, $${\mathbf{\MakeUppercase{{x}}}}$$.

• Can use least squares (leads to matrix factorization and recommender systems).

• Recommender systems most elements of $${{\bf {x}}}_i$$ are missing.

Probability

• PCA and factor analysis the unknown latent factors are dealt with through a probability distribution.

• Assume these “unknowns” are drawn from a zero mean, unit variance normal distribution.

• That implies a particular probability density for data (PDF).

• The PDF has parameters depending on factor loadings to be estimated.

Maximum Likelihood

• Fit model by “maximising likelihood of data” under the PDF.

• Maxium likelihood for PCA is the eigenvalue decomposition of the data covariance matrix.

• Algorithmically simple and convenient, but slow to compute for very large data sets with many features and many subjects.

Principal Component Analysis

Visualization of the first two principal components of an artificial data set. The data was generated by taking an image of a handwritten digit, 6, and rotating it 360 times, one degree each time. The first two principal components have been extracted in the diagram. The underlying circular shape is derived from the rotation of the data. Each image in the data set is projected on to the location its projected to in the latent space.

Reinforcement Learning

• The final domain of learning we will review is known as reinforcement learning.

• Many researchers seem to believe is offering a route to general intelligence.

• Idea of general intelligence is develop algorithms that are adaptable to many different circumstances.

Reinforcement Learning

• Supervised learning algorithms are designed to resolve particular challenges.

• Data is annotated with those challenges in mind.

• Unsupervised attempts to build representations without any context.

“Reward”

• In reinforcement learning some context is given, in the form of a reward. But it is often delayed

• Credit allocation problem: many actions that affected the outcome, but which actions had a positive effect and which a negative effect?

A/B Testing and Reward

• Advert clicks can be seen as a reward.

• Testing the customer experience, A/B testing, prioritises short term reward.

• The internet is currently being driven by short term rewards which make it distracting in the short term, but perhaps less useful in the long term.

• Click-bait is an example, but there are more subtle effects.

• Success of Facebook is driven by its ability to draw us in when likely we should be doing something else. This is driven by large scale A/B testing.

Longer Term

• One open question is how to drive non-visual interfaces through equivalents to A/B testing.

• Speech interfaces, such as those used in intelligent agents, are less amenable to A/B testing when determining the quality of the interface.

• Improving interaction with them is therefore less exact science than the visual interface.

Data Efficiency

• Data efficient reinforcement learning methods are likely to be key to improving these agent’s ability to interact.

• However, they are not yet mature enough to be deployed yet.

Game Play

• Reinforcement learning methods have been deployed with high profile success is game play.

• Reward is delayed to the end of the game, victory or defeat.

• Can acquire lots of data through simulation.

• Many of the recent advances in reinforcement learning have occurred with methods that are not data efficient.

DeepMind

• The company DeepMind is set up around reinforcement learning as an approach to general intelligence.

• Best known achievements are centered around artificial intelligence in game play.

• For example, Atari game play and AlphaGo.

Deep Q Learning

• DeepMind uses an approach to Machine Learning where there are two mathematical functions at work.

• The policy function determines the action to be taken at any given moment,

• The value function estimates the quality of a board position at any given time.

• In AlphaGo make use of convolutional neural networks for both these models.

AlphaGo

• Go was considered a challenge for artificial intelligence for two reasons.
1. Game tree has a very high `branching factor’.

In Go, there are so many legal moves that the game tree increases exponentially.

2. Evaluating the quality of any given board position was deemed to be very hard.[^chess].

• AlphaGo played more than 30,000,000 games to learn value and policy.

Model Based Approach

• An alternative approach to reinforcement learning is to use a prediction function.

• This is similar to classical control and field of system identification.

• Known as model based reinforcement learning.

Optimization Methods

• Reinforcement learning can also used to improve user experience.

• Reward is gained when the user buys a product from us.

• This makes it closely allied to the area of optimization.

• Optimization of our user interfaces is like reinforcement learning task, but normally approached through Bayesian optimization or bandit learning.

Optimization

• Normal optimization we have a mathematical representation of our objective function as a direct function of the parameters.

• In Bayesian Optimization we don’t.

• E.g. Examples in this form of optimization include:

1. What is the best user interface for presenting adverts?

2. What is the best design of wing for an F1 car?

3. Which product should I return top of the list in response to this user’s search?

Bayesian Optimization

• Can’t directly relate the parameters in the system of interest to our objective through a mathematical function.

• E.g. What is the mathematical function that relates a user’s experience to the probability that they will buy a product?

Bayesian Optimization

• Use machine learning to develop a surrogate model for the optimization task.

• Surrogate model is a prediction function that attempts to recreate the process we are finding hard to model.

• Try to simultaneously fit the surrogate model and optimize the process.

Surrogate Models

• Bayesian optimization methods use a surrogate model (normally a specific form of regression model).

• Use this to predict how the real system will perform.

• Optimize in the surrogate model.

Conclusion

• Introduce range of ML approaches.

• Focussed on where they use mathematical functions as a general overview.

Deployment

• Introduced almost in order of difficulty of deployment.

• Supervised learning requires more data annotation, but more straightforward to deploy.

• Therefore major focus with supervised learning should always be on maintaining data quality.

Where to Deploy

• If a typical person can do a mental task with less than one second of thought, we can probably automate it using AI either now or in the near future.[^ngquote]

Andrew Ng

Is this Right?

• Broadly agree with this quote but only in the context of supervised learning.

• The picture with regard to unsupervised learning and reinforcement learning is more clouded.

Getting Easier

• We seem to be moving beyond the era where very deep machine learning expertise is required to deploy methods.

• Just require a solid understanding of machine learning (say to Masters level)

Streaming Data

• Supervised machine learning solutions are normally trained offline.

• They do not adapt when deployed because this makes them less verifiable.

• This compounds the brittleness of our solutions.

Model Choice

• For all machine learning methods is the initial choice of useful classes of functions.

• The deep learning revolution is associated with a particular class of mathematical functions.

1. GOFAI has its origins in the birth of computer science so it is unsurprising that logic should be the dominant paradigm. There was also a sense in which pure logic felt like it mapped well on to human reasoning. In practice, for any complex real world situation, the number of logical rules required causes an explosion in the state space of the system. This makes logical systems intractable in many applications.

2. Using the test data in your training procedure is a major error in any machine learning procedure. It is extremely dangerous as it gives a misleading assessment of the model performance. The Baidu ImageNet scandal was an example of a team competing in the ImageNet challenge which did this. The team had announced via the publication pre-print server Arxiv that they had a world-leading performance on the ImageNet challenge. This was reported in the mainstream media. Two weeks later the challenge organizers revealed that the team had created multiple accounts for checking their test performance more times than was permitted by the challenge rules. This was then reported as “AI’s first doping scandal”. The team lead was fired by Baidu.


text:
# Find the Power to move a car on an incline

1. Jan 25, 2012

### BSElectrician

1. The problem statement, all variables and given/known data
What power is needed to move a 3000lb car up an 8 degree incline with a constant speed of 50mi/h against a frictionless force of 80lb?

ans. 68hp

2. Relevant equations

$P = Fv$

1hp = 550 lb-ft/s

3. The attempt at a solution
1st i draw free body diagram
summation of force in the horizontal direction
Fx = 0 = 3000sin(8) + 80 - Fn
Fn = 497.52lb

Power = velocity x force
= (50mi/h)(1/3600s)(1ft/0.3048m)(1000m/1km)(1km/0.62137mi)
= 36,485.56 lb-ft/s
= (36,485.56 lb-ft/s)(1hp/(550lb-ft/s))
= 66.34 horsepower
where did i get wronged?

2. Jan 25, 2012

### Delphi51

I get the same answer you have.

3. Jan 25, 2012

### LawrenceC

I think your answer is correct. I get the same result either by figuring the forces and multiplying by velocity (as you did) or by using the 80 pound force times velocity added to the change in potential energy per second.


text:
# Question about (probably false) elementary proof of Fermat's Last Theorem

Can somebody explain to me what is wrong with the following argument for a proof of Fermat's Last Theorem?

Suppose somewhere the following theorem has been proven:

Theorem:

If:

$\quad \quad \quad \quad x^p = y^p + z^p$

$\quad \quad \quad \quad \gcd(x,y,z) = 1$

$\quad \quad \quad \quad p \gt 2$ and prime

Then:

$\quad \quad \quad \quad x - y = r^p$,

$\quad \quad \quad \quad (x^p - y^p)/(x - y) = s^p$,

for some $r,s$ with $\gcd(r,s) = 1$.

We now proceed to prove some corollary:

Corollary:

$\quad$ There can be no solutions to our system.

Proof:

Because $\gcd(x - y,(x^p - y^p)/(x - y)) = \gcd(x - y,z^p/(x - y) = 1$, there exist some $a,b$ such that:

$\quad \quad \quad \quad a(x - y) + bz^p/(x - y) = 1$

$\quad \quad \quad \quad \implies a(x - y)^2 - (x - y) + bz^p = 0$

We now use the quadratic formula to show there can be no solutions to our system by infinitely generating smaller ones.

$\quad \quad \quad \quad D_{x - y} = 1 - 4abz^p = d^2$, for some d

$\quad \quad \quad \quad \implies (1 - d)(1 + d) = 4abz^p$

so $d$ is odd, say $2e + 1$ for some $e$.

$\quad \quad \quad \quad \implies -2e(2e + 2) = 4abz^p$

$\quad \quad \quad \quad \implies -e(e + 1) = abz^p$

$\quad \quad \quad \quad \implies e^2 + e + abz^p = 0$

$\quad \quad \quad \quad \implies D_e = 1 - 4abz^p = f^2$, for some f.

$\quad \quad \quad \quad \implies (1 - f)(1 + f) = 4abz^p$

so $f$ is odd, say $2g + 1$, for some $g$.

$\quad \quad \quad \quad \implies -2g(2g + 2) = 4abz^p$

$\quad \quad \quad \quad \implies -g(g + 1) = abz^p$

$\quad \quad \quad \quad \implies g^2 + g + abz^p = 0$

We see we keep ending up on the equation:

$\quad \quad \quad \quad u^2 + u + abz^p = 0$

Since we can infinitely repeat this procedure through calculation of the discriminant $D_u$, we find there is no smallest solution.

We conclude our system cannot have any solutions.

• Why is $x-y=r^p$? Is $p$ a prime here? – Dietrich Burde Jun 28 '17 at 12:30
• This couldn't be it as it would not fit in the margin. – Mikhail Katz Jun 28 '17 at 12:31
• @Niels How did you get from $a(x−y)+bz^p/(x−y)=1$ to $a(x−y)^2−(x−y)+bz^p=0$? – Toby Mak Jun 28 '17 at 12:32
• You want $p = n$? – Dirk Jun 28 '17 at 12:33
• @Dietrich: yes $p$ is prime. – user451710 Jun 28 '17 at 12:33

You will have $f = d$. Thus, your process will not be decreasing and it is totally ok (and not a contradiction) that it can be repeated infinitely.
When you write $d = 2e+1$, $\implies e = \dfrac{d-1}{2}$
On the other hand, $e = \dfrac{-1 \pm f}{2}$
So that, if $e$ is positive, $d=f$ and so on, all integers obtained are equal.


text:
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# The amount of time that three people worked on a special project was

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Math Expert
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The amount of time that three people worked on a special project was [#permalink]

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03 Jul 2015, 03:16
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The amount of time that three people worked on a special project was in the ratio of 2 to 3 to 5. If the project took 110 hours, how many more hours did the hardest working person work than the person who worked the least?

A. 55 hours
B. 44 hours
C. 33 hours
D. 22 hours
E. 11 hours

Kudos for a correct solution.
[Reveal] Spoiler: OA

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Re: The amount of time that three people worked on a special project was [#permalink]

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03 Jul 2015, 03:36
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Bunuel wrote:
The amount of time that three people worked on a special project was in the ratio of 2 to 3 to 5. If the project took 110 hours, how many more hours did the hardest working person work than the person who worked the least?

A. 55 hours
B. 44 hours
C. 33 hours
D. 22 hours
E. 11 hours

Kudos for a correct solution.

Solution -

Time is in the ratio 2:3:5
Total time taken for the project = 110hrs

1st person time taken = (2/10)*110 = 22 hrs
2nd person = (3/10)*110 = 33 hrs
3rd person = (5/10)*110 = 55 hrs
How many more hours did the hardest working person work than the person who worked the least = 55 - 22 = 33 hours
ANS C
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Re: The amount of time that three people worked on a special project was [#permalink]

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03 Jul 2015, 04:07
Let the persons be A, B, C.

Hours worked:
A = 2*110/10 = 22 hours
B = 3*110/10 = 33 hours
C = 5*110/10 = 55 hours

C is the hardest worker and A worked for the least number of hours. So the difference is 55-22 = 33 hours.
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Re: The amount of time that three people worked on a special project was [#permalink]

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03 Jul 2015, 06:30
1
KUDOS
Bunuel wrote:
The amount of time that three people worked on a special project was in the ratio of 2 to 3 to 5. If the project took 110 hours, how many more hours did the hardest working person work than the person who worked the least?

A. 55 hours
B. 44 hours
C. 33 hours
D. 22 hours
E. 11 hours

Kudos for a correct solution.

2+3+5=10--->110hours
So 1 unit (or x) of the ratio is 11 hours.

Hardest working - Least Working = 5x-2x=3x
x=11
Therefore, 3x=3*11=33

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The amount of time that three people worked on a special project was [#permalink]

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03 Jul 2015, 07:24
a LOT easier than I first thought, but that's what I suppose the question was testing, comprehension, etc. A simple ratio would solve the problem.

Time is given as a ratio: $$2x:3x:5x$$, sum $$10x$$
Work is given in hours: $$110$$ hours
Thus, each person works $$11$$ times their time ratio

Hardest working (most hours presumably) - person who has least amount of hours =
$$55 - 22 = 33$$

OR, differece of two ratios $$5-2 = 3$$. Ratio $$3x * 11 = 33$$. Since the integrity of the ratio is maintained, this method works too.

Thus, C.

Suppose back solving would be another approach, less efficient, mind you.
A. 55 hours, impossible since the range of the ratio limits the multiple to 3x
B. 44 hours, impossible since the range of the ratio limits the multiple to 3x
C. 33 hours, most correct answer
D. 22 hours, trick answer. If range is sought, then the range must not be less than 3x.
E. 11 hours, trick answer. If range is sought, then the range must not be less than 3x.

Thanks,
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Re: The amount of time that three people worked on a special project was [#permalink]

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03 Jul 2015, 08:55
1
KUDOS
Let x be common multiplier

So least hard working person worked for 2x hours
next person worked for 3x hours
most hard working person worked for 5x hours

In total the worked for 2x+3x+5x hours = 110
10x =110
x=11

So difference between least and most hard working person = 5(11) - 2(11) = 33

Ans C
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Re: The amount of time that three people worked on a special project was [#permalink]

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03 Jul 2015, 18:45
Bunuel wrote:
The amount of time that three people worked on a special project was in the ratio of 2 to 3 to 5. If the project took 110 hours, how many more hours did the hardest working person work than the person who worked the least?

A. 55 hours
B. 44 hours
C. 33 hours
D. 22 hours
E. 11 hours

Kudos for a correct solution.

Ratio of the Time taken by A:B:C = 2:3:5

i.e. If, A = 2x, then B = 3x and C = 5x

i.e. Total Time taken by all three of them together = 2x + 3x + 5x = 10x = 110 hours (Given)

i.e. x = 110/10 = 11

i.e. Time of A = 2x = 2*11 = 22 (Least Working Person)
i.e. Time of B = 3x = 3*11 = 33
i.e. Time of C = 5x = 5*11 = 55 (Hardest Working Person)

Extra No. of Hours worked by Hardest working Person more than Least Working Person = 55 - 22 = 33

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Re: The amount of time that three people worked on a special project was [#permalink]

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04 Jul 2015, 09:02
1
KUDOS
Bunuel wrote:
The amount of time that three people worked on a special project was in the ratio of 2 to 3 to 5. If the project took 110 hours, how many more hours did the hardest working person work than the person who worked the least?

A. 55 hours
B. 44 hours
C. 33 hours
D. 22 hours
E. 11 hours

Kudos for a correct solution.

Let the no of hours each employee worked be 2x,3x and 5x
Now, 2x+3x+5x=110
x=11
Therefore, difference betwen 2x and 5x=55-22=33 hours
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The amount of time that three people worked on a special project was [#permalink]

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04 Jul 2015, 09:10
Bunuel wrote:
The amount of time that three people worked on a special project was in the ratio of 2 to 3 to 5. If the project took 110 hours, how many more hours did the hardest working person work than the person who worked the least?

A. 55 hours
B. 44 hours
C. 33 hours
D. 22 hours
E. 11 hours

Kudos for a correct solution.

The ratios of time spent is 2:3:5. Thus, the times spent by each person is 2x, 3x and 5x, with the total being 10x

Given, 10x=110 >> x=11. The difference between the times spent by the hardest worker and the least worked worker = 5x-2x=3x= 3*11 = 33 hours. Thus C is the correct answer.

Bunuel, the language of this questions is vague. We need to assume that 'hardest' and 'least' work is directly proportional to the amount of time spent. A better way to phrase this question will be to ask what is the difference in the most time spent and the least time spent. The quantity or quality of work can not be determined by just relying on time spent.
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The amount of time that three people worked on a special project was [#permalink]

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04 Jul 2015, 10:10
C is correct
Time is in the ratio 2:3:5
Total time taken for the project = 110hrs

1st person time taken = (2/10)*110 = 22 hrs
2nd person = (3/10)*110 = 33 hrs
3rd person = (5/10)*110 = 55 hrs
Hours between hardest working person work than the person who worked the least = 55 - 22 = 33 hours
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Posts: 43785
Re: The amount of time that three people worked on a special project was [#permalink]

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06 Jul 2015, 05:25
Bunuel wrote:
The amount of time that three people worked on a special project was in the ratio of 2 to 3 to 5. If the project took 110 hours, how many more hours did the hardest working person work than the person who worked the least?

A. 55 hours
B. 44 hours
C. 33 hours
D. 22 hours
E. 11 hours

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

Use an equation with the Unknown Multiplier to represent the total hours put in by the three people:

2x + 3x + 5x= 110
10x = 110
x= 11

Therefore, the hardest working person put in 5(11) = 55 hours, and the person who worked the least put in 2(11) = 22 hours. This represents a difference of 55 - 22 = 33 hours.

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Re: The amount of time that three people worked on a special project was [#permalink]

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Re: The amount of time that three people worked on a special project was [#permalink]

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15 Mar 2017, 15:46
Bunuel wrote:
The amount of time that three people worked on a special project was in the ratio of 2 to 3 to 5. If the project took 110 hours, how many more hours did the hardest working person work than the person who worked the least?

A. 55 hours
B. 44 hours
C. 33 hours
D. 22 hours
E. 11 hours

We are given that the amount of time that three people worked on a special project was in the ratio of 2 to 3 to 5. We can create the following ratio:

person working least : person working 2nd least : person working most = 2x : 3x : 5x

Since the project took a total of 110 hours, we can create the following equation:

2x + 3x + 5x = 110

10x = 110

x = 11

Thus, the person who worked the most number of hours worked for 5 * 11 = 55 hours and the person who worked the least worked for 2 * 11 = 22 hours. The difference between the two values is 55 - 22 = 33 hours.

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Re: The amount of time that three people worked on a special project was   [#permalink] 15 Mar 2017, 15:46
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