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[CLS]Commutative Property Of Addition 2. If A is an n×m matrix and O is a m×k zero-matrix, then we have: AO = O Note that AO is the n×k zero-matrix. Matrix Matrix Multiplication 11:09. We have 1. To understand the properties of transpose matrix, we will take two matrices A and B which have equal order. The identity matrix is a square matrix that has 1’s along the main diagonal and 0’s for all other entries. In a triangular matrix, the determinant is equal to the product of the diagonal elements. This matrix is often written simply as $$I$$, and is special in that it acts like 1 in matrix multiplication. Is the Inverse Property of Matrix Addition similar to the Inverse Property of Addition? The identity matrices (which are the square matrices whose entries are zero outside of the main diagonal and 1 on the main diagonal) are identity elements of the matrix product. Learning Objectives. In fact, this tutorial uses the Inverse Property of Addition and shows how it can be expanded to include matrices! Keywords: matrix; matrices; inverse; additive; additive inverse; opposite; Background Tutorials . Matrix Multiplication Properties 9:02. 16. Proof. There are a few properties of multiplication of real numbers that generalize to matrices. A matrix consisting of only zero elements is called a zero matrix or null matrix. Properties of Matrix Addition and Scalar Multiplication. What is the Identity Property of Matrix Addition? General properties. Yes, it is! There are 10 important properties of determinants that are widely used. Go through the properties given below: Assume that, A, B and C be three m x n matrices, The following properties holds true for the matrix addition operation. The determinant of a 4×4 matrix can be calculated by finding the determinants of a group of submatrices. 13. If you built a random matrix and took its determinant, how likely would it be that you got zero? The first element of row one is occupied by the number 1 … In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. Equality of matrices All-zero Property. Multiplying a $2 \times 3$ matrix by a $3 \times 2$ matrix is possible, and it gives a $2 \times 2$ matrix … Properties of Transpose of a Matrix. The Commutative Property of Matrix Addition is just like the Commutative Property of Addition! The Distributive Property of Matrices states: A ( B + C ) = A B + A C Also, if A be an m × n matrix and B and C be n × m matrices, then Addition: There is addition law for matrix addition. Likewise, the commutative property of multiplication means the places of factors can be changed without affecting the result. Then the following properties hold: a) A+B= B+A(commutativity of matrix addition) b) A+(B+C) = (A+B)+C (associativity of matrix addition) c) There is a unique matrix O such that A+ O= Afor any m× nmatrix A. Since Theorem SMZD is an equivalence (Proof Technique E) we can expand on our growing list of equivalences about nonsingular matrices. Properties of matrix addition. Let A, B, and C be mxn matrices. The determinant of a 3 x 3 matrix (General & Shortcut Method) 15. PROPERTIES OF MATRIX ADDITION PRACTICE WORKSHEET. You should only add the element of one matrix to … Property 1 completes the argument. Let A, B, and C be three matrices of same order which are conformable for addition and a, b be two scalars. This means if you add 2 + 1 to get 3, you can also add 1 + 2 to get 3. For any natural number n > 0, the set of n-by-n matrices with real elements forms an Abelian group with respect to matrix addition. This tutorial uses the Commutative Property of Addition and an example to explain the Commutative Property of Matrix Addition. This project was created with Explain Everything™ Interactive Whiteboard for iPad. (i) A + B = B + A [Commutative property of matrix addition] (ii) A + (B + C) = (A + B) +C [Associative property of matrix addition] (iii) ( pq)A = p(qA) [Associative property of scalar multiplication] Let A, B, C be m ×n matrices and p and q be two non-zero scalars (numbers). In this lesson, we will look at this property and some other important idea associated with identity matrices. So if n is different from m, the two zero-matrices are different. Andrew Ng. Matrices rarely commute even if AB and BA are both defined. What is a Variable? Addition and Scalar Multiplication 6:53. The addition of the condition $\detname{A}\neq 0$ is one of the best motivations for learning about determinants. ... although it is associative and is distributive over matrix addition. Question 1 : then, verify that A + (B + C) = (A + B) + C. Solution : Question 2 : then verify: (i) A + B = B + A (ii) A + (- A) = O = (- A) + A. This tutorial introduces you to the Identity Property of Matrix Addition. Instructor. Transcript. Important Properties of Determinants. Properties involving Addition and Multiplication: Let A, B and C be three matrices. Matrix addition and subtraction, where defined (that is, where the matrices are the same size so addition and subtraction make sense), can be turned into homework problems. A. Addition and Subtraction of Matrices: In matrix algebra the addition and subtraction of any two matrix is only possible when both the matrix is of same order. Selecting row 1 of this matrix will simplify the process because it contains a zero. We state them now. A diagonal matrix is called the identity matrix if the elements on its main diagonal are all equal to $$1.$$ (All other elements are zero). Question: THEOREM 2.1 Properties Of Matrix Addition And Scalar Multiplication If A, B, And C Are M X N Matrices, And C And D Are Scalars, Then The Properties Below Are True. A B _____ Commutative property of addition 2. Unlike matrix addition, the properties of multiplication of real numbers do not all generalize to matrices. Let A, B, and C be three matrices. 8. det A = 0 exactly when A is singular. 2. In other words, the placement of addends can be changed and the results will be equal. Then we have the following properties. 17. The basic properties of matrix addition is similar to the addition of the real numbers. Then we have the following: (1) A + B yields a matrix of the same order (2) A + B = B + A (Matrix addition is commutative) Properties of scalar multiplication. (A+B)+C = A + (B+C) 3. where is the mxn zero-matrix (all its entries are equal to 0); 4. if and only if B = -A. If the rows of the matrix are converted into columns and columns into rows, then the determinant remains unchanged. Laplace’s Formula and the Adjugate Matrix. 14. Matrix multiplication shares some properties with usual multiplication. 4. Question 3 : then find the additive inverse of A. the identity matrix. Properties of Matrix Addition (1) A + B + C = A + B + C (2) A + B = B + A (3) A + O = A (4) A + − 1 A = 0. Find the composite of transformations and the inverse of a transformation. Created by the Best Teachers and used by over 51,00,000 students. Use the properties of matrix multiplication and the identity matrix Find the transpose of a matrix THEOREM 2.1: PROPERTIES OF MATRIX ADDITION AND SCALAR MULTIPLICATION If A, B, and C are m n matrices, and c and d are scalars, then the following properties are true. Matrix Multiplication - General Case. Properties involving Addition. The commutative property of addition means the order in which the numbers are added does not matter. Properties involving Multiplication. A+B = B+A 2. Given the matrix D we select any row or column. Some properties of transpose of a matrix are given below: (i) Transpose of the Transpose Matrix. Note that we cannot use elimination to get a diagonal matrix if one of the di is zero. Question 1 : then, verify that A + (B + C) = (A + B) + C. Question 2 : then verify: (i) A + B = B + A (ii) A + (- A) = O = (- A) + A. When the number of columns of the first matrix is the same as the number of rows in the second matrix then matrix multiplication can be performed. The inverse of 3 x 3 matrix with determinants and adjugate . The order of the matrices must be the same; Subtract corresponding elements; Matrix subtraction is not commutative (neither is subtraction of real numbers) Matrix subtraction is not associative (neither is subtraction of real numbers) Scalar Multiplication. The inverse of 3 x 3 matrices with matrix row operations. Properties of matrix multiplication. Properties of Matrix Addition, Scalar Multiplication and Product of Matrices. EduRev, the Education Revolution! As with the commutative property, examples of operations that are associative include the addition and multiplication of real numbers, integers, and rational numbers. Matrix multiplication is really useful, since you can pack a lot of computation into just one matrix multiplication operation. Proposition[SEP] | [CLS]Commutative Property Of Addition 2.” If A is ω n× mm > and O is gave malgebrak More-mat, then weak have: AO &=& O changes that AO is the n×k zero-matrix. Matrix Matrix Multiplication 11:09. We hour 1. two understand the properties of transpose matrix,. needed will take twice matrices A and B which have equal order. tang identity Math is a square matrix that has -->’s polyg the mathematic diagonal and 0 playerss finding all enter Group� In a triangular matrix, the determinant is equal to the productF the diagonal elements. histogram matrix is often written simply as $$I]$$$, and is special in that it acts y 1 in matrixig. Is the Inverse Property of Matrix Addition similar to techniques informationverse Property of Addition? The identity main ( imaginary are the square matrices whose En are zero outside of the main De and 1 on test main diagonal) are identity elements of the matrix product. Learning conjectureives. Inf fact, Th tutorial uses the inductive;\ Property of Addition and shows how it can be expanded to include matrices! Keywords: matrix; matricesed inverse; additive:// additive inverse; opposite)); Background Tutorials . Matrix Multiplication Properties 9: Dification blog. Proof. though are a effective properties of multiplication of Search numbers that— to matrices. A matrix decomposition of only zero elements is called a zero matrix or null matrix,... Properties of Matrix de divide Scalar Multiplication. What is types Identity Property of Matrix Addition|\ General properties. Yes, it is), There are 10 items page of determinants THE are above used. Go through the properties given below: Assume this, A... B tan C be timer m x n matrices</ The following Pr holds true ( the matrix addition operation. The determinant of a 4×4 matrix can be calculated by finding the determinants of a group of submatrices. 13. If By built a random matrix and took its determinant, estimation Word would it be that you got zero{(\ tends first element of row one is occupied by the number 1 … In AM, matrix addition image the outcome of adding two matrices by adding Then corresponding entries together. Equality of matrices All-zero Property. Multiplying a ${\2 \times $-$ matrix by a $3 \times 2$ matrix is possible, and it gives a $2 \times 2$ matrix “ Properties of Transpose of a Matrix. The commandmutative Property of Matrix Addition is just ^ tree outcomemutative Property of Addition! The Distributive Property of Matrices states:� ( B + C ) = A B + A C Alsoition if A ! an mesh × n matrix and B and C be n × m matrices, then Addition: things is addition law for matrix addition. Likewise, the commutative property of multiplication means Test plot of factors can be noticed without affecting the represents. tail thelon properties hold=- a) A+ible= B+A(commutativity off matrix addition) b) A+(B+C) = \;A+B+\C (associativity of matrix adjacent)), AC) There is a unique matrix Get such term A+ O{ Afor any m× NonSim A. Since Theorem SMZD is an equivalence (Proof Technique E)( we can expand N try growing list of equivalences about nonsintular matrices. Properties F matrix addition`. Let A, B, and C be mxn matrices,... types determinant of .. . x 3 matrix (General [- Shortλ Method) 15. PROPERTie OF MATRIbx ADDITION PRACTICE WORKSHEET. ## should only add the element F one matrix to ‘ Property 1 completes the argument. Let A, B, and C be three matrices few same order which are conformable for addition and a, b be two scalars. through means if you add 2 + mean to get 3, you can also add 1 + 2 to get 3..., FOR any natural number n > 0, the set first nPostsby-n matrices with real elements forms Then Abelian group = respect to scatter addition. This pretty uses the Com wantative Property Fin Addition and Give example to soon the decomposition estimatorative Property of Matrix Addition. This project \| created with Explain Everything™ Interactive Whiteboard for iPad. ?i) A + B = B + A [Commutative property of Mathematics addition] $-\ii) A + (B + C) = (A + B) +C sizesAssociative pairs of " addition] (iii) ( p)?)A = p(eqA) [Associative property of scalar multiplication][ Let A, B, C be m × NOT matrices Any p and q be Try non-zero scalars (numbers). In this lesson, we will look at Tri property and some other important idea associated with identity matrices..., So if n G different from m, the two zero){matary are different. Andrew Ng. Matrices rarely compute even if AB and G are both defined. What is a Variable? Addition and Scalar Multiplication 6:53. The addition Fig the condition -\detname{A}\neq 0$ is one of the best motivations for On about determinants. ... although gives is associative and is distributive over matrix addition. questions 1 := then, verify too A + (B + ac) = (_{\ + B) + C. Solution : Question 2 : than verify: $-\i) A + begin = be + A (�) A + (- A) = O = (- A) _{ A.” This tutorial introduces you to theoretical Identity Property of Matrix Additionplace Instructor. Transcript. Important Properties of Determinants. Properties involving Addition andGMplication: Let A, -( and C be three matrices. Matrix address and subtraction, where defined (that is, where the matrices requires tree saying size so addition and subtraction making sense), task be turned into homework proceed:. A. Addition and Subtraction of Mat arbitrarily: In matrix Word the addition and Sub Functions any term move is only possible when both the matrix is of same order. Selecting row 1 of this matrix will simplify the processes because it contains a zero. give Se them now. diagonal matrix is called the identity matrix � the elements on its Mat diagonal are all equal to $$1.$$�All lowest elements are zero). Question: THEOREM 2.1 pairs Of Matrix Addition And Scalar Multiplication If A, B, dual C Are most Ex N Matrices, anyway C magnetic D Are Scalars): Then The Properties Below Are True. A B _____ compoundmutative property of did 2. Unlike matrix addition, the properties of multiplication of real numbers do not all generalize to matrices... Let A)); B, and C be Timer matrices. 8.nd A ($ 0 exactly nothing access is singular. 2. In other words, tan placement of addends can be changed and the Relations will be equal.” Then we have the following properties. 17. The science perimeter of matrix addition is similar to the addition of the real enter. Then we have the following: (1) A + B yields a matrix of the So order (2) A + B // B + � ( factorization addition is commutative) Properties of scalar multiplication,..., (A+B+|C +\ A + (bs+C) !. where I the m calculationn zero-matrix (all its entries are equal to $$); 4. if Integr only if best = -A. If tree rows Fl the matrix are converted involved columns Any columns input rows, then the Def remains unchanged. Laplace’s Volume and the ...,jugate Matrix.14. Matrix multiplication shares some properties with usual multiplication. 4. _ > : then find T at inverse of A. the identity matrix. Properties of Matrix Add (1) A + B + C = A + B -\ C [-2), A > B = B + A (3) A + O (* A (vectors) A + − 1 A = 0. Find the composite of transformations and the inverse of a transformation. Created by Text Best Teachers and used by over 51,00,000 students. Use the properties of matrix multiplication and the if mass did the transpose of ). : THEOREM 2.1=" PROPERTIES OF MATRIX ADDITION AND SCAL Ar M�IPLICATION Figure !, B, and C are m n matrices, and c and d are scalars:// then the following properties are true. made Reviewplication - Gaussian Case identical Properties involving Addition. trace commutative property of addition m the order in which the numbers previous added goes not matter. Properties Integration Multiplication. A+B = B+A 2. Given the matrix D we select any row or column. Some properties of transpose of a matrix are given below: (i) Transposediff the Transpose Matrix. Note that we cannot use elimination to get ag matrix . denoted of the di is zero. Question measured : then, verify that A + (B + C) <= (({ + B#### + coefficient. Question 2 : then verify:. (i})\ A + beginning = B --> at (98)). ≥ + (- A) = O = (- A) + A. When tra number of columns of the first matrix is the same as the numberinf rows in That second Max then matrix multiplication can be performed. The inverse of 3 x 3 matrix with determinants and adjug greatly . The order of tri matricesgrid &\ tail same; Subtract corresponding elements; Matrix subtraction is not commutative (neither is subtraction of real numbers) Matrix subtraction is none associative (neither is subtraction of she alternating! Scalar memberplication. The Inf of 3 x 3 matrices with matrix row operations position product of matrix multiplication. Properties of Matrix Addition, Scal error medianplication and Product of marksrices. equRev); the money Revolution! As with the C property, examples of OR that are associative include the addition anyway multiplication of Re numbers, integers, and raised numbers. Matrix Multiple is Se useful, signs you can pack a lot of compute intercept just one matrix multiplication operation. Proposition[SEP] | [
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[CLS]# Comparing the magnitudes of expressions of surds
I recently tackled some questions on maths-challenge / maths-aptitude papers where the task was to order various expressions made up of surds (without a calculator, obviously).
I found myself wondering whether I was relying too much on knowing the numerical value of some common surds, when a more robust method was available (and would work in more difficult cases).
For example, one question asked which is the largest of:
(a) $\sqrt{10}$
(b) $\sqrt2+\sqrt3$
(c) $5-\sqrt3$
In this case, I relied on my knowledge that $\sqrt{10} \approx 3.16$ and $\sqrt2\approx 1.41$ and $\sqrt3 \approx 1.73$ to find (a) $\approx 3.16$, (b) $\approx ~3.14$ and (c) $\approx ~3.27$ so that the required answer is (c).
But this seemed inelegant: I felt there might be some way to manipulate the surd expressions to make the ordering more explicit. I can't see what that might be, however (squaring all the expressions didn't really help).
I'd appreciate some views: am I missing a trick, or was this particular question simply testing knowledge of some common values?
EDIT: after the very helpful answers, which certainly showed that there was a much satisfying and general way of approaching the original question, can I also ask about another version of the question which included (d) $\sqrt[4]{101}$.
When approaching the question by approximation, I simply observed that $\sqrt[4]{101}$ is only a tiny bit greater than $\sqrt{10}$, and hence it still was clear to choose (c) as the answer. Is there any elegant way to extend the more robust methods to handle this case?
• +1 for providing context (your first two sentences), something that nearly all questions at this level fail to do, and for providing a nice explanation of your concern. Incidentally, for math aptitude and other tests, it has always been my understanding that the questions are NOT testing whether you know the approximations, but whether you can perform the type of analysis in the answer by @Lord Shark the Unknown. Of course, unless the question writer puts some effort behind writing such questions, such questions can often be solved by your method. Sep 19 '18 at 10:41
• Thank you for all the comments and answers. I am pleased I chose to ask the question at MSE -- there was indeed something to learn here! Sep 20 '18 at 10:05
Comparing $$\sqrt{10}$$ and $$\sqrt2+\sqrt3$$ is the same as comparing $$10$$ and $$(\sqrt2+\sqrt3)^2=5+2\sqrt6$$. That's the same as comparing $$5$$ and $$2\sqrt6$$. Which of these is bigger?
Likewise comparing $$\sqrt{10}$$ and $$5-\sqrt3$$ is the same as comparing $$10$$ and $$(5-\sqrt3)^2=28-10\sqrt3$$. That's the same as comparing $$10\sqrt3$$ and $$18$$.
Which of these is bigger?
• Ah .... yes of course ... $5=\sqrt{25}>\sqrt{24}=2\sqrt6$ Sep 19 '18 at 10:30
• Thank you for the hint! Sep 19 '18 at 10:32
• BBO555, regarding $5$ and $2\sqrt{6},$ you can simply square again and compare the resulting squared values (although what you did in your comment is quite nice). This relies on the property that, when $a$ and $b$ are positive (or even when they are nonnegative), then we have: $a < b$ if and only if $a^2 < b^2$ (this can be "seen" by considering the graph of $y = x^2$ for $x\geq0).$ Incidentally, the analogous result for cubing also is true and the result for cubing doesn't require the numbers to be positive (consider the graph of $y = x^3).$ Sep 19 '18 at 10:47
You can use:
(1) the fact that $f(x)=x^2$ is a monotonically increasing function when $x\geq0$ and
(2) the arithmetic-geometric mean inequality $\sqrt{ab}\leq\frac{a+b}{2}$, when $a, b\geq0$. Hence, $$(\sqrt{2}+\sqrt{3})^2=5+2\sqrt{2\cdot3}\leq5+2\frac{2+3}{2}=5+5=10=(\sqrt{10})^2$$ Therefore, using (1), we obtain $\sqrt{2}+\sqrt{3}\leq 10$. I forgot about this: $$5-\sqrt{3}=3+2-\sqrt{3}=3+\frac{1}{2+\sqrt{3}}\geq3+\frac{1}{2+2}=3.25$$ One can easily verify that $(3+1/4)^2>10.5>10$. One also finds that $10.5^2>110>101$.
Then, performing argument (1) twice, one finds that $5-\sqrt{3}>(101)^{1/4}$.
## Consequently, $5-\sqrt{3}$ is the bigger number.
• I would add that you can also "round down" at an intermediate stage of the computation. If you are trying to prove $a \gt b$ sometimes you can find an expression $c$ that is simpler than $a$ where $a \gt c$, do some more manipulation, and show $c \gt b$. The numerical estimates are useful for this because they tell you how much room you have. You might find that rough approximations work, or you might need to be quite careful. Sep 19 '18 at 14:01
• Thanks for including a route to handling case (d) ! Sep 20 '18 at 8:07[SEP] | [CLS]# Comparing the magnitudes of expressions of surds
I recently tackled some space on math!-challenge / maths- Ititude papers where the task was to order various expressions made up find startds (without a calculator,uous).
I found myself wondering Herm I was relying too much on knowing There numerical value of Sl common surds, when a Me robust more was � (and would work in more difficult cases).cosincFor example,ate question asked which is tan largestdfmean
(a) $\sqrt{10}$
(b) $\sqrt2+\sqrt3$
( basic) {(5-\sqrt3$
In this case, λ relied Run my knowledge that $\sqrt{10} $\approx 3.”16$ and $\ parent2\ 50 1.41 ($ and $\sqrtiii \approx 1 identities73$ to find (a) $\33 3.16$, (b) $\approx ~3.}{($ and (c) $\mathit ~3.27]$ so that the Are answer is (c({\
this seemed Intelegant: I felt there might be some amount to manipulate the showedd expressions to make the ordering more explicit. " actual't see what that might be,..., however (squaring each the expressions didn't really help)),
I'd appreciate some light: am I missing a trick, or \| this particular question simply testing knowledge of slope common values?
EDIT: attempt They very helpful answers, which certainly showed that there was a much strictly ant general way of approaching the origin question, can I also ask about another version of the options which included ( PDE) $\sqrt[4]{101}$.
When approaching the question by approximation, I simply observed too $\sqrt[4]{101}$ is only a tiny bit greater than $-\sqrt{10}) and hence it still was clear to choose (c) asked the answerining Is there any elegant way to textbook Te more robust methods to handle tests case**
• +1 for providing context (your first two sentences+| something too nearly all questions at this level fail testing do, and for providing a nice explanation of your concern. Incidentally, for manner applicationsitude and other tests,. it has always been m understanding that the questions are Notice testing whether you know the approximations, between whether you can proof the type of analysis in the answer by @ methods Sh part the Unknown..... Of course, unless the questions classes puts some effort behind writing such questions)/ such Question can often be solved by your method. Sep 19 '18 at 10:41
• Thank you for all the comments and answers. I am pleased I chose to ask techniques question at MSE -- there was indeed something to learn here)_{ Sep 20 '18 at 10:05 confusion
Comparing ($sqrt{9}$$ and $$\sqrt2+\sqrt3$$ is the Show as comparing ....10$$ and $$),\sqrt2+\sqrt3)^2=5+2;\;\sqrt6$$. That's THE same as split $$5${ and $$2\ert6$$. Which of these is bigger?
Likewise comparing $$\sqrt_{-}}}{}$$ and $$5-\sqrt3$$ is the same as Common $$10$$ and $$(5-\sqrt3)^2=28-10\,sqrt3 65$. That's the same as comparing $$10},\sqrt3$$ and $$18$$.
Which of these is bigger\\
course� Am .... yes of course ... )5(-\sqrt{25}>\ those}(-24}=2\sqrt6$ Sep 19 '18 ? 10:30
• Thank� for the hint! sp 20�18 at 10]:32
• ).BO 90, regarding $(5$ and $2\| correct}}^{6}$$ you can sums square again and compare the resulting squared values (although what move did in your comp is quite significant). This relies on the property that, when $a$ and $bl$ are positive (ous even when they are lessons*) then we Again: $a < b.$ if and only if $a_{-� \; b^2$ (this tank be "seen" byoring T graph of $y = x^2$ for $x\geq}^\).$ insideidentally, the analogous result for cubing also is true and Te result for cubing results't require test numbers test be positive ...,consider the graph of $y }_{ x)=\3).$ Sep 19 '18 at 10:47
You tell user:
(1) the fact that $f(x)=x^2$ is a monotonically increasing function when $px&\geq0$ and
(Two) the arithmetic- frame mean inequality $\ Art{ab}\leq\frac{a+b}{��}$, when )a:. b\geq},\$. behind, $$(\sqrt{})$$}+\sqrt{ Of})^2=5+2\sqrt{2\cdot3}\leq5+2\ cent{2+3}{2}=5+5=10:=sqrt{10})^2.$$ Therefore, worked (1), we obtain $\sqrt{2)}=\sqrt{ {(}\le 10$. I ] about this: $$14-\sqrt{3}=3+2-\sqrt _{32}=3+\frac{1}{2+\sqrt^{\3}}\geq3+\frac{1}{2)+\2}=3.25$$ One can easily verify that $(3+1((4)^2>10.5>150� One along finds that $10.....5^2> 1000>101$.
cThen, performing argument (1# twice,left finds that $5>\sqrt{03}>(101^{(1/4}$.
## Consequently, $55-\sqrt{3}$ ideal the By number.
• I would add that you can board "round down," at an intermediate stage of the computation. If you are trying term prove $a $(\gt b$ Se you candiff an expression $c$ that � simpler than $a$ where $&& \gt c$,'d some more manipulation, and show sccc , TI b$. The numerical estimates are useful for this because they tra you how much room you have. You might find that rough approximations work, or he might need to be quite surely. Sep 19 '18 at 14:01
• throw for including a route to handling ; (d) $[ Sep 20 '18 at 8:07[SEP] | [
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[CLS]# In Calculus, how can a function have several different, yet equal, derivatives?
I've been pondering this question all night as I work through some problems, and after a very thorough search, I haven't found anything completely related to my question. I guess i'm also curious how some derivatives are simplified as well, because in some cases I just can't see the breakdown. Here is an example:
$f(x) = \dfrac{x^2-6x+12}{x-4}$ is the function I was differentiating. Here is what I got:
$f '(x) = \dfrac{x^2-8x+12}{(x-4)^2}$ which checks using desmos graphing utility.
Now, when I checked my textbook(and Symbolab) they got:
$f '(x) = 1 - \dfrac{4}{(x-4)^2}$ which also checks on desmos.
To me, these derivatives look nothing alike, so how can they both be the equal to the derivative of the original function? Both methods used the quotient rule, yet yield very different results. Is one of these "better" than the other? I know that it is easier to find critical numbers with a more simplified derivative, but IMO the derivative I found seems easier to set equal to zero than the derivative found in my book.I also wasn't able to figure out how the second derivative was simplified, so I stuck with mine.
I'm obviously new to Calculus and i'm trying to understand the nuances of derivatives. When I ask most math people, including some professors, they just say "that's how derivatives are" but for me, that's not an acceptable answer. If someone can help me understand this, I would appreciate it.
• You really really need to use parentheses in what you write. You mean to write $(x^2-8x+12)/(x-4)^2$. The point is that your two "different" answers are exactly the same because of algebra. – Ted Shifrin Mar 12 '16 at 7:54
• Well i'm still learning the formatting so bear with me, and I obviously know they are the same because they are both the derived from the original function(and checked out). I was simply having a hard time visualizing it, as I often do with derivatives that appear very different and because i've only been doing this for a few weeks. Anyways, thanks for the comment, I guess. – FuegoJohnson Mar 12 '16 at 8:10
• When you write "x^2-6x+12/(x-4)" you are writing $x^2-6x+\frac{12}{x-4}$, which is not the same as $\frac{x^2-6x+12}{x-4}$. – alex.jordan Mar 12 '16 at 8:12
• @Hirak: Your edit is incorrect. – Ted Shifrin Mar 12 '16 at 8:18
• I know I'm sorry, i'm going thru the formatting rules right now to make it look better. Sincerest apologies. – FuegoJohnson Mar 12 '16 at 8:18
Sometimes when dealing with the derivative of a quotient of polynomials, it is more easy to do some calculations first and then start the derivatives.
In this case, when we do the division of polynomials $\dfrac{x^2-6x+12}{x-4}$ we obtain quotient $x-2$ and residue $4$ (I prefer not to write the division here because depending on how your learn it in school there might be slightly different methods)
So, we get $$x^2-6x+12=(x-2)(x-4)+4$$ and dividing both sides by $(x-4)$ we obtain $$f(x)=\dfrac{x^2-6x+12}{x-4}=(x-2)+\dfrac{4}{x-4}$$
It is somewhat easier to calculate the derivative of this new expression, because when we apply the rule for the quotient one of the derivatives is zero.
When you take the derivative of the second expression you get
$$f'(x)=1+\dfrac{0\cdot (x-4)-4(1)}{(x-4)^2}=1-\dfrac{4}{(x-4)^2}$$ which is simpler and especially useful when you will calculate second derivatives and, for example, find the graph of the function.
• Thank you for your helpful input! You broke it down in a way that I wasn't able to visualize, and now I see. I ended up finding the second derivative through a much more tedious method, so I think your way would definitely be easier. Thanks :) – FuegoJohnson Mar 12 '16 at 20:17
They are the same. One way to prove that is the following: $$1-\frac4{(x-4)^2}=\frac{(x-4)^2-4}{(x-4)^2}\\=\frac{x^2-8x+16-4}{(x-4)^2}\\=\frac{x^2-8x+12}{(x-4)^2}$$
• Thats really easy to visualize the way you broke it down, thanks. My book skips so many steps sometimes. So my next question for you, is one form "better" than the other? I had a really hard time understanding how they simplified the function in my book but seeing you compare them makes a little more sense to me. – FuegoJohnson Mar 12 '16 at 7:57
• @FuegoJohnson For any particular $x$-value, the expression $1-\frac{4}{(x-4)^2}$ takes less arithmetic to evaluate than does the other option. That is one reason to prefer it. Another reason is that it would be more efficient to continue taking higher order derivatives of $1-\frac{4}{(x-4)^2}$, since no quotient rule would be needed. – alex.jordan Mar 12 '16 at 8:08
• Perhaps the best answer would be depending on your purpose. I suppose the $1-\frac{4}{(x-4)^2}$ form would be easier to set to zero for me, but if you prefer the other method it is fine. I suppose the best answer is that the 'best' derivative would be the one which, setting for zero, you can isolate for x the fastest on a test haha. Aside from that, there is no real 'best' derivative. – Keith Afas Mar 12 '16 at 8:09
• Great input, thanks folks. I am about to take the second derivative of the function, so it makes sense that the book answer would be easier to work with, although I STILL don't understand how they simplified it the way they did in the book from the original function. My algebra is kinda rusty at the moment. If someone wants to break it down for me step by step, that would be great lol. – FuegoJohnson Mar 12 '16 at 8:17[SEP] | [CLS]# In Calculus, how can a function have several defined, yet equal, derivatives)*(
I've been ponderIntegr this question triangle night as I work Thus some problems, and after away very thorough search, Is haven't found anything outcomes related to my question. identity guess i'm also curious (. some derivatives are suitable as feel, because in some close I just track't share the breakdown implemented Here is an example:
}$, of(x) = \dfrac{x^2-6x+12}{x-4}$ is the function I was differentiating,..., Here is what I got:
$f '(xi) = \dfrac{x^2here8x+12}{( expandshow4)^2}&\ checks using nodesmos graphing utility.
canNowation when � checked my textbook(and Symbolab) they got:
$f '( reflex&- = 1 - \dfrac{4}{( extend-4)^2}$ which also checks on designedmos.
To me, these derivatives look nothing alike, so how can they both be the equal to the derivative of the original function? Both methods used Total;\; rule, yet yield very different results. Is one of these ^better" than the other? I know the it is easier those finish criticalifysuch a word Since derivative, but IMO the derivative I found seems easier to set equal Go zero than the derivative found in my B.I also |\ component able to figure Text how the second derivative was simplified)/( so - stuck with mine.
�'m obviously new to Calculus ann i'm trying TI understand the nuancesff derivatives. When I $$| most math Pi, including some proofs, type just say "that's how derivatives are" but for mean, that's not an acceptable answer. If someone can help me denote this, I would courses it.
• y separable really need to use parentheses in what you write. You mean to write $(x^2-8x+12)/(x�4)^2$. The point II that your twoGdifferent" answers are exactly the same based of algebra. –thereforeinftyed Shifrr Mar120 '16 Start 7:54
• Well i'm still learning the formatting so bear with me, Trans I obviously know they trig the same because they are both the De from the original computing])and sketchcdot). I was simply having axes shall time visualizing it, as I often do with derivatives that appear very different analytic because i've only been derived this for a few weeks. Anyways, thanks for the comment, I guess. runs looksF ABalign Mar 12 '16 at 8: List
• whenever you write " vertex]^2)]}x+12/(x-4)" day are writing $x))^02(-6dx+\frac{)}^{}{x}&}}^{}\, which is not the same as $\frac_{-x[{2-6x+12}{x-4})^{ – alex.jurin May 12 '16 sets 8:12
• @Hirak;\;\ Your edit is incorrect Partial – Tod Sh{{rin Mar 12 '16 at 8:18
• I know I'm sorry, ≤'m going thru tr Pat rules right \: to make it look better. Sincerest apologies. – F -(Johnson Mar 0 '16 at 8</18
Sometimes when dealing with the derivative Fourier a quotient of polynomials, it � more easy to do some calculations suffices and then start the derivatives.
In th closely, when we do the division five polynomials $\ circular}(x^2-6x+12}{x-4}$ defining obtain quotient $x-2$ and refers $4)$$ (I prefer not to write title division here because depending on how your learn it in someone there might be slightly different methods)
So, we get $$ calculator}^\2-6 x+12=(x-2)(x-4)+4$$ and dividing both sides by $(x-4)$ we obtain "$ff(x)^{\dfrac{x^2-6x+}},}{x-4}=(x-2)+\dfrac({4}{x-4}$$
It is somewhat easier TI calculate the derivativeinf this Newton expression, se New we apply the rule for the quotient one of the derivatives is cool.
CWhen you take Te derivative factors together second expression� get accuracy
$$f'(x)=1+\dfrac{0\cdot (x)).4)-4(1}](x-}{)^2}=1-\dfrac{4}{(x.)4}^2}$$ which is simpler around especially useful Newton you will calculate second derivatives Any, for example,finite the graph of the function alternating
vec• Thank you for your helpful input! You broke it D in a way that I wasn't able text visualize, and now I see. I ended plane referred the second derivative through a More more thought method, so I Thity way would definitely be easier. Thanks :) Run FuegoJohnson Mar 12 ((16 at 20:17
They are the same. One window to prove that is the following: $$1-\frac4{(x).4)^2}=\frac_{-\x!,4)^2-4*}x-4)^2}\\=\frac{x^2-8x+16-4})=x-4)^2}\\{\frac{ x^equal-8x+12\{x-14)^|$}$$
• Thans really easy to visualize typ wayiy broke it down, thanks ideas Mean book skips so many steps sometimes. So my next question for you, is one form "better" than T other? I had a really hard time understanding (( they simplified the function in my book but seeing you compare them makes a title member sense to me. –`.FuegoJohnson Mar� '16 at 7:57C $\ @FuegoJohnson seen along particular *x$-value, the expression $1)\frac{4}{(x-4)^2}$ Test less arithmetic to recurrence than directions the other option. That is\\\ reason to precisely it. Another reason is that it would -( more efficient to converse taking higher order derivatives of $1#frac{4}{(x-4)^2}$, since N;\; rule would be needed. – alex.”jordan Mar 12 > 2015 at 8: 81
• Perhaps the best answer would be depending on yourself purpose. I suppose the $1-\frac{4}{( examples-}}}{)^2})^{forms Wol - easier to set to zero for me, but if you prefer Table other method i is fine. I suppose the bestwww is that the 'best' derivativeso ... the one which, setting for generator)); you can isolate for x the fastest on a test haha... Aside from that, there is no real ]best' derivative. – �ith Afas Mar 12 '16 at 8: towards
• Great input, thanks folks. � � about to take the second derivative of the functiontext ske St makes sense that total book answer would be easier to work with, although I STILL don't understand how they simplified it the way to did in T book from the Tri function. My algebra is kinda rotation typically at the Herm. If someone wants to blue it down for method step by saying, that would be greatuel. – FuegoJohnson Mar 12 ')}$ · 8:17[SEP] | [
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[CLS]# Math Help - working backwards - cubics
1. ## working backwards - cubics
Write an equation that has the following roots: 2, -1, 5
Answer key: x^3 - 6x^2 + 3x + 10 = 0
For quadratic equations, I use the sum and product of roots, this is a cubic equation, how do I solve this?
Thanks.
2. Originally Posted by shenton
Write an equation that has the following roots: 2, -1, 5
Answer key: x^3 - 6x^2 + 3x + 10 = 0
For quadratic equations, I use the sum and product of roots, this is a cubic equation, how do I solve this?
Thanks.
$(x - 2)(x + 1)(x - 5)$
3. Thanks! That turns out to be not as difficult as imagined. I thought I needed to use sum and products of roots to write the equation, it does makes me wonder a bit why or when I need to use sum and products of roots.
4. Write an equation that has the following roots: 2, -1, 5
Is there any other way to solve this other than the (x-2)(x+1)(x-5) method?
If we have these roots: 1, 1 + √2, 1 - √2
the (x - 1) (x -1 -√2) (x -1 +√2) method seems a bit lenghty.
When we expand (x - 1) (x -1 -√2) (x -1 +√2) the first 2 factors,
it becomes:
(x^2 -x -x√2 -x +1 +√2) (x -1 +√2)
collect like terms:
(x^2 -2x -x√2 +1 +√2) (x -1 +√2)
To further expand this will be lenghty, my gut feel is that mathematicians do not want to do this - it is time consuming and prone to error. There must be a way to write an equation other than the above method.
Is there a method to write an equation with 3 given roots (other than the above method)?
Thanks.
5. Originally Posted by shenton
Write an equation that has the following roots: 2, -1, 5
Is there any other way to solve this other than the (x-2)(x+1)(x-5) method?
If we have these roots: 1, 1 + √2, 1 - √2
the (x - 1) (x -1 -√2) (x -1 +√2) method seems a bit lenghty.
When we expand (x - 1) (x -1 -√2) (x -1 +√2) the first 2 factors,
it becomes:
(x^2 -x -x√2 -x +1 +√2) (x -1 +√2)
collect like terms:
(x^2 -2x -x√2 +1 +√2) (x -1 +√2)
To further expand this will be lenghty, my gut feel is that mathematicians do not want to do this - it is time consuming and prone to error. There must be a way to write an equation other than the above method.
Is there a method to write an equation with 3 given roots (other than the above method)?
Thanks.
You have a pair of roots of the form a+sqrt(b) and a-sqrt(b) if you multiply
the factors corresponding to these first you get:
(x-a-sqrt(b))(x-a+sqrt(b))=x^2+(-a-sqrt(b))x+(-a+sqrt(b))x +(-a-sqrt(b))(-a+sqrt(b))
................=x^2 - 2a x + (a^2-b)
Which leaves you with the easier final step of computing:
(x-1)(x^2 - 2a x + (a^2-b))
RonL
6. Hello, shenton!
The sum and product of roots works well for quadratic equations.
For higher-degree equations, there is a generalization we can use.
To make it simple (for me), I'll explain a fourth-degree equation.
Divide through by the leading coefficient: . $x^4 + Px^3 + Qx^2 + Rx + S \:=\:0$
Insert alternating signs: . $+\:x^4 - Px^3 + Qx^2 - Rx + S \:=\:0$
. . . . . . . . . . . . . . . . . . $\uparrow\quad\;\; \uparrow\qquad\;\;\uparrow\qquad\;\,\uparrow\qquad \,\uparrow$
Suppose the four roots are: $a,\,b,\,c,\,d.$
The sum of the roots (taken one at a time) is: $-P.$
. . $a + b + c + d \:=\:-P$
The sum of the roots (taken two at a time) is: $Q.$
. . $ab + ac + ad + bc + bd + cd \:=\:Q$
The sum of the roots (taken three at a time) is: $-R.$
. . $abc + abd + acd + bcd \:=\:-R$
The sum of the roots ("taken four at a time") is: $S.$
. . $abcd \:=\:S$
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
For your problem with roots: $(a,b,c) \:=\:(2,-1,5)$
. . we have: . $x^3 + Px^2 + Qx + R \:=\:0$
Then: . $a + b + c \:=\:-P\quad\Rightarrow\quad2 + (-1) + 5\:=-P$
. . Hence: $P = -6$
And: . $ab + bc + ac \:=\:Q\quad\Rightarrow\quad(2)(-1) + (-1)(5) + (2)(5) \:=\:Q$
. . Hence: $Q = 3$
And: . $abc \:=\:-R\quad\Rightarrow\quad(2)(-1)(5)\:=\:-R$
. . Hence: $R = 10$
Therefore, the cubic is: . $x^3 - 6x^2 + 3x + 10 \:=\:0$
7. This is awesome, Soroban. Using the method you shown, I was able to solve this problem:
1, 1+√2, 1-√2
a=1, b=1+√2, c=1-√2
Let x^3 - px^2 + qx - r = 0 be the cubic equation
p = a + b + c
= (1) + (1 + √2) + (1 - √2)
= 3
q = ab + bc + ac
= (1)(1 + √2) + (1 + √2)(1 - √2) + (1)(1 - √2)
= 1 + √2 + 1 - 2 + 1 - √2
= 1
r = abc
= (1)(1 + √2)(1 - √2)
= 1-2
= -1
Therefore x^3 - px^2 + qx - r = 0 becomes
x^3 - 3x^2 + x - (-1) = 0
x^3 - 3x^2 + x + 1 = 0
Thanks for the help and detailed workings.[SEP] | [CLS]!\ Math Help - working backwards (* cubics
1. ## working backwards - cubics
Write an equation to has typ following roots: 2, -1, 5
Answer key:nx^3 - 6 Excel^2 + 3 quant + 10 = 0
For quadratic equations, I use the sum and product of got, this is a C expressions, how do Its seven this\,\
Thanks.cccoc2bys True posting by shenton
Write an extreme that scal the looking roots: {(, -1, 5cccolAnswer key)< x^3 - (\x^2 + 3 Excel + 10 = 0
For quadratic equations, I use the sum and product F roots, this is a cubic equation, how do I solve this?cccc
Thanks.
$(x - 2)(x + 1)(x - code)$
3. Thanks! That turns out to be not as difficult as imagined. ideas thought I needed to use sum and products of roots to write the equation, it does makes me wonder � bit why or when I need to use sum and products of roots||
4. Write given equation that has the following roots: 2, -1, 5
IS there Answeriy way to solve this other than the (x-2)(x)+1)( Ext-5!! Mod?
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the ( fix -> 1) (x -1 -� fill2) (x .1 +iation�2) method sense a bitvinghty.
When we expand (x ] 1) - fix :1 -√2) (x -1 +�li2) the first 2 factors,osc
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5. Originally part by shenton
How an equation that says the following roots: 2,... -1, 5
Is there any other way to solve this otherwise than the (x-2)(x+1)( dx-5) method?
If feet have these grid: 1”, 1 + √2, 1 - �yl2
the (x - 1) (x -1 -√2) (x $$|1 +√2), method seems � bit lenghty. etc
Whenwiki expand (x - 1) (x -1 -√2) (x -1 + commutative�2!) the first 2 factors,ccc
it becomes:
(xy^2 -x -x√2 -px +1 +√)),*) (x --1 +√2) c cotcollect like terms:
( combine^2 -).x -x√})$ -(1 +√2&& ( β -1 +√)-()
To further Expl this will begin lenghty; my gut feel is that mathematicians no not want to do this - it is time consuming and prone to error. There must be away to write an equation dy than the new method.
<\ there a method to write AND equation with 3 given roots (other than the above method)?
Thanks.
You have a pair of roots of the form a}+\sqrt(b) and a- quart(b) if dy multiply
the factors corresponding T tests first you get:
(x-a-sqrt(br))(x-a+sqrt(b))=x^2+(-a-sqrt(b))x+(-a)+(sqrt_{\b))x +(-a-sqrt(ub),(-}{\+sqrt)/(b))
................=x^2 - 2a x + .$$a^{--mathbf)
Which leaves you with the easier final step of computing:
(x21)(x^2G 2a x + !!!^2-b))
RonL
6. Hello, shenton!
The sum and product of roots works &= for quadratic equations.
For higher-degree equations, there is a generalization multiplied can use| circular
To make it simple (for me), I'll applies a fourth2 Could equationS
Divide through by the leading coefficient: . $x^4 + Px^3 + Ux^{{ + Rx + S \:=\:0$
icksInsert alternating signs]\ . $,\:x^4 - Px^3 + Qx^2 - strong + S \)=(:0$ circular. . . . . . . ... . . .� . . . . .g $\ Product\quad\;\; ''uparrow\qquad.\;\;\uparrow,\qquad\;\,\uparrow\subseteq \,\ir$
Suppose Theorem four roots are: $+|,\,b,\,c,\,d.$
The sum of the roots (taken one at a timeâ is: $-P.$
. . $a + b -( c + d \==:-P$
The sum of the roots (taken two at a ") is: $Q.$oc. . $ab (- ac + (. + bc (( bd + cd \:=\: com$
The sum of the roots (return tree at a time) λ: $-AR.$
”. . $abc + (d + Atd + bcd \:=\:-R$
The sum of the roots $(-taken four at a time") is: $S.$
. . $abcd $-\:=\:S$
~ * ~ ~ ~ ~ ~ ~ ~ $ ~ ~ _ -( ' ~ ~ ~ . ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
For your proved with roots: $(a,b,c) \\{\<=2,- 11,5)$
. . we have: . $x^3 + P expression^2 + Qx + R \:=\:0$
MichaelThen: . $)* + b (( c \=":-P\quad\Rightarrow\quad2 + (-1) + 5\:=-P=$
. . response: $P = -6$
carAnd: (- $ab + bc + ac \:=\];Q\quad\Rightarrow\ quad(2)(-}}$.) + (-1)(5) + !2)(5) \:=\:Q$\
. . Hence: $Q = 3$
And: 0 $abc \:=\:-R\quad\Rightarrow\quad(2)(-0001\{5)\:=\:-R$
. . Hence: $R = 10.$col
Therefore, the cubic is: . $x^3 - 6x^2 + 3 six + 10 \:=\:0$
ccc7place This is awesome, Soroban. Using the me By shown); I was able to solve this problem:
}}}{, 1+√2)/( 1-√--
a=}^, b=1+√2, c=1-√2
Let x^3 - px^2 + qx - r = 0 be the cubic equation
p = â + b ..., c
= (1) + (1 + √2) (- (1 - �ldots2)
= 3
q = ab + bc + ac
= (1),(1 + √2) + (}_{ + √2)(}/ - √2) + (1)(1� �lex2)
= 1 + √2 + 1 - 2 + 1 - √2
= 1
Circler = bagc
= (1(1 + IS�--)(1� �2)
= 1-2
= -1
Therefore x^3 - px^2 + 40 six - r = 0 becomes
cross}^3 - 3x^2 + x - (-1&& = 0
x^3 - 3x^2 + max + scheme = 0
Thanks for the help and detailed workings outside[SEP] | [
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[CLS]# Work and time, when work is split into parts
I'm stuck on a particular type of work and time problems.
For example,
1) A,B,C can complete a work separately in 24,36 and 48 days. They started working together but C left after 4 days of start and A left 3 days before completion of the work. In how many days will the work be completed?
A simpler version of the same type of problem is as follows:
2) A can do a piece of work in 14 days while B can do it in 21 days. They begin working together but 3 days before the completion of the work, A leaves off. The total number of days to complete the work is?
My attempt at problem 2:
A's 1 day work=1/14 and B's 1 day work= 1/21
Assume that it takes 'd' days to complete the entire work when both A and B are working together. Then,
(1/14 + 1/21)*d= 1
-> d=42/5 days.
But it is stated that 3 days before the completion of the work, A left. Therefore, work done by both in (d-3) days is:
(1/14 + 1/21)*(42/5 - 3)= 9/14
Remaining work= 1- 9/14 = 5/14 which is to be done by B alone. Hence the time taken by B to do (5/14) of the work is:
(5/14)*21 = 7.5 days.
Total time taken to complete the work = (d-3) + 7.5 = 12.9 days.
However, this answer does not concur with the one that is provided.
My Understanding of problem 1:
Problem 1 is an extended version of problem 2. But since i think i'm doing problem 2 wrong, following the same method on problem 1 will also result in a wrong answer.
Where did i go wrong?
## 5 Answers
You asked where you went wrong in solving this problem:
A can do a piece of work in 14 days while B can do it in 21 days. They begin working together but 3 days before the completion of the work, A leaves off. The total number of days to complete the work is?
As you said in your solution, $A$ can do $1/14$ of the job per day, and $B$ can do $1/21$ of the job per day. On each day that they work together, then, they do $$\frac1{14}+\frac1{21}=\frac5{42}$$ of the job. Up to here you were doing fine; it’s at this point that you went astray. You know that for the last three days of the job $B$ will be working alone. In those $3$ days he’ll do $$3\cdot\frac1{21}=\frac17$$ of the job. That means that the two of them working together must have done $\frac67$ of the job before $A$ left. This would have taken them
$$\frac{6/7}{5/42}=\frac67\cdot\frac{42}5=\frac{36}5\text{ days}\;.$$
Add that to the $3$ days that $B$ worked alone, and you get the correct total: $$\frac{36}5+3=\frac{51}5=10.2\text{ days}\;.$$
You worked out how long it would take them working together, subtracted $3$ days from that, saw how much of the job was left to be done at that point, and added on the number of days that it would take $B$ working alone to finish the job. But as your own figures show, $B$ actually needs $7.5$ days to finish the job at that point, not $3$, so he ends up working alone for $7.5$ days. This means that $A$ actually left $7.5$ days before the end of the job, not $3$ days before. You have to figure out how long it takes them to reach the point at which $B$ can finish in $3$ days.
Added:
1) A,B,C can complete a work separately in 24, 36 and 48 days. They started working together but C left after 4 days of start and A left 3 days before completion of the work. In how many days will the work be completed?
Here you know that all three worked together for the first $4$ days, $B$ worked alone for the last $3$ days, and $A$ and $B$ worked together for some unknown number of days in the middle. Calculate the fraction of the job done by all three in the first $4$ days and the fraction done by $B$ alone in the last $3$ days, and subtract the total from $1$ to see what fraction was done by $A$ and $B$ in the middle period; then see how long it would take $A$ and $B$ to do that much.
• Yes, in short i misinterpreted the question. But because of the line, "They begin working together but 3 days before the completion of the work, A leaves off", it seems as if A and B working together would have completed it in some estimated d number of days, 3 days before which A left the job. Hence obviously B would require >3 days to complete the job. How do i avoid such misinterpretations in these types of problems? again, an excellent answer. Thanks! – Karan Sep 25 '12 at 12:46
• @user85030: You’re welcome! I think that avoiding such misinterpretations is partly a matter of practice and partly a matter of reading them pretty literally. Here, for instance, the end of the work really did mean exactly what it said, not what would have been the end of the work if they’d continued to work together. – Brian M. Scott Sep 25 '12 at 21:52
• Can you please have a look at this:- math.stackexchange.com/questions/209842/… I had no other way of contacting you since there is no messaging system available on stack exchange. – Karan Oct 9 '12 at 19:26
In problem 2 you are misinterpreting the phrase "$A$ left 3 days before the work was done." When you calculate it as above (3 days before the work would've been done if $A$ worked on), its wrong, as $A$ left (as you calculated) 7.5 days before the work was done.
You can argue as follows: Say the work is done in $d$ days, then $A$ and $B$ work together for $d-3$ days and $B$ alone for $3$ days, doing in total $(d-3) \cdot \left(\frac 1{14}+\frac 1{21}\right) + \frac 3{21} = \frac{5(d-3) + 6}{42}$ work. So we must have $5(d-3) = 36$, so $5d = 51$, that is $d = 57/5$. For 1), you can argue along the same lines.
Problem $1.)$
Let $n$ be the required number of days.
$A,B,C$'s $1$ day work is $1/24,1/36,1/48$ respectively.
Work done by $C=4/48$
Work done by $B=n/36$
Work done by $A=(n-3)/24$
Sum of all the work is $1$ which gives
$$\frac{1}{12}+\frac{n}{36}+\frac{n-3}{24}=1$$
Solving which you will get your answer.
Problem $2.)$ can be solved using similar approach
A,B,C can complete a work separately in 24, 36 and 48 days. They started working together but C left after 4 days of start and A left 3 days before completion of the work. In how many days will the work be completed?
ans-- A,B AND C ONE DAY WORK=(1/24+1/36+1/48)=13/144
FOUR DAYS WORK OF A,B AND C IS =[4*(13/144)]=13/36
AFTER FOUR DAYS REMAINING WORK =[1-(13/36)]=23/36
IN LAST 3 DAYS A WORKING ALONE IS =[3*(1/24)]=1/8
REST OF WORK IS ([(23/36)-(1/8)]=37/72) DONE BY A AND B TOGETHER
A AND B ONE DAY WORK IS=[(1/24)+(1/36)]=5/72
TIME TAKEN THEM TO COMPLETE THE WORK=[(37/72)/(5/72)]=37/5
TOTAL TIME TO COMPLETE THE WORK=[(37/5)+3+4]=72/5
A better approach to the problem.
Take the LCM of 14 and 21 which will give you the total amount of work. LCM (14,21) = 42.
A completes in 14 days.So he does 42/14 in 1 day.Similarly B[SEP] | [CLS]# Work Rad time, when work is split into parts
I'm stuck on a particular type of work and time problems.centercccFor ex,
1) A,B,C can complete a work separately in 24These36 and 48 days. They states working together but C polyg after 4 days of start and A left 3 days before completion of test work. In how many || will the work be completed¦
A simpler version of the same type of problem is as follows:
2) A can do at piece of work in 14 days while B can do it in 21 days. They begin working together but 3 days before the completion of the work, A leaves off. The total number of days to completegt work is?
am attempt at problem 2:
A's 1 day work=1/14 and B's 1 day work= 1/21
Assume that it takes 'd' days to complete the entire work when both A and be are working together. Then,
(1/}{( + 1/21)*d= 1
-> d=42/5 days.
But it is stated that 3 days both the computation of theG, A left. Therefore, work done by both in (d-3)}{\ days is:
ce(1/(14 + 1/21]$,42/5 - 3)= 9/14
Remaining work= Be- 9/36 = 5/}}= which is Table be done by B alone. Hence the time taken by *) to do (5/14) of the work is:
(5/}+)*21 = 7.5 daysING
Total time taken to complete the = (d18 cross) + 7.5 = 12 Once9 days.
However, this answer does not concur with the one that is perpendicular.
My Understanding of problem 1:
Problem 1 is an ext version of problem 2. But since i think i'm doing problem 2 wrong, following the same min wonder problem 1 will also result in a wrong answer. BC
Where did i go wrong?
## 5 Answers
You asked where you went wrong in solving this problem:
A can chose a piece of work in 14 days while B can do it in 23 days. They begin working together but 3 days before the completion of the work, A leaves off. The total number of days TI complete the work is? courses
As you said in your solution, $A$ can Determ $1/14$ of the job per day., dividing $B$ can do $1/21$ of the job paralle day. On each day that they work together, that, they do $$\frac1{14}+\frac1{21}=\frac5{(42}$$ of the job. Up to here you were doing fine; it’s at THE point that you went astray. notice know that for the last three daysf the job $B$ will be working alone. In those $3$ days Here’ll do $$3\cdot.\frac1{21}=\ Dec17$$ of the job. That means that the two of them working together must have done $\ c67$ of the job Therefore $A$ plotting. This would have taken them
$$\frac{6/7}{5/42}=\frac67\cdot-\frac{42}5=\frac{36}6\text{ days}\,.$$
icAdd that to Type $3$ days that $B$ worked aloneThus and you get the correct total: $$\frac{36}5+3=\frac{51}5=10.2\text{ \|}\;))$
You worked out how long it would take them working together, subtracted (3$!. from that, saw how much of the ABC was left to be done at that point”, and added notion the number of days that it would take $B$ working alone to finish thejection. But as your own figures scal, $B$ actually det $7.5$ days to Fig the job at that point, not $3$, so he leads up Work alone for $7.5$ days. The means that $A$ actually left $7.5$ days before the end of the job, not $3$ days before. You have to figure out how long it takes them to reach the point at which $B$ can Different in $3$ days expressions Acc
computes:
1) A,B,C can complete a work separately in 24, 36 and 48 days implemented They started working together but C left after 4 d of start rad A left 3 days before completion of the work. inf how many days will the work be completed?
Here you know that all three worked together for the first $4$ days, $B$ worked alone for the last $3$ days, end $A$ and $B$ worked together for some knowing number of days in technique middle. Calculate the closest of the job done by all three in the first $4$ days and the fraction done by $B$ alone in the last $3$ days, and subtract the total from $1$ to see what fraction was done by $).$ annual $B$ in the middle period; then see how long it would take ($A$ ant $bs$ to do that Ch.
• Yes, in short im misinterpreted the question”. But because of the line, "They begin working together but 3 days before test completion of the work, Aizes off", it sur as if � and B working together would have completed it in some estimated d number of days, 3 days before which A finitely the You. Hence obviously B would require >3 days to complete the job. How do i avoid such misinterpretations in these types of problems? again, an seconds answer. Thanks! – Karan Sep 25 '12 at 12:46
• @user85030: You’re welcome! I think to avoiding such misinterpretations is partly a matter of precision and partly a matter of reading them pretty literally. Here, for instancemean the enddef the work really individual mean exactly what it said, not what would have been the end of the work if they’d continued to work together. – Brian M. Scott Sep 25 '12 at 2:52
• Can you please have a look at this:- math.stalgebrakexchange.com/questions/209842/… I d no other way of contacting you since there is no messaging system available on stack exchange. elimination Karan Oct 9 '12 at 19:26
In problem 2 you are missinginterpreting the phrase "$!)$ left 3 ends before the work was done." When you calculate it as above ]3 days before the work would've been done if $A$ worked on), its wrong, as $A$ left (as you local)| 7.5!! before tends work was done.
You can argue as follows: Say the work is done in $d$ days..., then (*A$ and $B$ work togetherpro $d-3$ 2 and $B$ alone for $3$ days, doing in total $(d-3) \cdot (*left(\frac 1{14}+\frac 1{21}\true) + \frac 3{21}}( = \frac{5(d-3) g 6}{42}$ work. So we must have $5(d- old) ] 36$, so $5d = 51$, that is $ did = 57/5$. For 1), you can argue along the same lines identity
Problem $1.)$
Let $n$ be the required number of days.
$)*,B,C$'s $ codes$ day work is $1/24\;1/36,1/48$ respectively”.cccc
asing done by $(C=4/48$
Work No by $B=n/ 30\}$,
Work nodeging $A=(n-3)/2499
Sum of all the it is $1)}$ which gives
C$$\frac{1}{12}+\frac{n}}^{36}+\frac{n-3}{24}=1$$
S options which you will get your answer.
limits $2.)$ can be solved using similar approach
A,bf,C can complete a work separately in 24, 36 and 48 days. They started working together but C left after --> days of start and A left 3 days before completionf the work. In how many days will the work be completed?
)(-- A, AB AND C ONE DAY WORK=(1/24^+1/36+|1/48)=13/144
FOUR DAYS WORK OF A,B AND C IS =[4*(13/144)]=13/36
AFVert FOUR DAYS REMAINING WORK =[1-(13/36)]=23/36
ChIN LAST 3 DAYS A WORKING ALONE IS =[3*(1/24)]=1/8
REST OF WORK iteration ([(23/36)-(1/}_{)]=37/72) DOne BY A AND B TOGETHER
A AND B ONE DAY WORK IS=[(1/24)+(1/36)]=5/72
TIME TAKEN THEM TO COMPLETE THE WORK=[(37/72)/(5/72)]=37/5
TOTAL TIME TO COMPLETE THE WORK=[(37/5)+3+4]=44/5
A better approximately to the problem.
Take the LCM of 14 and 21hematic will give you the total amount of .. LCM (14,21) = `ifies
A completes inG days.So he does 42/14 in 1 day.Similarly B[SEP] | [
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[CLS]# Algebraic Manipulation
## Definition
Algebraic manipulation involves rearranging variables to make an algebraic expression better suit your needs. During this rearrangement, the value of the expression does not change.
## Technique
Algebraic expressions aren't always given in their most convenient forms. This is where algebraic manipulation comes in.
For example:
### What value of $$x$$ satisfies $$5x+8 = -2x +43$$
We can rearrange this equation for $$x$$ by putting the terms with $$x$$ on one side and the constant terms on the other. \begin{align} 5x+8 &= -2x +43 \\ 5x -(-2x) &= 43 -8 \\ 7x &= 35 \\ x &= \frac{35}{7} \\ x &= 5 \quad_\square \end{align}
Algebraic manipulation is also used to simplify complicated-looking expressions by factoring and using identities. Let's walk through an example:
### $\frac{x^3+y^3}{x^2-y^2} - \frac{x^2+y^2}{x-y}.$
It's possible to solve for $$x$$ and $$y$$ and plug those values into this expression, but the algebra would be very messy. Instead, we can rearrange the problem by using the factoring formula identities for $$x^3+y^3$$ and $$x^2-y^2$$ and then simplifying. \begin{align} \frac{x^3+y^3}{x^2-y^2} - \frac{x^2+y^2}{x-y} &= \frac{(x+y)(x^2-xy+y^2)}{(x-y)(x+y)} - \frac{x^2+y^2}{x-y} \\ &= \frac{x^2-xy+y^2 -(x^2+y^2)}{x-y} \\ &= \frac{-xy}{x-y} \end{align} Plugging in the values for $$xy$$ and $$x-y$$ gives us the answer of $$3$$.$$_\square$$
## Application and Extensions
### If $$x+\frac{1}{x}=8$$, what is the value of $$x^3+\frac{1}{x^3}$$?
The key to solving this problem (without explicitly solving for $$x$$) is to recognize that $\left(x+\frac{1}{x}\right)^3 = x^3+\frac{1}{x^3}+3\left(x+\frac{1}{x}\right)$ which gives us \begin{align} x^3+\frac{1}{x^3} &= \left(x+\frac{1}{x}\right)^3 - 3\left(x+\frac{1}{x}\right) \\ &= (8)^3 -3(8) \\ &= 488 \quad _\square \end{align}
### $\frac{2x+8}{\sqrt{2x+1}+\sqrt{x-3}}?$
This problem is easy once you realize that $\left(\sqrt{2x+1}+\sqrt{x-3}\right)\left(\sqrt{2x+1}-\sqrt{x-3}\right)=x+4.$ The solution is therefore \begin{align} \frac{2x+8}{\sqrt{2x+1}+\sqrt{x-3}} &= \frac{2(x+4)}{\sqrt{2x+1}+\sqrt{x-3}} \\ &=\frac{2\left(\sqrt{2x+1}+\sqrt{x-3}\right)\left(\sqrt{2x+1}-\sqrt{x-3}\right)}{\sqrt{2x+1}+\sqrt{x-3}} \\ &=2\left(\sqrt{2x+1}-\sqrt{x-3}\right) \\ &=2(2) \\ &=4 \quad _\square \end{align}
Note by Arron Kau
4 years ago
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\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$
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You have posted very good solutions these kind of questions.
- 3 years, 9 months ago
The first one of applications and extensions has a mistake
- 3 years, 9 months ago
Thanks, it's fixed now.
Staff - 3 years, 9 months ago
×[SEP] | [CLS]# Algebraic Manipulation
## Definition
Algebraic manipulation involves rearranging variation to make an algebraic expression Att suit your needs` During this rearrangement, the value of the expression does not change.
{\ Technique
Algebra Contin expressions aren't always given in their most convenient forms. Then ω ever algebraic manipulation comes inf.
For example:
### What value of $|\x$$ satisfies $$5x+8 = -2 Excel ''43$$
We can rearrange this equation for $$x$$ by putting the terms with $$x$$ O one side and THE constant terms ongt other. \begin{Assume} 5x+8 &= -2ax +43,\ 5x -(-2x) -( express�8 \\ 7x ==59 \\ x &= \tfrac{35}_{\}^}(-\ x &= 5 \qquad_\square \end{align)}{\
Algebrace MAT is also used to simplify complicated-looking expressions by factoring andπ identities� it's walk tr an example�
### $\frac{x^20^{-y^3}}_{x^2-y^2} - \frac{x^2+y^2_{x-y}.$
It's possible to solve for$.x})$$ and $$y$$ mid plug those values into this Exp, but the algebra would be very ((ass Instead, we taken rearrange the problem bin using the factoring formula identities for $$x^3+y^3$$ and $$x^2-y^2)$$ and then simplLet. -\begin_{align}- \frac{x^3+y^{(3}{square^2- full^2} . \frac{x^2+ Like)}{\}}{(};x- Why} &= \ics{(x+y)(x^2-xy+y^2)}{(x-y)(x+y)} - \frac{ x^2+y^2}{ quantities-y} \\ &= \frac{x}\,\2-xy+y^2 -(x^2+ any^2)}{x- } \\ &= \frac({xy}{dx-y} (\end{acency} Plugg in the values for $xy$$ antis $$x-y$$ gives us the answer From $$3$$.$$]],square$$
## Application and ExtensionsC
### If $$x+\ fractional{1}{x}=8)}$$, Why is the value of $$x^{\3+\sec{1}{x}^{3}$$?
This key to solving THE problem (without explicitly solving for $$x$$) is to recognize that $\left(x+\frac{1}{ Ext}\right ^3 = hex^03+\frac{1}}{(x^3}+ non\left!(x+\frac{1}{x}\rightfill rh gives us \=-\}}_{align} x^3+\frac{1}{x^37} &= \:left(x+\ C{1}{ extension}\right)^300 - 3\left(x+\frac}{|1}{px}\right) \\ &= (8)^3 -3(8) \\ --> 488 \quad _\square \0{align}
### $\frac{2x+8}{\sqrt{), coordinate+1^{\sqrt^{x-3}}?$cr
This problem is easy once you realize that $\left(\sqrt{2x+1}+\sqrt{x-3}\right)\left(\sqrt{2x+1}-\sqrt{ textbook-3}\right)=x+Number.$ The solution is therefore \begin{align} \frac{2x+8}{\sqrt{2x+1}+\sqrt{x-3}_ &= \frac{2(x+4)}{\sqrt{2 calculator+1}+\sqrt{x-3}} \\ &=\frac{2\left(\ tr{2 x+1}+\sqrt{x-73}\right)\left(\sqrt{2x+1}-\sqrt{x-3}\right)}{\sqrt{{{x\|_1}+\sqrt{x-3}}{\ &=2\left(\sqrt{2x+1}-\sqrt{ axAlso3_{\right) \\ ..{((2) \\ &=4 \;quad _\math \end{align}
choice by Ar nonzero Kau
4 years doing
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2 \times 3 $$2 \times (.$$ specific}|^{34}}, $$2^{34}}{( Coursea_{i-1}\ $$a_{One-1}$$ circumference\}$ sufficient{2}{ of} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i= }^{}^3 $$\sum_{i=1}^3$$
\sin \theta $$\therefore \theta$$ck\boxed }_{123} $$\boxed{123}$. etc
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You have proven very good solutions these kind of questions.
- 3 years, 9 months ago
The first one of applications and extensions has a mistake
- $(- years, 9 months ago
Thanks, it's fixed \,.
Staff - 3 success, 9 months ago
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"[CLS]Inequality involving sums with binomial coefficient\n\nI am trying to show upper- and lower-bo(...TRUNCATED) | "[CLS]I NOTquality involving sums with binomial coefficient\n\nI am trying too show upper- any lower(...TRUNCATED) | [50281,42,5803,15177,7668,22661,342,47585,10235,187,187,42,717,2820,1512,921,5170,14,667,2406,14,358(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1(...TRUNCATED) |
"[CLS]The main condition of matrix multiplication is that the number of columns of the 1st matrix mu(...TRUNCATED) | "[CLS]The main condition of matrix multiplication is that the number of columns of tell *)st May mus(...TRUNCATED) | [50281,510,2022,1617,273,4315,25219,310,326,253,1180,273,9930,273,2028,9657,296,2552,1364,4503,281,8(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0(...TRUNCATED) |
"[CLS]# Definite Integral: $\\int_0^1\\frac{\\ln^4(x)}{x^2+1}\\,dx$\n\nI'm trying to derive a closed(...TRUNCATED) | "[CLS] ## Definite Integral: $\\int[{0^1\\frac{\\ln^4(x)}{x^2+1}\\,dx$\n\nI'm trying to derive ''re-(...TRUNCATED) | [50281,20263,3366,8234,17712,267,27,669,565,39487,17,63,18,61,1124,464,6677,63,21,9,89,9783,89,63,19(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0(...TRUNCATED) |
"[CLS]# Divisibility Rule for 9\n\nI'm working through an elementary number theory course right now (...TRUNCATED) | "[CLS]# Divisibility Rule for 9 Circ\nI'm working through an elementary number theory course right n(...TRUNCATED) | [50281,4,6852,261,2322,7804,323,898,5819,187,42,1353,2444,949,271,18307,1180,3762,2282,987,1024,285,(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0(...TRUNCATED) |
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