Word problem and their solution

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This article is about algorithmic word problems in mathematics and computer science. For other uses, see Word problem.

In computational mathematics, a word problem is the problem of deciding whether two given expressions are equivalent with respect to a set of rewriting identities. A prototypical example is the word problem for groups, but there are many other instances as well. A deep result of computational theory is that answering this question is in many important cases undecidable.[1]

Background and motivation[edit]

In computer algebra one often wishes to encode mathematical expressions using an expression tree. But there are often multiple equivalent expression trees. The question naturally arises of whether there is an algorithm which, given as input two expressions, decides whether they represent the same element. Such an algorithm is called a solution to the word problem. For example, imagine that x,y,z are symbols representing real numbers — then a relevant solution to the word problem would, given the input {displaystyle (xcdot y)/zmathrel {overset {?}{=}} (x/z)cdot y}, produce the output EQUAL, and similarly produce NOT_EQUAL from {displaystyle (xcdot y)/zmathrel {overset {?}{=}} (x/x)cdot y}.

The most direct solution to a word problem takes the form of a normal form theorem and algorithm which maps every element in an equivalence class of expressions to a single encoding known as the normal form — the word problem is then solved by comparing these normal forms via syntactic equality.[1] For example one might decide that {displaystyle xcdot ycdot z^{-1}} is the normal form of {displaystyle (xcdot y)/z}, {displaystyle (x/z)cdot y}, and {displaystyle (y/z)cdot x}, and devise a transformation system to rewrite those expressions to that form, in the process proving that all equivalent expressions will be rewritten to the same normal form.[2] But not all solutions to the word problem use a normal form theorem — there are algebraic properties which indirectly imply the existence of an algorithm.[1]

While the word problem asks whether two terms containing constants are equal, a proper extension of the word problem known as the unification problem asks whether two terms t_{1},t_{2} containing variables have instances that are equal, or in other words whether the equation t_{1}=t_{2} has any solutions. As a common example, {displaystyle 2+3mathrel {overset {?}{=}} 8+(-3)} is a word problem in the integer group ℤ,
while {displaystyle 2+xmathrel {overset {?}{=}} 8+(-x)} is a unification problem in the same group; since the former terms happen to be equal in ℤ, the latter problem has the substitution {xmapsto 3} as a solution.

History[edit]

One of the most deeply studied cases of the word problem is in the theory of semigroups and groups. A timeline of papers relevant to the Novikov-Boone theorem is as follows:[3][4]

  • 1910: Axel Thue poses a general problem of term rewriting on tree-like structures. He states «A solution of this problem in the most general case may perhaps be connected with unsurmountable difficulties».[5][6]
  • 1911: Max Dehn poses the word problem for finitely presented groups.[7]
  • 1912: Dehn presents Dehn’s algorithm, and proves it solves the word problem for the fundamental groups of closed orientable two-dimensional manifolds of genus greater than or equal to 2.[8] Subsequent authors have greatly extended it to a wide range of group-theoretic decision problems.[9][10][11]
  • 1914: Axel Thue poses the word problem for finitely presented semigroups.[12]
  • 1930 – 1938: The Church-Turing thesis emerges, defining formal notions of computability and undecidability.[13]
  • 1947: Emil Post and Andrey Markov Jr. independently construct finitely presented semigroups with unsolvable word problem.[14][15] Post’s construction is built on Turing machines while Markov’s uses Post’s normal systems.[3]
  • 1950: Alan Turing shows the word problem for cancellation semigroups is unsolvable,[16] by furthering Post’s construction. The proof is difficult to follow but marks a turning point in the word problem for groups.[3]: 342 
  • 1955: Pyotr Novikov gives the first published proof that the word problem for groups is unsolvable, using Turing’s cancellation semigroup result.[17][3]: 354  The proof contains a «Principal Lemma» equivalent to Britton’s Lemma.[3]: 355 
  • 1954 – 1957: William Boone independently shows the word problem for groups is unsolvable, using Post’s semigroup construction.[18][19]
  • 1957 – 1958: John Britton gives another proof that the word problem for groups is unsolvable, based on Turing’s cancellation semigroups result and some of Britton’s earlier work.[20] An early version of Britton’s Lemma appears.[3]: 355 
  • 1958 – 1959: Boone publishes a simplified version of his construction.[21][22]
  • 1961: Graham Higman characterises the subgroups of finitely presented groups with Higman’s embedding theorem,[23] connecting recursion theory with group theory in an unexpected way and giving a very different proof of the unsolvability of the word problem.[3]
  • 1961 – 1963: Britton presents a greatly simplified version of Boone’s 1959 proof that the word problem for groups is unsolvable.[24] It uses a group-theoretic approach, in particular Britton’s Lemma. This proof has been used in a graduate course, although more modern and condensed proofs exist.[25]
  • 1977: Gennady Makanin proves that the existential theory of equations over free monoids is solvable.[26]

The word problem for semi-Thue systems[edit]

The accessibility problem for string rewriting systems (semi-Thue systems or semigroups) can be stated as follows: Given a semi-Thue system T:=(Sigma ,R) and two words (strings) u,vin Sigma ^{*}, can u be transformed into v by applying rules from R? Note that the rewriting here is one-way. The word problem is the accessibility problem for symmetric rewrite relations, i.e. Thue systems.[27]

The accessibility and word problems are undecidable, i.e. there is no general algorithm for solving this problem.[28] This even holds if we limit the systems to have finite presentations, i.e. a finite set of symbols and a finite set of relations on those symbols.[27] Even the word problem restricted to ground terms is not decidable for certain finitely presented semigroups.[29][30]

The word problem for groups[edit]

Given a presentation {displaystyle langle Smid {mathcal {R}}rangle } for a group G, the word problem is the algorithmic problem of deciding, given as input two words in S, whether they represent the same element of G. The word problem is one of three algorithmic problems for groups proposed by Max Dehn in 1911. It was shown by Pyotr Novikov in 1955 that there exists a finitely presented group G such that the word problem for G is undecidable.[31]

The word problem in combinatorial calculus and lambda calculus[edit]

One of the earliest proofs that a word problem is undecidable was for combinatory logic: when are two strings of combinators equivalent? Because combinators encode all possible Turing machines, and the equivalence of two Turing machines is undecidable, it follows that the equivalence of two strings of combinators is undecidable. Alonzo Church observed this in 1936.[32]

Likewise, one has essentially the same problem in (untyped) lambda calculus: given two distinct lambda expressions, there is no algorithm which can discern whether they are equivalent or not; equivalence is undecidable. For several typed variants of the lambda calculus, equivalence is decidable by comparison of normal forms.

The word problem for abstract rewriting systems[edit]

Solving the word problem: deciding if x{stackrel {*}{leftrightarrow }}y usually requires heuristic search (red, green), while deciding xdownarrow =ydownarrow is straightforward (grey).

The word problem for an abstract rewriting system (ARS) is quite succinct: given objects x and y are they equivalent under {stackrel {*}{leftrightarrow }}?[29] The word problem for an ARS is undecidable in general. However, there is a computable solution for the word problem in the specific case where every object reduces to a unique normal form in a finite number of steps (i.e. the system is convergent): two objects are equivalent under {stackrel {*}{leftrightarrow }} if and only if they reduce to the same normal form.[33]
The Knuth-Bendix completion algorithm can be used to transform a set of equations into a convergent term rewriting system.

The word problem in universal algebra[edit]

In universal algebra one studies algebraic structures consisting of a generating set A, a collection of operations on A of finite arity, and a finite set of identities that these operations must satisfy. The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras.[1]

The word problem on free Heyting algebras is difficult.[34]
The only known results are that the free Heyting algebra on one generator is infinite, and that the free complete Heyting algebra on one generator exists (and has one more element than the free Heyting algebra).

The word problem for free lattices[edit]

Example computation of x∧z ~ x∧z∧(x∨y)

x∧z∧(x∨y) ≤~ x∧z
by 5. since x∧z ≤~ x∧z
by 1. since x∧z = x∧z
 
 
x∧z ≤~ x∧z∧(x∨y)
by 7. since x∧z ≤~ x∧z and x∧z ≤~ x∨y
by 1. since x∧z = x∧z by 6. since x∧z ≤~ x
by 5. since x ≤~ x
by 1. since x = x

The word problem on free lattices and more generally free bounded lattices has a decidable solution. Bounded lattices are algebraic structures with the two binary operations ∨ and ∧ and the two constants (nullary operations) 0 and 1. The set of all well-formed expressions that can be formulated using these operations on elements from a given set of generators X will be called W(X). This set of words contains many expressions that turn out to denote equal values in every lattice. For example, if a is some element of X, then a ∨ 1 = 1 and a ∧ 1 = a. The word problem for free bounded lattices is the problem of determining which of these elements of W(X) denote the same element in the free bounded lattice FX, and hence in every bounded lattice.

The word problem may be resolved as follows. A relation ≤~ on W(X) may be defined inductively by setting w ≤~ v if and only if one of the following holds:

  1.   w = v (this can be restricted to the case where w and v are elements of X),
  2.   w = 0,
  3.   v = 1,
  4.   w = w1 ∨ w2 and both w1 ≤~ v and w2 ≤~ v hold,
  5.   w = w1 ∧ w2 and either w1 ≤~ v or w2 ≤~ v holds,
  6.   v = v1 ∨ v2 and either w ≤~ v1 or w ≤~ v2 holds,
  7.   v = v1 ∧ v2 and both w ≤~ v1 and w ≤~ v2 hold.

This defines a preorder ≤~ on W(X), so an equivalence relation can be defined by w ~ v when w ≤~ v and v ≤~ w. One may then show that the partially ordered quotient set W(X)/~ is the free bounded lattice FX.[35][36] The equivalence classes of W(X)/~ are the sets of all words w and v with w ≤~ v and v ≤~ w. Two well-formed words v and w in W(X) denote the same value in every bounded lattice if and only if w ≤~ v and v ≤~ w; the latter conditions can be effectively decided using the above inductive definition. The table shows an example computation to show that the words x∧z and x∧z∧(x∨y) denote the same value in every bounded lattice. The case of lattices that are not bounded is treated similarly, omitting rules 2 and 3 in the above construction of ≤~.

Example: A term rewriting system to decide the word problem in the free group[edit]

Bläsius and Bürckert
[37]
demonstrate the Knuth–Bendix algorithm on an axiom set for groups.
The algorithm yields a confluent and noetherian term rewrite system that transforms every term into a unique normal form.[38]
The rewrite rules are numbered incontiguous since some rules became redundant and were deleted during the algorithm run.
The equality of two terms follows from the axioms if and only if both terms are transformed into literally the same normal form term. For example, the terms

{displaystyle ((a^{-1}cdot a)cdot (bcdot b^{-1}))^{-1}mathrel {overset {R2}{rightsquigarrow }} (1cdot (bcdot b^{-1}))^{-1}mathrel {overset {R13}{rightsquigarrow }} (1cdot 1)^{-1}mathrel {overset {R1}{rightsquigarrow }} 1^{-1}mathrel {overset {R8}{rightsquigarrow }} 1}, and
{displaystyle bcdot ((acdot b)^{-1}cdot a)mathrel {overset {R17}{rightsquigarrow }} bcdot ((b^{-1}cdot a^{-1})cdot a)mathrel {overset {R3}{rightsquigarrow }} bcdot (b^{-1}cdot (a^{-1}cdot a))mathrel {overset {R2}{rightsquigarrow }} bcdot (b^{-1}cdot 1)mathrel {overset {R11}{rightsquigarrow }} bcdot b^{-1}mathrel {overset {R13}{rightsquigarrow }} 1}

share the same normal form, viz. 1; therefore both terms are equal in every group.
As another example, the term 1cdot (acdot b) and bcdot (1cdot a) has the normal form acdot b and bcdot a, respectively. Since the normal forms are literally different, the original terms cannot be equal in every group. In fact, they are usually different in non-abelian groups.

Group axioms used in Knuth–Bendix completion

A1 1cdot x =x
A2 x^{{-1}}cdot x =1
A3     (xcdot y)cdot z =xcdot (ycdot z)
Term rewrite system obtained from Knuth–Bendix completion

R1 1cdot x {displaystyle rightsquigarrow x}
R2 x^{{-1}}cdot x {displaystyle rightsquigarrow 1}
R3 (xcdot y)cdot z rightsquigarrow xcdot (ycdot z)
R4 {displaystyle x^{-1}cdot (xcdot y)} {displaystyle rightsquigarrow y}
R8 {displaystyle 1^{-1}} {displaystyle rightsquigarrow 1}
R11 {displaystyle xcdot 1} {displaystyle rightsquigarrow x}
R12 {displaystyle (x^{-1})^{-1}} {displaystyle rightsquigarrow x}
R13 {displaystyle xcdot x^{-1}} {displaystyle rightsquigarrow 1}
R14 {displaystyle xcdot (x^{-1}cdot y)} {displaystyle rightsquigarrow y}
R17     {displaystyle (xcdot y)^{-1}} {displaystyle rightsquigarrow y^{-1}cdot x^{-1}}

See also[edit]

  • Conjugacy problem
  • Group isomorphism problem

References[edit]

  1. ^ a b c d Evans, Trevor (1978). «Word problems». Bulletin of the American Mathematical Society. 84 (5): 790. doi:10.1090/S0002-9904-1978-14516-9.
  2. ^ Cohen, Joel S. (2002). Computer algebra and symbolic computation: elementary algorithms. Natick, Mass.: A K Peters. pp. 90–92. ISBN 1568811586.
  3. ^ a b c d e f g Miller, Charles F. (2014). Downey, Rod (ed.). «Turing machines to word problems» (PDF). Turing’s Legacy: 330. doi:10.1017/CBO9781107338579.010. hdl:11343/51723. ISBN 9781107338579. Retrieved 6 December 2021.
  4. ^ Stillwell, John (1982). «The word problem and the isomorphism problem for groups». Bulletin of the American Mathematical Society. 6 (1): 33–56. doi:10.1090/S0273-0979-1982-14963-1.
  5. ^ Müller-Stach, Stefan (12 September 2021). «Max Dehn, Axel Thue, and the Undecidable». p. 13. arXiv:1703.09750 [math.HO].
  6. ^ Steinby, Magnus; Thomas, Wolfgang (2000). «Trees and term rewriting in 1910: on a paper by Axel Thue». Bulletin of the European Association for Theoretical Computer Science. 72: 256–269. CiteSeerX 10.1.1.32.8993. MR 1798015.
  7. ^ Dehn, Max (1911). «Über unendliche diskontinuierliche Gruppen». Mathematische Annalen. 71 (1): 116–144. doi:10.1007/BF01456932. ISSN 0025-5831. MR 1511645. S2CID 123478582.
  8. ^ Dehn, Max (1912). «Transformation der Kurven auf zweiseitigen Flächen». Mathematische Annalen. 72 (3): 413–421. doi:10.1007/BF01456725. ISSN 0025-5831. MR 1511705. S2CID 122988176.
  9. ^ Greendlinger, Martin (June 1959). «Dehn’s algorithm for the word problem». Communications on Pure and Applied Mathematics. 13 (1): 67–83. doi:10.1002/cpa.3160130108.
  10. ^ Lyndon, Roger C. (September 1966). «On Dehn’s algorithm». Mathematische Annalen. 166 (3): 208–228. doi:10.1007/BF01361168. hdl:2027.42/46211. S2CID 36469569.
  11. ^ Schupp, Paul E. (June 1968). «On Dehn’s algorithm and the conjugacy problem». Mathematische Annalen. 178 (2): 119–130. doi:10.1007/BF01350654. S2CID 120429853.
  12. ^ Power, James F. (27 August 2013). «Thue’s 1914 paper: a translation». arXiv:1308.5858 [cs.FL].
  13. ^ See History of the Church–Turing thesis. The dates are based on On Formally Undecidable Propositions of Principia Mathematica and Related Systems and Systems of Logic Based on Ordinals.
  14. ^ Post, Emil L. (March 1947). «Recursive Unsolvability of a problem of Thue» (PDF). Journal of Symbolic Logic. 12 (1): 1–11. doi:10.2307/2267170. JSTOR 2267170. S2CID 30320278. Retrieved 6 December 2021.
  15. ^ Mostowski, Andrzej (September 1951). «A. Markov. Névožmoinost’ nékotoryh algoritmov v téorii associativnyh sistém (Impossibility of certain algorithms in the theory of associative systems). Doklady Akadémii Nauk SSSR, vol. 77 (1951), pp. 19–20». Journal of Symbolic Logic. 16 (3): 215. doi:10.2307/2266407. JSTOR 2266407.
  16. ^ Turing, A. M. (September 1950). «The Word Problem in Semi-Groups With Cancellation». The Annals of Mathematics. 52 (2): 491–505. doi:10.2307/1969481. JSTOR 1969481.
  17. ^ Novikov, P. S. (1955). «On the algorithmic unsolvability of the word problem in group theory». Proceedings of the Steklov Institute of Mathematics (in Russian). 44: 1–143. Zbl 0068.01301.
  18. ^ Boone, William W. (1954). «Certain Simple, Unsolvable Problems of Group Theory. I». Indagationes Mathematicae (Proceedings). 57: 231–237. doi:10.1016/S1385-7258(54)50033-8.
  19. ^ Boone, William W. (1957). «Certain Simple, Unsolvable Problems of Group Theory. VI». Indagationes Mathematicae (Proceedings). 60: 227–232. doi:10.1016/S1385-7258(57)50030-9.
  20. ^ Britton, J. L. (October 1958). «The Word Problem for Groups». Proceedings of the London Mathematical Society. s3-8 (4): 493–506. doi:10.1112/plms/s3-8.4.493.
  21. ^ Boone, William W. (1958). «The word problem» (PDF). Proceedings of the National Academy of Sciences. 44 (10): 1061–1065. Bibcode:1958PNAS…44.1061B. doi:10.1073/pnas.44.10.1061. PMC 528693. PMID 16590307. Zbl 0086.24701.
  22. ^ Boone, William W. (September 1959). «The Word Problem». The Annals of Mathematics. 70 (2): 207–265. doi:10.2307/1970103. JSTOR 1970103.
  23. ^ Higman, G. (8 August 1961). «Subgroups of finitely presented groups». Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 262 (1311): 455–475. Bibcode:1961RSPSA.262..455H. doi:10.1098/rspa.1961.0132. S2CID 120100270.
  24. ^ Britton, John L. (January 1963). «The Word Problem». The Annals of Mathematics. 77 (1): 16–32. doi:10.2307/1970200. JSTOR 1970200.
  25. ^ Simpson, Stephen G. (18 May 2005). «A Slick Proof of the Unsolvability of the Word Problem for Finitely Presented Groups» (PDF). Retrieved 6 December 2021.
  26. ^ «Subgroups of finitely presented groups». Mathematics of the USSR-Sbornik. 103 (145): 147–236. 13 February 1977. doi:10.1070/SM1977v032n02ABEH002376.
  27. ^ a b Matiyasevich, Yuri; Sénizergues, Géraud (January 2005). «Decision problems for semi-Thue systems with a few rules». Theoretical Computer Science. 330 (1): 145–169. doi:10.1016/j.tcs.2004.09.016.
  28. ^ Davis, Martin (1978). «What is a Computation?» (PDF). Mathematics Today Twelve Informal Essays: 257–259. doi:10.1007/978-1-4613-9435-8_10. ISBN 978-1-4613-9437-2. Retrieved 5 December 2021.
  29. ^ a b Baader, Franz; Nipkow, Tobias (5 August 1999). Term Rewriting and All That. Cambridge University Press. pp. 59–60. ISBN 978-0-521-77920-3.
  30. ^
    • Matiyasevich, Yu. V. (1967). «Простые примеры неразрешимых ассоциативных исчислений» [Simple examples of undecidable associative calculi]. Doklady Akademii Nauk SSSR (in Russian). 173 (6): 1264–1266. ISSN 0869-5652.
    • Matiyasevich, Yu. V. (1967). «Simple examples of undecidable associative calculi». Soviet Mathematics. 8 (2): 555–557. ISSN 0197-6788.

  31. ^ Novikov, P. S. (1955). «On the algorithmic unsolvability of the word problem in group theory». Trudy Mat. Inst. Steklov (in Russian). 44: 1–143.
  32. ^ Statman, Rick (2000). «On the Word Problem for Combinators». Rewriting Techniques and Applications. Lecture Notes in Computer Science. 1833: 203–213. doi:10.1007/10721975_14. ISBN 978-3-540-67778-9.
  33. ^ Beke, Tibor (May 2011). «Categorification, term rewriting and the Knuth–Bendix procedure». Journal of Pure and Applied Algebra. 215 (5): 730. doi:10.1016/j.jpaa.2010.06.019.
  34. ^ Peter T. Johnstone, Stone Spaces, (1982) Cambridge University Press, Cambridge, ISBN 0-521-23893-5. (See chapter 1, paragraph 4.11)
  35. ^ Whitman, Philip M. (January 1941). «Free Lattices». The Annals of Mathematics. 42 (1): 325–329. doi:10.2307/1969001. JSTOR 1969001.
  36. ^ Whitman, Philip M. (1942). «Free Lattices II». Annals of Mathematics. 43 (1): 104–115. doi:10.2307/1968883. JSTOR 1968883.
  37. ^ K. H. Bläsius and H.-J. Bürckert, ed. (1992). Deduktionsssysteme. Oldenbourg. p. 291.; here: p.126, 134
  38. ^ Apply rules in any order to a term, as long as possible; the result doesn’t depend on the order; it is the term’s normal form.
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How to solve word problemsWord problems can be intimidating and overwhelming for children and parents alike. They require children to read at grade level while solving a complex puzzle. Empower your child to tackle those tricky problems by teaching a systematic approach for solving them. Whether it’s a one-step or multi-step word problem, the simple strategies listed below will take the guesswork out of the equation. 😉

3-Step System

1. Read: Read the problem and decide what the question is asking.

  • Read the problem 2 times or more.
  • Underline or circle key words, phrases, and numbers. Draw a line through irrelevant information.

2. Plan: Think about what the story is asking you to do. What information are you given, and what do you need to find out?

  • Draw a picture.
  • Circle or underline key words. (Use highlighters or crayons to color-code key numbers and phrases.)
  • Write out the question in your own words.

3. Solve: What strategy could you use to find the missing information: addition, subtraction, multiplication, or division?

  • Write a number sentence and solve.
  • Use counters.
  • Create charts.

 Check your work by explaining your reasoning. Does your answer make sense?

Download this free strategy checklist from Math Fundamentals to help your child solve word problems. Word problem strategies

Different Strategies to Solve Word Problems

Everyone learns in a different way. What makes sense to one individual often isn’t the easiest option for another. Incorporating different strategies to solve word problems can help your child discover what strategy works best for him or her. A few tips to use are:

1. Circle numbers in a story and underline key phrases.

Color coding is a fun method to incorporate to help children decide what operation the question is asking for. Assign a color to each operation and highlight the phrase that identifies it. For example, red links to addition and blue links to subtraction.

2. Incorporate a key word list.

Key word lists are best used for teaching younger children how to solve word problems. As math curriculum advances, children should not be dependent on a key word list to solve a problem. The questions get trickier.

Addition
In all
Together
Total
Altogether
Combine
Sum
Join

Subtraction
Difference
Fewer
How many more
How much more
Left
Remain
Less

3. Visuals

If your child is a visual learner, drawing a picture or using counters can help him or her understand what the problem is asking. Use number lines, charts, or counters or draw a picture.

4. Write your own word problem.
Knowing what is needed to write a word problem is the first step in identifying key words to solve a story. Take turns writing your own word problems with your child and exchange them to solve.

5. Stay organized.

It is important to write clearly and keep work space neat so children can read and follow their own computations. Many children need a separate piece of paper to allow them enough space to solve and understand their answer. Graphing paper is a great option to help students record neat work.


Download this free sample word problem from Math Fundamentals, grade 1.

How to solve a two-step word problem

In a two-step word problem children are being asking to solve two related equations. These can get tricky for children to understand when they transition from one-step to two-step problems. Help your child understand his or her relationships within two-step word problems with these strategies:

1. Circle important information.

Circle numbers and important phrases that ask questions. The number sentences needed to solve these equations are hidden in those asking questions. Identify the first and second questions needed to solve.

2. Distinguish the two parts of the problem.

First identify the first step of the first part of the word problem. Write a number sentence and solve.

3. Use the answer from the first-step solution to the whole problem.

Use the answer from the first question to help you solve the next equation. What operation does the second question require?

Check your work by explaining your reasoning. What was the question answered? Is the answer reasonable for the question being asked?


Download this free sample two-strategy word problem from Math Fundamentals, grade 2


Download this free sample multi-strategy word problem from Math Fundamentals, grade 4

Evan-Moor’s Math Fundamentals is a great resource for training students how to solve word problems in 3 simple steps. It provides step-by-step directions for solving questions and guides children with helpful visuals and key phrases.

Check out Daily Word Problems for consistent practice solving word problems.

For more fun math tips and strategies check out our Math- Ideas, Activities and Lessons Pinterest Board.

Save these tips and Pin It now!

Heather Foudy is a certified elementary teacher with over 7 years’ experience as an educator and volunteer in the classroom. She enjoys creating lessons that are meaningful and creative for students. She is currently working for Evan-Moor’s marketing and communications team and enjoys building learning opportunities that are both meaningful and creative for students and teachers alike.

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Here are some examples of mixed operations word problems. These problems are slightly more challenging,
but they also illustrate how helpful the block diagrams can be. The block diagrams or tape diagrams
(Common Core) can be used to help solve word problems that would usually require algebra.

Example:
Two bowls and three plates cost $1421. The cost of the plate is half the cost of the bowl. What is
the cost of the bowl?

Solution:
Step 1: Draw a block diagram to illustrate the number of bowls
and plates. (In this diagram, the bowls are shown as orange blocks and the plates as blue blocks.)

Step 2: Since a bowl costs twice as much as a plate, we can
replace one orange block (bowl) with two blue blocks (plate).

Step 3: Looking at the block diagram, find the cost of each plate.
7 blue blocks = 1421
1 blue block = 1421 ÷ 7 = 203
The cost of each plate is $203.
The cost of each bowl is 203 × 2 = $406.

Example:
A factory makes 4250 bars of chocolate. There were three kinds of chocolate bars – creamy,
milky and white. The number of white chocolate bars was 715 more than the number of milky chocolate
bars. The number of creamy chocolate bars was 5 times the number of milky chocolate bars. How many
creamy chocolate bars did the factory make?

Solution:
Step 1: Draw a block diagram to illustrate the different types of
chocolate bars. (In this diagram, the creamy chocolate bars are shown as orange blocks, the milky
chocolate bars as blue blocks and the white chocolate bars as red blocks.)

solve difficult word problems

Step 2: Since the number of creamy chocolate bars was 5 times the
number of milky chocolate bars, we can replace one orange block (creamy) with 5 blue blocks (milky).
Since the number of white chocolate bars was 715 more than the number of milky chocolate bars, we
replace one red block with one blue block + 715.

Step 3: Looking at the block diagram, find the number of milky
chocolate bars (blue block)
4250 – 715 = 3535
7 blue blocks = 3535
1 blue block = 3535 ÷ 7 = 505
The number of milky chocolate bars made was 505.

Step 4: Calculate the number of creamy chocolate bars.
The number of creamy chocolate bars was 5 times the number of milky chocolate bars = 5 × 505 = 2525

How to solve word problems with four types of bar models — Comparing, Taking Out, Combining, Missing Part?

Examples:

  1. Each week, Dwight calculated that he spends 2/3 of his allowance on food and 1/9 on video games.
    How much did he spend on food and video games?
  2. Marvin bought 1/2 pound of gummy bears. He ate 1/3 pound on his way to school. How much of gummy bears
    does Marvin have left?
  3. A recipe needs 7/8 teaspoons of salt and 2/5 teaspoons of sugar. How much more salt is needed than sugar?
  4. For the last hour Mr. Negron was awake, he spent 4/8 of the hour making homework and 1/8 of the hour
    talking. He spent the rest of the hour watching TV. What fraction of the hour did he spend watching TV?
  • Show Video Lesson

Example:
Julian and Stacey need 8 liters of water to fill a tank. Stacey filled the tank with 3 11/12 liters of water.
Julian poured 1 2/5 liters less than Stacey into the tank. How much water is still needed to fill the tank?

  • Show Video Lesson

How to solve part-whole word problems with bar modeling?

Examples:

  1. Danika bought some nail polish for $51. This was 3/5 of her money. How much money did she have to begin with?
  2. Henry bought 520 cookies for the math competition. If 3/4 were eaten, how many did he have left over?
  3. Betsy made cupcakes to bring to work. After the Science Department ate 2/7 and the Math Department ate 64,
    she had 1/7 left. How many cupcakes did Betsy make?
  4. Fran sold 108 chocolate chip and Snickerdoodle cookies at her lemonade stand. If she made 36 Snickerdoodle
    cookies, how many chocolate chip cookies did she make? How much money did she earn selling the chocolate chip
    cookies if she sold them in bags of 6 for $1.25 and she sold all of the bags.
  5. Two lbs. of nactarines and one lb. of bananas cost $3.15. Two lbs. of nectarines and three lbs. of bananas
    cost $5.65. Find the cost of one lb. of bananas.
  6. Two tennis balls and one racquet cost $75. Two tennis balls and three racquets cost $215. Find the cost of
    one racquet.
  7. Carter raised money for a walk-a-thon. On Monday, he earned $0.75 per lap. On Tuesday, he earned $0.80 a
    lap and on Wednesday, he earned $0.95 a lap. If he walked 36 laps on Monday, 50% as many laps on Wednesday
    as on Monday, how much money did Carter earn for the walk-a-thon?
  • Show Video Lesson

How to solve comparison word problems with bar modeling?

Examples:

  1. Alex, Sonia and Tini have a total of $580. Sonia has $120 more than Alex, and Toni has $190 more than Sonia.
    How much money does Toni have?
  2. Trevor bought a ski jacket, gloves and helmet on clearance for $242 altogether. The jacket cost $71 more
    than the gloves while the gloves cost $27 less than the helmet. How much did the jacket cost?
  3. A hard drive costs twice as much as a 64 GB jump drive and a 32 GB jump drive costs half as much as a 64 GB
    jump drive. If the hard drive costs $140, find the total cost.
  4. Tracy earned 3/4 as much as Julie. Shannon earned 2/3 as much as Tracy. Julie earned $44. How much less
    did Shannon earn than Tracy?
  5. Sadie spent twice as much money as Kim. Kim spent $150 less than Jason. Jason spent $45 more than Sadie.
    How much money did they spend altogether?
  6. Harlis is five years younger than Janae. Sherry is three times as old as Janae. Together, Harlis’ and
    Janae’s total age is 20 years less than Sherry’s age. How old is Sherry?
  7. Josh has 3/4 as many jelly beans as Lisa. Lisa has 4/5 as many jelly beans as Curt. If Curt has 28 more
    jelly beans than Josh, how many jelly beans do they have altogether?
  • Show Video Lesson

Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.

Mathway Calculator Widget

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You can solve many real world problems with the help of math. In order to familiarize students with these kinds of problems, teachers include word problems in their math curriculum. However, word problems can present a real challenge if you don’t know how to break them down and find the numbers underneath the story. Solving word problems is an art of transforming the words and sentences into mathematical expressions and then applying conventional algebraic techniques to solve the problem.

  1. Image titled Solve Word Problems in Algebra Step 1

    1

    Read the problem carefully.[1]
     A common setback when trying to solve algebra word problems is assuming what the question is asking before you read the entire problem. In order to be successful in solving a word problem, you need to read the whole problem in order to assess what information is provided, and what information is missing.[2]

  2. Image titled Solve Word Problems in Algebra Step 2

    2

    Determine what you are asked to find. In many problems, what you are asked to find is presented in the last sentence. This is not always true, however, so you need to read the entire problem carefully.[3]
     Write down what you need to find, or else underline it in the problem, so that you do not forget what your final answer means.[4]
     In an algebra word problem, you will likely be asked to find a certain value, or you may be asked to find an equation that represents a value.

    • For example, you might have the following problem: Jane went to a book shop and bought a book. While at the store Jane found a second interesting book and bought it for $80. The price of the second book was $10 less than three times the price of he first book. What was the price of the first book?
    • In this problem, you are asked to find the price of the first book Jane purchased.

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  3. Image titled Solve Word Problems in Algebra Step 3

    3

    Summarize what you know, and what you need to know. Likely, the information you need to know is the same as what information you are asked to find. You also need to assess what information you already know. Again, underline or write out this information, so you can keep track of all the parts of the problem. For problems involving geometry, it is often helpful to draw a sketch at this point.[5]

    • For example, you know that Jane bought two books. You know that the second book was $80. You also know that the second book cost $10 less than 3 times the price of the first book. You don’t know the price of the first book.

  4. Image titled Solve Word Problems in Algebra Step 4

    4

    Assign variables to the unknown quantities. If you are being asked to find a certain value, you will likely only have one variable. If, however, you are asked to find an equation, you will likely have multiple variables. No matter how many variables you have, you should list each one, and indicate what they are equal to.[6]

  5. Image titled Solve Word Problems in Algebra Step 5

    5

    Look for keywords.[7]
     Word problems are full of keywords that give you clues about what operations to use. Locating and interpreting these keywords can help you translate the words into algebra.[8]

    • Multiplication keywords include times, of, and factor.[9]
    • Division keywords include per, out of, and percent.[10]
    • Addition keywords include some, more, and together.[11]
    • Subtraction keywords include difference, fewer, and decreased.[12]

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  1. Image titled Solve Word Problems in Algebra Step 6

    1

    Write an equation. Use the information you learn from the problem, including keywords, to write an algebraic description of the story.[13]

  2. Image titled Solve Word Problems in Algebra Step 7

    2

    Solve an equation for one variable. If you have only one unknown in your word problem, isolate the variable in your equation and find which number it is equal to. Use the normal rules of algebra to isolate the variable. Remember that you need to keep the equation balanced. This means that whatever you do to one side of the equation, you must also do to the other side.[14]

  3. Image titled Solve Word Problems in Algebra Step 8

    3

    Solve an equation with multiple variables. If you have more than one unknown in your word problem, you need to make sure you combine like terms to simplify your equation.

  4. Image titled Solve Word Problems in Algebra Step 9

    4

    Interpret your answer. Look back to your list of variables and unknown information. This will remind you what you were trying to solve. Write a statement indicating what your answer means.[15]

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  1. Image titled Solve Word Problems in Algebra Step 10

    1

    Solve the following problem. This problem has more than one unknown value, so its equation will have multiple variables. This means you cannot solve for a specific numerical value of a variable. Instead, you will solve to find an equation that describes a variable.

    • Robyn and Billy run a lemonade stand. They are giving all the money that they make to a cat shelter. They will combine their profits from selling lemonade with their tips. They sell cups of lemonade for 75 cents. Their mom and dad have agreed to double whatever amount they receive in tips. Write an equation that describes the amount of money Robyn and Billy will give to the shelter.

  2. Image titled Solve Word Problems in Algebra Step 11

    2

    Read the problem carefully and determine what you are asked to find.[16]
     You are asked to find how much money Robyn and Billy will give to the cat shelter.

  3. Image titled Solve Word Problems in Algebra Step 12

    3

    Summarize what you know, and what you need to know. You know that Robyn and Billy will make money from selling cups of lemonade and from getting tips. You know that they will sell each cup for 75 cents. You also know that their mom and dad will double the amount they make in tips. You don’t know how many cups of lemonade they sell, or how much tip money they get.

  4. Image titled Solve Word Problems in Algebra Step 13

    4

    Assign variables to the unknown quantities. Since you have three unknowns, you will have three variables. Let x equal the amount of money they will give to the shelter. Let c equal the number of cups they sell. Let t equal the number of dollars they make in tips.

  5. Image titled Solve Word Problems in Algebra Step 14

    5

    Look for keywords. Since they will “combine” their profits and tips, you know addition will be involved. Since their mom and dad will “double” their tips, you know you need to multiply their tips by a factor of 2.

  6. Image titled Solve Word Problems in Algebra Step 15

    6

    Write an equation. Since you are writing an equation that describes the amount of money they will give to the shelter, the variable x will be alone on one side of the equation.

  7. Image titled Solve Word Problems in Algebra Step 16

    7

    Interpret your answer. The variable x equals the amount of money Robyn and Billy will donate to the cat shelter. So, the amount they donate can be found by multiplying the number of cups of lemonade they sell by .75, and adding this product to the product of their tip money and 2.

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Add New Question

  • Question

    How do you solve an algebra word problem?

    Daron Cam

    Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary’s College.

    Daron Cam

    Academic Tutor

    Expert Answer

    Carefully read the problem and figure out what information you’re given and what that information should be used for. Once you know what you need to do with the values they’ve given you, the problem should be a lot easier to solve.

  • Question

    If Deborah and Colin have $150 between them, and Deborah has $27 more than Colin, how much money does Deborah have?

    Donagan

    Let x = Deborah’s money. Then (x — 27) = Colin’s money. That means that (x) + (x — 27) = 150. Combining terms: 2x — 27 = 150. Adding 27 to both sides: 2x = 177. So x = 88.50, and (x — 27) = 61.50. Deborah has $88.50, and Colin has $61.50, which together add up to $150.

  • Question

    Karl is twice as old Bob. Nine years ago, Karl was three times as old as Bob. How old is each now?

    Donagan

    Let x be Bob’s current age. Then Karl’s current age is 2x. Nine years ago Bob’s age was x-9, and Karl’s age was 2x-9. We’re told that nine years ago Karl’s age (2x-9) was three times Bob’s age (x-9). Therefore, 2x-9 = 3(x-9) = 3x-27. Subtract 2x from both sides, and add 27 to both sides: 18 = x. So Bob’s current age is 18, and Karl’s current age is 36, twice Bob’s current age. (Nine years ago Bob would have been 9, and Karl would have been 27, or three times Bob’s age then.)

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  • Word problems can have more than one unknown and more the one variable.

  • The number of variables is always equal to the number of unknowns.

  • While solving word problems you should always read every sentence carefully and try to extract all the numerical information.

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Article SummaryX

To solve word problems in algebra, start by reading the problem carefully and determining what you’re being asked to find. Next, summarize what information you know and what you need to know. Then, assign variables to the unknown quantities. For example, if you know that Jane bought 2 books, and the second book cost $80, which was $10 less than 3 times the price of the first book, assign x to the price of the 1st book. Use this information to write your equation, which is 80 = 3x — 10. To learn how to solve an equation with multiple variables, keep reading!

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A Picture is Worth a Thousand Words

“Today, we are going to solve math word problems.” When students hear this from their math teacher, their faces drop, sweat starts to form on their foreheads and they refuse to make eye contact. As a math teacher, I understand. I understand the anxiety and want to make the learning process with word problems more enjoyable. This is where reading, writing and, math collide and the real world of math begins. Let’s talk about how to solve word problems with pictures.

When students struggle with word problems in school, they also have difficulty tackling them for homework. Most students want the “one-size-fits-all” formula for word problems but unfortunately, that does not exist. However, drawing a picture will help students visualize the problem and will start advancing their learning stage from the concrete to the abstract. Let’s take a look at how to solve a word problem using pictures:

Steps for Solving Word Problems using Pictures

  1. Read the entire problem: Get all the facts – Underline key word
  2. Answer the question: What am I looking for?
  3. Draw a picture or diagram: Visualize as a real world situation
  4. Solve the problem: Set up the equation and solve
  5. Check your solution: Is this answer reasonable?

Word Problem Examples – Example A

Cody has 6 pencils on his desk, Jonah has 4 more than Cody and Vinny has three less pencils than Jonah. How many pencils are there in all?

  • Read the entire problem  √
  • What am I looking for? How many pencils do Cody, Jonah and Vinny have altogether?  √
  • Draw a picture or diagram √Math Word Problems 1
  • Solve the problem    √

      6 + 10 + 7 = 23 pencils

  • Check your solution  √

      This answer is reasonable

Example B

There are 3 fish tanks labeled   X, Y, and Z. Y weighs 6 times as much as X and twice as much as Z. If Z is 36 lbs. heavier than X, find the total weight of X, Y and Z.

  • Read the entire problem  √
  • What am I looking for?  What is thetotal weight of fish tanks X, Y and Z?  √
  • Draw a picture or diagram √

Math Word Problems 2

  • Solve the problem    √

18 + 108 + 54 = 180 pounds

  • Check your solution  √

      This answer is reasonable

Visualizing a word problem with pictures is a strategy that will help motivate many students to begin the process of solving them.  This works especially for students that may become bored by the excess amount of words instead of numbers or for those students who become overwhelmed by the information and want to break it down into a simpler form.   Another benefit of using pictures when solving word problems involves communicating the results.  The pictures act as justification for answers, make the problems easier to understand and therefore, secure the learning process.

Meet our Guest Blogger: Jan Rowe

Jan is one of Educational Connections’ top tutors. She has twelve years of classroom teaching experience and holds a Virginia and Florida teaching license in middle school mathematics and elementary education. Jan has been with Educational Connections for over a year, working with over twenty students. Her tutoring goal is to help each student understand their learning style so they can improve the speed and quality of that learning. When she is not tutoring or teaching, Jan loves to play scrabble, go hiking and play Frisbee with her new puppy.

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