And in a math word problem

From Wikipedia, the free encyclopedia

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 are symbols representing real numbers — then a relevant solution to the word problem would, given the input , produce the output EQUAL, and similarly produce NOT_EQUAL from .

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 is the normal form of , , and , 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 containing variables have instances that are equal, or in other words whether the equation has any solutions. As a common example, is a word problem in the integer group ℤ,
while is a unification problem in the same group; since the former terms happen to be equal in ℤ, the latter problem has the substitution 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 and two words (strings) , can be transformed into by applying rules from ? 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 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]

The word problem for an abstract rewriting system (ARS) is quite succinct: given objects x and y are they equivalent under ?^[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 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

, and

share the same normal form, viz. ; therefore both terms are equal in every group.
As another example, the term and has the normal form and , 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
A2
A3

Term rewrite system obtained from Knuth–Bendix completion

R1
R2
R3
R4
R8
R11
R12
R13
R14
R17

See also[edit]

• Conjugacy problem
• Group isomorphism problem

References[edit]

Solving Word Problems in Mathematics

     Learn what word problems are and how to solve them in 7 easy steps.

     Real life math problems don’t usually look as simple as 3 + 5 = ?. Instead, things are a bit more complex. To show this, sometimes, math curriculum creators use word problems to help students see what happens in the real world. Word problems often show math happening in a more natural way in real life circumstances. 
     As a teacher, you can share some tips with your students to show that in everyday life they actually solve such problems all the time, and it’s not as scary as it may seem. 

     As you know, word problems can involve just about any operation: from addition to subtraction and division, or even multiple operations simultaneously.

If you’re a teacher, you may sometimes wonder how to teach students to solve word problems. It may be helpful to introduce some basic steps of working through a word problem in order to guide students’ experience. So, what steps do students need for solving a word problem in math?

Steps of Solving a Word Problem

     To work through any word problem, students should do the following:

     1. Read the problem: first, students should read through the problem once.
     2. Highlight facts: then, students should read through the problem again and highlight or underline important facts such as numbers or words that indicate an operation.
     3. Visualize the problem: drawing a picture or creating a diagram can be helpful. 

     Drawing the picture can be a big help in figuring this out. However, they can also look for the clues in the words such as:

          – Addition: add, more, total, altogether, and, plus, combine, in all;

          – Subtraction: fewer, than, take away, subtract, left, difference;

          – Multiplication: times, twice, triple, in all, total, groups;

     These key words may be very helpful when learning how to determine the operation students need to perform, but we should still pay attention to the fact that in the end it all depends on the context of the wording. The same word can have different meanings in different word problems. 

     Here they should learn to identify the steps they need to perform first to solve the problem, whether it’s a simple or a complex sentence. 
     6. Solve the problem: then, students can solve the number sentence and determine the solution. For example, 3 + 8 = 11.
     7. Check the answer: finally, students should check their work to make sure that the answer is correct.

     These 7 steps will help students get closer to mastering the skill of solving word problems. Of course, they still need plenty of practice. So, make sure to create enough opportunities for that! 

     At Happy Numbers, we gradually include word problems throughout the curriculum to ensure math flexibility and application of skills. Check out how easy it is to learn how to solve word problems with our visual exercises! 

     Word problems can be introduced in Kindergarten and be used through all grades as an important part of an educational process connecting mathematics to real life experience. 

     Happy Numbers introduces young students to the first math symbols by first building conceptual understanding of the operation through simple yet engaging visuals and key words. Once they understand the connection between these keywords and the actions they represent, they begin to substitute them with math symbols and translate word problems into number sentences. In this way, students gradually advance to the more abstract representations of these concepts.

     From the beginning, visualization helps the youngest students to understand the concepts of addition, subtraction, and even more complex operations. Even if they don’t draw the representations by themselves yet, students learn the connection between operations they need to perform in the problem and the real-world process this problem describes. 

     Next, students organize data from the word problem and pictures into a number sentence. To diversify the activity, you can ask students to match a word problem with the number sentence it represents.  

     Solving measurement problems is also a good way of mastering practical math skills. This is an example where students can see that math problems are closely related to real-world situations. Happy Numbers applies this by introducing more complicated forms of word problems as we help students advance to the next skill. By solving measurement word problems, students upgrade their vocabulary, learning such new terms as “difference” and “sum,” and continue mastering the connection between math operations and their word problem representations. 

     Later, students move to the next step, in which they learn how to create drawings and diagrams by themselves. They start by distributing light bulbs equally into boxes, which helps them to understand basic properties of division and multiplication. Eventually, with the help of Dino, they master tape diagrams! 

To see the full exercise, follow this link.

     The importance of working with diagrams and models becomes even more apparent when students move to more complex word problems. Pictorial representations help students master conceptual understanding by representing a challenging multi-step word problem in a visually simple and logical form. The ability to interact with a model helps students better understand logical patterns and motivates them to complete the task. 

     Having mentioned complex word problems, we have to show some of the examples that Happy Numbers uses in its curriculum. As the last step of mastering word problems, it is not the least important part of the journey. It’s crucial for students to learn how to solve the most challenging math problems without being intimidated by them. This only happens when their logical and algorithmic thinking skills are mastered perfectly, so they easily start talking in “math” language. 

This article is for parents who think about how to help with math and support their children. The math word problems below provide a gentle introduction to common math operations for schoolers of different grades.

What are math word problems?

During long-time education, kids face various hurdles that turn into real challenges. Parents shouldn’t leave their youngsters with their problems. They need an adult’s possible help, but what if the parents themselves aren’t good at mathematics? All’s not lost. You can provide your kid with different types of support. Not let a kid burn the midnight oil! Help him/ her to get over the challenges thanks to these captivating math word examples.

Math word problems are short math questions formulated into one or several sentences. They help schoolers to apply their knowledge to real-life scenarios. Besides, this kind of task helps kids to understand this subject better.

Addition for the first and second grades

These math examples are perfect for kids that just stepped into primary school. Here you find six easy math problems with answers:

1. Peter has eight apples. Dennis gives Peter three more. How many apples does Peter have in all?

2. Ann has seven candies. Lack gives her seven candies more. How many candies does Ann have in all?

3. Walter has two books. Matt has nine books. If Matt gives all his books to Walter, how many books will Walter have?

4. There are three crayons on the table. Albert puts five more crayons on the table. How many crayons are on the table?

5. Bill has nine oranges. His friend has one orange. If his friend gives his orange to Bill, how many oranges will Bill have?

6. Jassie has four leaves. Ben has two leaves. Ben gives her all his leaves. How many leaves does Jessie have in all?

Subtraction for the first and second grades

1. There were three books in total at the book shop. A customer bought one book. How many books are left?

2. There are five pizzas in total at the pizza shop. Andy bought one pizza. How many pizzas are left?

3. Liza had eleven stickers. She gave one of her stickers to Sarah. How many stickers does Liza have?

4. Adrianna had ten stones. But then she left two stones. How many stones does Adrianna have?

5. Mary bought a big bag of candy to share with her friends. There were 20 candies in the bag. Mary gave three candies to Marissa. She also gave three candies to Kayla. How many candies were left?

6. Betty had a pack of 25 pencil crayons. She gave five to her friend Theresa. She gave three to her friend Mary. How many pencil crayons does Betty have left?

Multiplication for the 2nd grade and 3rd grade

See the simple multiplication word problems. Make sure that the kid has a concrete understanding of the meaning of multiplication before.

Bill is having his friends over for the game night. He decided to prepare snacks and games.

1. He makes mini sandwiches. If he has five friends coming over and he made three sandwiches for each of them, how many sandwiches did he make?

2. He also decided to get some juice from fresh oranges. If he used two oranges per glass of juice and made six glasses of juice, how many oranges did he use?

3. Then Bill prepared the games for his five friends. If each game takes 7 minutes to prepare and he prepared a total of four games, how many minutes did it take for Bill to prepare all the games?

4. Bill decided to have takeout food as well. If each friend and Bill eat three slices of pizza, how many slices of pizza do they have in total?

Mike is having a party at his house to celebrate his birthday. He invited some friends and family.

1. He and his mother prepared cupcakes for dessert. Each box had 8 cupcakes, and they prepared four boxes. How many cupcakes have they prepared in the total?

2. They also baked some cookies. If they baked 6 pans of cookies, and there were 7 cookies per pan, how many cookies did they bake?

3. Mike planned to serve some cold drinks as well. If they make 7 pitchers of drinks and each pitcher can fill 5 glasses, how many glasses of drinks are they preparing?

4. At the end of the party, Mike wants to give away some souvenirs to his 6 closest friends. If he gives 2 souvenir items for each friend, how many souvenirs does Mike prepare?

Division: best for 3rd and 4th grades

1. If you have 10 books split evenly into 2 bags, how many books are in each bag?

2. You have 40 tickets for the fair. Each ride costs 2 tickets. How many rides can you go on?

3. The school has $20,000 to buy new equipment. If each piece of equipment costs $100, how many pieces can the school buy in total?

4. Melissa has 2 packs of tennis balls for $10 in total. How much does 1 pack of tennis balls cost?

5. Jack has 25 books. He has a bookshelf with 5 shelves on it. If Jack puts the same number of books on each shelf, how many books will be on each shelf?

6. Matt is having a picnic for his family. He has 36 cookies. There are 6 people in his family. If each person gets the same number of cookies, how many cookies will each person get?

Division with remainders for fourth and fifth grades

1. Sarah sold 35 boxes of cookies. How many cases of ten boxes, plus extra boxes does Sarah need to deliver?

2. Candies come in packages of 16. Mat ate 46 candies. How many whole packages of candies did he eat, and how many candies did he leave? 46 candies divided by 16 candies = 2 packages and 2 candies left over.

3. Mary sold 24 boxes of chocolate biscuits. How many cases of ten boxes, plus extra boxes does she need to deliver?

4. Gummy bears come in packages of 25. Suzie and Tom ate 30 gummy bears. How many whole packages did they eat? How many gummy bears did they leave?

5. Darel sold 55 ice-creams. How many cases of ten boxes, plus extra boxes does he need to deliver?

6. Crackers come in packages of 8. Mat ate 20 crackers. How many whole packages of crackers did he eat, and how many crackers did he leave?

Mixed operations for the fifth grade

These math word problems involve four basic operations: addition, multiplication, subtraction, and division. They suit best for the fifth-grade schoolers.

200 planes are taking off from the airport daily. During the Christmas holidays, the airport is busier — 240 planes are taking off every day from the airport.

1. During the Christmas holidays, how many planes take off from the airport in each hour if the airport opens 12 hours daily?

2. Each plane takes 220 passengers. How many passengers depart from the airport every hour during the Christmas holidays? 20 x 220 = 4400.

3. Compared with a normal day, how many more passengers are departing from the airport in a day during the Christmas holidays?

4. During normal days on average 650 passengers are late for their plane daily. During the Christmas holidays, 1300 passengers are late for their plane. That’s why 14 planes couldn’t take off and are delayed. How many more passengers are late for their planes during Christmas week?

5. According to the administration’s study, an additional 5 minutes of delay in the overall operation of the airport is caused for every 27 passengers that are late for their flights. What is the delay in the overall operation if there are 732 passengers late for their flights?

Extra info math problems for the fifth grade

1. Ann has 7 pairs of red socks and 8 pairs of pink socks. Her sister has 12 pairs of white socks. How many pairs of socks does Ann have?

2. Kurt spent 17 minutes doing home tasks. He took a 3-minute snack break. Then he studied for 10 more minutes. How long did Kurt study altogether?

3. There were 15 spelling words on the test. The first schooler spelled 9 words correctly. Miguel spelled 8 words correctly. How many words did Miguel spell incorrectly?

4. In the morning, Jack gave his friend 2 gummies. His friend ate 1 of them. Later Jack gave his friend 7 more gummies. How many gummies did Jack give his friend in all?

5. Peter wants to buy 2 candy bars. They cost 8 cents, and the gum costs 5 cents. How much will Peter pay?

Finding averages for 5th grade

We need to find averages in many situations in everyday life.

1. The dog slept 8 hours on Monday, 10 hours on Tuesday, and 900 minutes on Wednesday. What was the
average number of hours the dog slept per day?

2. Jakarta can get a lot of rain in the rainy season. The rainfall during 6 days was 90 mm, 74 mm, 112 mm, 30 mm, 100 mm, and 44 mm. What was the average daily rainfall during this period?

3. Mary bought 4 books. The prices of the first 3 books were $30, $15, and $18. The average price she paid for the 4 books was $25 per books. How much did she pay for the 4th books?

Ordering and number sense for the 5th grade

1. There are 135 pencils, 200 pens, 167 crayons, and 555 books in the bookshop. How would you write these numbers in ascending order?

2. There are five carrots, one cabbage, eleven eggs, and 15 apples in the fridge. How would you write these numbers in descending order?

3. Peter has completed exercises on pages 279, 256, 264, 259, and 192. How would you write these numbers in ascending order?

4. Mary picked 32 pants, 15 dresses, 26 pairs of socks, 10 purses. Put all these numbers in order.

5. The family bought 12 cans of tuna, 23 potatoes, 11 onions, and 33 pears. Put all these numbers in order.

Fractions for the 6th-8th grades

1. Jannet cooked 12 lemon biscuits for her daughter, Jill. She ate up 4 biscuits. What fraction of lemon biscuits did Jill eat?

2. Guinet travels a distance of 7 miles to reach her school. The bus covers only 5 miles. Then she has to walk 2 miles to reach the school. What fraction of the distance does Guinet travel by bus?

3. Bob has 24 pencils in a box. Eighteen pencils have #2 marked on them, and the 6 are marked #3. What fraction of pencils are marked #3?

4. My mother places 15 tulips in a glass vase. It holds 6 yellow tulips and 9 red tulips. What fraction of tulips are red?

5. Bill owns 14 pairs of socks, of which 7 pairs are white, and the rest are brown. What fraction of pairs of socks are brown?

6. Bred spotted a total of 39 birds in an aviary at the Zoo. He counted 18 macaws and 21 cockatoos. What fraction of macaws did Bred spot at the aviary?

Decimals for the 6th grade

Write in words the following decimals:

• 0,004
• 0,07
• 2,1
• 0,725
• 46,36
• 2000,19

Comparing and sequencing for the 6th grade

1. The older brother picked 42 apples at the orchard. The younger brother picked only 22 apples. How many more apples did the older brother pick?

2. There were 16 oranges in a basket and 66 oranges in a barrel. How many fewer oranges were in the basket than were in the barrel?

3. There were 40 parrots in the flock. Some of them flew away. Then there were 25 parrots in the flock. How many parrots flew away?

4. One hundred fifty is how much greater than fifty-three?

5. On Monday, the temperature was 13°C. The next day, the temperature dropped by 8 degrees. What was the temperature on Tuesday?

6. Zoie picked 15 dandelions. Her sister picked 22 ones. How many more dandelions did her sister pick than Zoie?

Time for the 4th grade

1. The bus was scheduled to arrive at 7:10 p.m. However, it was delayed for 45 minutes. What time was it when the bus arrived?

2. My mother starts her 7-hour work at 9:15 a.m. What time does she get off from work?

3. Jack’s walk started at 6:45 p.m. and ended at 7:25 p.m. How long did his walk last?

4. The school closes at 9:00 p.m. Today, the school’s principal left 15 minutes after the office closed, and his secretary left the office 25 minutes after he left. When did the secretary leave work?

5. Suzie arrives at school at 8:20 a.m. How much time does she need to wait before the school opens? The school opens at 8:35 a.m.

6. The class starts at 9:15 a.m.. The first bell will ring 20 minutes before the class starts. When will the first bell ring?

Money word problems for the fourth grade

1. James had $20. He bought a chocolate bar for $2.30 and a coffee cup for $5.50. How much money did he have left?

2. Coffee mugs cost $1.50 each. How much do 7 coffee mugs cost?

3. The father gives $32 to his four children to share equally. How much will each of his children get?

4. Each donut costs $1.20. How much do 6 donuts cost?

5. Bill and Bob went out for takeout food. They bought 4 hamburgers for $10. Fries cost $2 each. How much does one hamburger with fries cost?

6. A bottle of juice costs $2.80, and a can is $1.50. What would it cost to buy two cans of soft drinks and a bottle of juice?

Measurement word problems for the 6th grade

The task is to convert the given measures to new units. It best suits the sixth-grade schoolers.

• 55 yd = ____ in.
• 43 ft = ____ yd.
• 31 in = ____ ft.
• 29 ft = ____ in.
• 72 in = ____ ft.
• 13 ft = ____ yd.
• 54 lb = ____ t.
• 26 t = ____ lb.
• 77 t = ____ lb.
• 98 lb = ____ t.
• 25 lb = ____ t.
• 30 t = ____ lb.

Ratios and percentages for the 6th-8th grades

It is another area that children can find quite difficult. Let’s look at simple examples of how to find percentages and ratios.

1. A chess club has 25 members, of which 13 are males, and the rest are females. What is the ratio of males to all club members?

2. A group has 8 boys and 24 girls. What is the ratio of girls to all children?

3. A pattern has 4 red triangles for every 12 yellow triangles. What is the ratio of red triangles to all triangles?

4. An English club has 21 members, of which 13 are males, and the rest are females. What is the ratio of females to all club members?

5. Dan drew 1 heart, 1 star, and 26 circles. What is the ratio of circles to hearts?

6. Percentages of whole numbers:

• 50% of 60 = …
• 100% of 70 = …
• 90% of 70 = …
• 20% of 30 = …
• 40% of 10 = …
• 70% of 60 = …
• 100% of 20 = …
• 80% of 90 = …

Probability and data relationships for the 8th grade

1. John ‘s probability of winning the game is 60%. What is the probability of John not winning the game?

2. The probability that it will rain is 70%. What is the probability that it won’t rain?

3. There is a pack of 13 cards with numbers from 1 to 13. What is the probability of picking a number 9 from the pack?

4. A bag had 4 red toy cars, 6 white cars, and 7 blue cars. When a car is picked from this bag, what is the probability of it being red or blue?

5. In a class, 22 students like orange juice, and 18 students like milk. What is the probability that a schooler likes juice?

Geometry for the 7th grade

The following task is to write out equations and find the angles. Complementary angles are two angles that sum up to 90 degrees, and supplementary angles are two angles that sum up to 180 degrees.

1. The complement of a 32° angle = …

2. The supplement of a 10° angle = …

3. The complement of a 12° angle = …

4. The supplement of a 104° angle = …

Variables/ equation word problems for the 5th grades

1. The park is 𝑥 miles away from Jack’s home. Jack had to drive to and from the beach with a total distance of 36 miles. How many miles is Jack’s home away from the park?

2. Larry bought some biscuits which cost $24. He paid $x and got back $6 of change. Find x.

3. Mike played with his children on the beach for 90 minutes. After they played for x minutes, he had to remind them that they would be leaving in 15 minutes. Find x.

4. At 8 a.m., there were x people at the orchard. Later at noon, 27 of the people left the orchard, and there were 30 people left in the orchard. Find x.

Travel time word problems for the 5th-7th grades

1. Tony sprinted 22 miles at 4 miles per hour. How long did Tony sprint?

2. Danny walked 15 miles at 3 miles per hour. How long did Danny walk?

3. Roy sprinted 30 miles at 6 miles per hour. How long did Roy sprint?

4. Harry wandered 5 hours to get Pam’s house. It is 20 miles from his house to hers. How fast did Harry go?

About 25% of your total SAT Math section will be word problems, meaning you will have to create your own visuals and equations to solve for your answers. Though the actual math topics can vary, SAT word problems share a few commonalities, and we’re here to walk you through how to best solve them.

This post will be your complete guide to SAT Math word problems. We’ll cover how to translate word problems into equations and diagrams, the different types of math word problems you’ll see on the test, and how to go about solving your word problems on test day.

Feature Image: Antonio Litterio/Wikimedia

What Are SAT Math Word Problems?

A word problem is any math problem based mostly or entirely on a written description. You will not be provided with an equation, diagram, or graph on a word problem and must instead use your reading skills to translate the words of the question into a workable math problem. Once you do this, you can then solve it.

You will be given word problems on the SAT Math section for a variety of reasons. For one, word problems test your reading comprehension and your ability to visualize information.

Secondly, these types of questions allow test makers to ask questions that’d be impossible to ask with just a diagram or an equation. For instance, if a math question asks you to fit as many small objects into a larger one as is possible, it’d be difficult to demonstrate and ask this with only a diagram.

Translating Math Word Problems Into Equations or Drawings

In order to translate your SAT word problems into actionable math equations you can solve, you’ll need to understand and know how to utilize some key math terms. Whenever you see these words, you can translate them into the proper mathematical action.

For instance, the word «sum» means the value when two or more items are added together. So if you need to find the sum of a and b, you’ll need to set up your equation like this: a+b.

Also, note that many mathematical actions have more than one term attached, which can be used interchangeably.

Here is a chart with all the key terms and symbols you should know for SAT Math word problems:

Key Terms	Mathematical Action
Sum, increased by, added to, more than, total of	+
Difference, decreased by, less than, subtracted from	−
Product, times, __ times as much, __ times as many (a number, e.g., “three times as many”)	* or x
Divided by, per, __ as many, __ as much (a fraction, e.g., “one-third as much”)	/ or ÷
Equals, is, are, equivalent	=
Is less than	<
Is greater than	>
Is less than or equal to	≤
Is greater than or equal to	≥

Now, let’s look at these math terms in action using a few official examples:

Though the hardest SAT word problems might look like Latin to you right now, practice and study will soon have you translating them into workable questions.

Typical SAT Word Problems

Word problems on the SAT can be grouped into three major categories:

• Word problems for which you must simply set up an equation
• Word problems for which you must solve for a specific value
• Word problems for which you must define the meaning of a value or variable

Below, we look at each world problem type and give you examples.

Word Problem Type 1: Setting Up an Equation

This is a fairly uncommon type of SAT word problem, but you’ll generally see it at least once on the Math section. You’ll also most likely see it first on the section.

For these problems, you must use the information you’re given and then set up the equation. No need to solve for the missing variable—this is as far as you need to go.

Almost always, you’ll see this type of question in the first four questions on the SAT Math section, meaning that the College Board consider these questions easy. This is due to the fact that you only have to provide the setup and not the execution.

Word Problem Type 2: Solving for a Missing Value

The vast majority of SAT Math word problem questions will fall into this category. For these questions, you must both set up your equation and solve for a specific piece of information.

Most (though not all) word problem questions of this type will be scenarios or stories covering all sorts of SAT Math topics, such as averages, single-variable equations, and ratios. You almost always must have a solid understanding of the math topic in question in order to solve the word problem on the topic.

You might also get a geometry problem as a word problem, which might or might not be set up with a scenario, too. Geometry questions will be presented as word problems typically because the test makers felt the problem would be too easy to solve had you been given a diagram, or because the problem would be impossible to show with a diagram. (Note that geometry makes up a very small percentage of SAT Math.)

Word Problem Type 3: Explaining the Meaning of a Variable or Value

This type of problem will show up at least once. It asks you to define part of an equation provided by the word problem—generally the meaning of a specific variable or number.

To help juggle all the various SAT word problems, let’s look at the math strategies and tips at our disposal.

SAT Math Strategies for Word Problems

Though you’ll see word problems on the SAT Math section on a variety of math topics, there are still a few techniques you can apply to solve word problems as a whole.

#1: Draw It Out

Whether your problem is a geometry problem or an algebra problem, sometimes making a quick sketch of the scene can help you understand what exactly you’re working with. For instance, let’s look at how a picture can help you solve a word problem about a circle (specifically, a pizza):

As for geometry problems, remember that you might get a geometry word problem written as a word problem. In this case, make your own drawing of the scene. Even a rough sketch can help you visualize the math problem and keep all your information in order.

#2: Memorize Key Terms

If you’re not used to translating English words and descriptions into mathematical equations, then SAT word problems might be difficult to wrap your head around at first. Look at the chart we gave you above so you can learn how to translate keywords into their math equivalents. This way, you can understand exactly what a problem is asking you to find and how you’re supposed to find it.

There are free SAT Math questions available online, so memorize your terms and then practice on realistic SAT word problems to make sure you’ve got your definitions down and can apply them to the actual test.

#3: Underline and/or Write Out Important Information

The key to solving a word problem is to bring together all the key pieces of given information and put them in the right places. Make sure you write out all these givens on the diagram you’ve drawn (if the problem calls for a diagram) so that all your moving pieces are in order.

One of the best ways to keep all your pieces straight is to underline your key information in the problem, and then write them out yourself before you set up your equation. So take a moment to perform this step before you zero in on solving the question.

#4: Pay Close Attention to What’s Being Asked

It can be infuriating to find yourself solving for the wrong variable or writing in your given values in the wrong places. And yet this is entirely too easy to do when working with math word problems.

Make sure you pay strict attention to exactly what you’re meant to be solving for and exactly what pieces of information go where. Are you looking for the area or the perimeter? The value of x, 2x, or y?

It’s always better to double-check what you’re supposed to find before you start than to realize two minutes down the line that you have to begin solving the problem all over again.

#5: Brush Up on Any Specific Math Topic You Feel Weak In

You’re likely to see both a diagram/equation problem and a word problem for almost every SAT Math topic on the test. This is why there are so many different types of word problems and why you’ll need to know the ins and outs of every SAT Math topic in order to be able to solve a word problem about it.

For example, if you don’t know how to find an average given a set of numbers, you certainly won’t know how to solve a word problem that deals with averages!

Understand that solving an SAT Math word problem is a two-step process: it requires you to both understand how word problems work and to understand the math topic in question. If you have any areas of mathematical weakness, now’s a good time to brush up on them—or else SAT word problems might be trickier than you were expecting!

All set? Let’s go!

Test Your SAT Math Word Problem Knowledge

Finally, it’s time to test your word problem know-how against real SAT Math problems:

Word Problems

Answers: C, B, A, 1160

Answer Explanations

Aaaaaaaaaaand time for a nap.

Key Takeaways: Making Sense of SAT Math Word Problems

Word problems make up a significant portion of the SAT Math section, so it’s a good idea to understand how they work and how to translate the words on the page into a proper expression or equation. But this is still only half the battle.

Though you won’t know how to solve a word problem if you don’t know what a product is or how to draw a right triangle, you also won’t know how to solve a word problem about ratios if you don’t know how ratios work.

Therefore, be sure to learn not only how to approach math word problems as a whole, but also how to narrow your focus on any SAT Math topics you need help with. You can find links to all of our SAT Math topic guides here to help you in your studies.

What’s Next?

Want to brush up on SAT Math topics? Check out our individual math guides to get an overview of each and every topic on SAT Math. From polygons and slopes to probabilities and sequences, we’ve got you covered!

Running out of time on the SAT Math section? We have the know-how to help you beat the clock and maximize your score.

Been procrastinating on your SAT studying? Learn how you can overcome your desire to procrastinate and make a well-balanced prep plan.

Trying to get a perfect SAT score? Take a look at our guide to getting a perfect 800 on SAT Math, written by a perfect scorer.

Have friends who also need help with test prep? Share this article!

An algebraic sentence when written in equation form involves algebraic expressions (which contain variables such as letters in the alphabet), constants, and an equal symbol. Each algebraic sentence may contain a combination of algebraic expressions and constants, or with just two or more algebraic expressions. Almost always, the word “is” in an algebraic sentence denotes the symbol of equality.

In our example above, the algebraic sentence, “Five more than twice a number is forty-three“, is translated and written into its equation form: 2x + 5 = 43.

But before we delve into solving word problems that involve algebraic sentences, it’s crucial that we become familiar with how to translate and write algebraic expressions.

---

Algebraic Expressions

Learning how to write and translate algebraic expressions is the foundation in writing algebraic sentences. Eventually, we will use this knowledge to write algebraic equations where we solve for the value of the unknown variable.

What is an algebraic expression?

Think of an algebraic expression as the equivalent of a phrase in the English language. It does not convey a complete thought but is an important building block in constructing a sentence. Algebraic expressions consist of numbers, variables, and arithmetic operations.

Let’s take a quick look at some math phrases that were translated and written into algebraic expressions.

• the sum of a number and 8 large{,,, to ,,,, {color{red}{c + 8}}}

• 3 less than a number large{,,, to ,,,, {color{red}{y — 3}}}

• the product of 45 and m large{,,, to ,,,,{color{red}{45m}}}

• a number divided by 2 large{,,, to ,,,,Large {color{red}{x over 2}}}

---

Translating and Writing Algebraic Sentences

Once you know how to translate math phrases into algebraic expressions, it’ll be easy for you to translate and write algebraic sentences in an equation form as well.

For instance, how do you write “the difference of 16 and k is seven” into an algebraic equation?

Let’s deconstruct this algebraic sentence.

You may notice that we translated “the difference of 16 and k” into an algebraic expression, 16-k, then used the equal symbol (=) in place of the word “is”.

As I mentioned at the beginning of this lesson, the word “is” in an algebraic sentence, most of the time, signifies the equality symbol which is the case in this example.

Therefore, we can write this algebraic sentence in equation form as 16 — k = 7.

Now that we are confident in translating algebraic sentences and writing them in an equation format, it’s time for us to take one step further. We will not only translate and write algebraic sentences into algebraic equations but also proceed to solve them.

---

Examples of Algebraic Sentences Word Problems

The main key when solving word problems with algebraic sentences is to accurately translate the algebraic expressions then set up and write each algebraic equation correctly. In doing so, we can ensure that we are solving the right equation and as a result, will get the correct answer for each word problem.

Example 1: Six more than seven times a number is thirty-four. Find the number.

First, let’s deconstruct the algebraic sentence. It’s important that we identify and separate the algebraic expressions from the constants as well as determine if the problem suggests equality between the terms.

For the unknown value, we’ll use large{x} as our variable.

Note: Since addition is commutative, changing the order of the addends on the left side of the equation does not change the sum. Therefore, we may also write the algebraic expression as 7x+6 or the algebraic equation as 7x + 6 = 34 instead.

However, for our discussion, we will use 6 + 7x = 34 as our equation.

The original problem asks us to find the number which in this case is large{x}. So our next step is to solve for large{x} in our equation, 6 + 7x = 34.

Solution:

Now, we’ll evaluate the value of large{x = 4} to verify if the number we found satisfies the original algebraic sentence.

Remember that we are asked to find the identity of “the number” in our algebraic sentence and NOT the value of the variable large{x}. Therefore, it will be incorrect to say that large{x=4} is the answer. This is a common mistake that we always need to pay attention to.

We’ll replace the variable large{x} with the number 4 to see if the algebraic expression on the left side of the equation also results to 34.

Yes, it does! This means that number 4 is the answer.

---

Example 2: The difference between three times a number and five is sixteen. Find the number.

Right away, the word “difference” in our algebraic sentence gives us a clue that we will be using the subtraction operation. But, unlike, in addition, the order of the terms within an expression matters in subtraction. So we have to make sure that the terms in our algebraic expression are set up in the correct order.

Another keyword is “times”, which suggests that 3 is being multiplied to a number whose value is currently unknown.

This time, let’s use large{g} as our variable. Start by translating the algebraic expression correctly then continue to write the equation.

Now that we have our equation, let’s find out what the unknown value is by solving for the variable, large{g}.

Solution:

As discussed in our first example, it is important that we verify if the result of our solution makes the algebraic equation true. For this one, I will leave it up to you to evaluate the value of large{g = 7}. Remember to replace the unknown value of the variable large{g} with the number 7 to see if the left side of the equation is also equal to 16.

Since the original algebraic sentence requires us to find the number, then the answer is number 7.

---

Example 3: A number decreased by half of the number is four. Find the number.

This algebraic sentence is quite interesting because we don’t just have one unknown value, but two! First, we have the “number” (the unknown) then the “half of the number”, which means one-half of the unknown.

The keyword “half of” also indicates multiplying something by one-half while the “decreased by” tells us that we will be subtracting one term from the other.

To organize our thoughts, let’s deconstruct this algebraic sentence into its meat and potatoes.

Being able to break apart our algebraic sentence into its basic parts allows us to read the sentence thoroughly and understand the relationship between its quantities.

After setting up our algebraic equation, our next step is to solve it for large{m}.

Solution:

We need to evaluate the value of large{m = 8} to check if the number we got makes our algebraic equation true.

Great! The values on both sides of the equation are equal to each other. Therefore, the number 8 is the correct answer to our original algebraic sentence.

---

Example 4: Four times the sum of twice a number and six is thirty-two. Find the number.

Looking closely, we see a few keywords that would help us in translating this algebraic sentence into an equation.

• times – means that we’ll be multiplying 4 by the quantity, “sum of twice a number and six”
• sum – means that we need to add the terms, “twice a number” and six
• twice – means that a number (the unknown value) is multiplied by 2

With these in mind, let’s write our algebraic equation.

Notice that we used the variable large{d} in our equation to stand for our unknown value. Let’s now proceed and solve for large{d} and afterward, check if the value we get indeed makes the equation true.

Solution:

Evaluate the value of large{d = 1}:

Looks like everything checks out, so the answer to our algebraic sentence is the number 1.

---

Example 5: Two-thirds of the sum of three times a number and six is ten. What is the number?

This problem involves a fraction and an algebraic expression as you can tell. The keyword “of” indicates multiplication, so it tells us that we need to multiply the fraction large{2 over 3} by the sum of the two given quantities (“three times a number” and six).

Let’s move on and write our algebraic equation. We’ll use the variable large{y} as the placeholder for our “number” whose value is currently unknown.

Awesome! We are able to translate our algebraic sentence into an equation. It’s time for us to solve for large{y} then I’ll leave it up to you to do a check if the number we find as the value of large{y} makes the algebraic equation true.

Solution:

Once you evaluate large{y = 3}, you’ll see that the answer to our word problem is indeed the number 3.

---

Example 6: Twice the difference of a number and three is four more than the number. Find the number.

We have an example here that is a little different from the word problems we’ve covered so far. In this algebraic sentence, both sides of the equation contain an algebraic expression. However, the keywords present will help us translate it correctly into an equation so there’s nothing to worry about.

Keywords:

• twice – means that we will multiply the quantity (difference of a number and three) by 2
• difference – tells us to use the subtraction operation between our unknown value (“number”) and three
• more than – means that we need to add four to our unknown value

Let’s solve for our unknown value, large{a}, then verify that our result makes the algebraic equation true.

Solution:

Evaluate the value of large{a = 10}:

Perfect! The value we got for large{a} which is 10, made our equation equal on both sides. Thus, the answer is the number 10.

---

You might also be interested in:

Algebraic Expressions

Algebraic Expressions Worked Examples

Find below a wide variety of hard word problems in algebra. Most tricky and tough algebra word problems are covered here.  If you can solve these, you can probably solve any algebra problems.Teachers! Feel free to select from this list and give them to your students to see if they have mastered how to solve tough algebra problems.Find out below how you can print these problems. You can also purchase a solution if needed.

1. The
cost of petrol rises by 2 cents a liter. last week a man bought 20
liters at the old price. This week he bought 10 liters at the new
price. Altogether, the petrol costs $9.20. What was the old price for
1 liter?

2. Teachers
divided students into groups of 3. Each group of 3 wrote a report
that had 9 pictures in it. The students used 585 pictures altogether.
How many students were there in all?

3. Vera
and Vikki are sisters. Vera is 4 years old and Vikki is 13 years old.
What age will each sister be when Vikki is twice as old as Vera?

4. A
can do a work in 14 days and working together A and B can do the same
work in 10 days. In what time can B alone do the work?

5. 7
workers can make 210 pairs of cup in 6 days. How many workers are required
to make 450 pairs of cup in 10 days?

6. Ten
years ago the ratio between the ages of Mohan and Suman was 3:5. 11
years hence it will be 11:16. What is the present age of Mohan?

7. The
ratio of girls to boys in class is 9 to 7 and there are 80 students in
the class. How many girls are in the class?

8.  One
ounce of solution X contains only ingredients a and b in a ratio of 2:3.
One ounce of solution Y contains only ingredients a and b in a ratio of
1:2. If solution Z is created by mixing solutions X and Y in a ratio of
3:11, then 2520
ounces of solution Z contains how many ounces of a?

9.  This
week Bob puts gas in his truck
when the tank was about half empty.
Five days later, bob puts gas again when the tank was about three
fourths full. If Bob Bought 24 gallons of gas, how many gallons does
the tank hold?

10. A
commercial
airplane flying with a speed of 700 mi/h is detected 1000 miles
away with a radar. Half an hour later an interceptor plane flying with
a speed of 800 mi/h is dispatched. How long will it take the interceptor
plane to meet with the other plane?

11. There
are 40 pigs and chickens in a farmyard. Joseph counted 100 legs in
all. How many pigs and how many chickens are there?

12. The
top of a box is a rectangle with a perimeter of 72 inches. If the box
is 8 inches high, what dimensions will give the maximum volume?

13. You
are raising money for a charity. Someone made a fixed donation of
500. Then, you require each participant to make a pledge of 25
dollars. What is the minimum amount of money raised if there are 224 participants.

14. The
sum of two positive numbers is 4 and the sum of their squares
is 28.
What are the two numbers?

15. Flying
against the jet stream, a jet travels 1880 mi in 4 hours. Flying with
the jet stream, the same jet travels 5820 mi in 6 hours. What is the
rate of the jet in still air and what is the rate of the jet stream?

16. Jenna
and her friend, Khalil, are having a contest to see who can save the
most money. Jenna has already saved $110 and every week she saves an
additional $20. Khalil has already saved $80 and every week he saves an
additional $25. Let x represent the number of weeks and y represent the
total amount of money saved. Determine in how many weeks Jenna and Khalil
will have the same amount of money.

17. The
sum of three consecutive terms of a geometric sequence is 104 and their
product is 13824.find the terms.

18. The
sum of the first and last of four consecutive odd integers is 52. What
are the four integers?

19. A
health club charges a one-time initiation fee and
a monthly fee. John paid
100 dollars for 2 months of membership. However, Peter paid 200 for 6
months of membership. How much will Sylvia pay for 1 year of
membership?

20. The
sum of two positive numbers is 4 and the sum of their cubes is 28. What
is the product of the two numbers?

21. A
man selling computer parts realizes that when he sells 16 computer parts,
his earning is $1700. When he sells 56 computer parts, his earning
is $4300. What will the earning be if the man sells 30 computers parts?

22. A
man has 15 coins in his pockets. These coins are dimes and quarters
that add
up 2.4 dollars. How many quarters and how many dimes does the man have?

23. The
lengths of the sides of a triangle are in the ratio 4:3:5. Find
the lengths
of the sides if the perimeter is 18 inches.

24. The
ratio of base to height of a equilateral
triangle is 3:4. If the area of
the triangle is 6, what is the perimeter of the triangle?

25. The
percent return rate of a growth fund, income fund, and money market are
10%, 7%, and 5% respectively. Suppose you have 3200 to invest and you want
to put twice as much in the growth fund as in the money market to maximize
your return. How should you invest to get a return of 250 dollars
in 1 year?

26. A
shark was caught whose tail weighted 200 pounds. The head of the
shark weighted
as much as its tail plus half its body. Its body weighted as much
as its head and tail. What is the weight of the shark?

27. The
square root of a
number plus two is the same as the
number. What
is the
number?

28. Suppose
you have a coupon worth 6 dollars off any item at a mall. You go to
a store at the mall that offers a 20% discount. What do you need to
do to
save the most money?

29. Suppose
your grades on three math exams are 80, 93, and 91. What grade do you
need on your next exam to have at least a 90 average on the four exams?

30. Peter
has a photograph that is 5 inches wide and 6 inches long. She enlarged
each side by the same amount. By how much was the photograph enlarged
if the new area is 182 square inches?

31. The
cost to produce a book is 1200 to get started plus 9
dollars per book. The
book sells for 15 dollars each. How many books must be sold to make a profit?

32. Store
A sells CDs
for 2 dollars each if you pay a one-time
fee of 104
dollars. Store B offers 12 free CDs and charges 10 dollars for each additional
CD. How many CD must you buy so it will cost the same under both
plans?

33. When
4 is added to two numbers, the ratio is 5:6. When 4 is subtracted
from the two numbers, the ratio is 1:2. Find the two numbers.

34. A
store owner wants to sell 200 pounds of pistachios and walnuts mixed
together. Walnuts cost 4 dollars per pound and pistachio cost 6
dollars per
pounds. How many pounds of each type of nuts should be mixed if the
store owner will charge 5 dollars for the mixture?

35. A
cereal box manufacturer makes 32-ounce boxes of cereal. In a perfect
world, the box will be 32-ounce every time is made. However, since the
world is not perfect, they allow a difference of 0.06 ounce. Find the
range of acceptable size for the cereal box.

36. A
man weighing 600 kg has been losing 3.12% of his weight each month
with some
heavy exercises and eating the right food. What will the man weigh after
20 months?

37. An
object is thrown into the air at a height of 60 feet. After 1 second and
2 second, the object is 88 feet and 84 feet in the air respectively. What
is the initial speed of the object?

38. A
transit is 200 feet from the base of a building. There is man
standing on top of the building. The angles of elevation from the top
and bottom of the man are 45 degrees and 44 degrees. What is the
height of the man?

39. A
lemonade consists of 6% of lemon juice and a strawberry juice
consists of 15% pure fruit juice. How much of each kind should be
mixed together to get a 4 Liters of a 10% concentration of fruit
juice?

40. Ellen
can wash her car in 60 minutes. Her older sister Sarah can do the
same job in 45 minutes. How long will it take if they wash the car
together?

41. A
plane flies 500 mi/h. The plane can travel 1100 miles with the wind
in the same amount of time as it travels 900 miles against the wind.
What is the speed of the wind?

42. A
company produces boxes that are 5 feet long, 4 feet wide, and 3 feet
high. The company wants to increase each dimension by the same amount
so that the new volume is twice as big. How much is the increase in
dimension?

43. James
invested half of his money in land, a tenth in stock, and a twentieth
in saving bonds. Then, he put the remaining 21000 in a CD. How much
money did James saved or invested?

44. The
motherboards for a desktop computer can be manufactured for 50
dollars each.
The development cost is 250000. The first 20 motherboard are samples and
will not be sold. How many salable motherboards will yield an average cost
of 6325 dollars?

45. How
much of a 70% orange juice drink must be mixed with 44 gallons of a
20% orange juice drink to obtain a mixture that is 50% orange juice?

46. A
company sells nuts in bulk quantities. When bought in bulk, peanuts
sell for $1.20 per pound, almonds for $ 2.20 per pound, and cashews
for $3.20 per pound. Suppose a specialty shop wants a mixture of 280
pounds that will cost $2.59 per pound. Find the number of pounds of
each type of nut if the sum of the number of pounds of almonds and
cashews is three times the number of pounds of peanuts. Round your
answers to the nearest pound.

47. A
Basketball player has successfully made 36 of his last 48 free
throws. Find
the number of consecutive free throws the player needs to increase his
success rate to 80%.

48. John
can wash cars 3 times as fast as his son Erick. Working together,
they need to wash 30 cars in 6 hours. How many hours will it takes
each of them working alone?

49. In
a college, about 36% of student are under 20 years old and 15% are
over 40 years old. What is the probability that a student chosen at
random is under 20 years old or over 40 years?

50. Twice
a number plus the square root of the number is twelve minus the
square root of the number.

51. The
light intensity, I , of a light bulb varies inversely as the square
of the
distance from the bulb. A a distance of 3 meters from the bulb, I = 1.5 W/m^2 .
What is the light intensity at a distance of 2 meters from the bulb?

52. The
lengths of two sides of a triangle are 2 and 6. find the range of
values for the possible lengths of the third side.

53. Find
three consecutive integers such that one half of their sum is between
15 and 21.

54. After
you open a book, you notice that the
product of the
two page numbers on
the facing pages is 650. What are the two page numbers?

55. Suppose
you start with a number. You multiply the number by 3, add 7, divide
by ½, subtract 5, and then divide by 12. The result is 5. What
number did you start with?

56. You
have 156 feet of fencing to enclose a rectangular garden. You want
the garden to be 5 times as long as it is wide. Find the dimensions
of the garden.

57. The
amount of water a dripping faucet wastes water varies directly with
the amount of time the faucet drips. If the faucet drips 2 cups of
water every 6 minutes, find out how long it will take the faucet to
drip 10.6465 liters of water.

58. A
washer costs 25% more than a dryer. If the store clerk gave a 10% discount
for the dryer and a 20% discount for the washer, how much is the
washer before the discount if you paid 1900 dollars.

59. Baking
a tray of blueberry muffins takes 4 cups of milk and 3 cups of wheat
flour. A tray of pumpkin muffins takes 2 cups of milk and 3 cups of
wheat flour. A baker has 16 cups of milk and 15 cups of wheat flour.
You make 3 dollars profit per tray of blueberry muffins and 2 dollars
profit per tray of pumpkin muffins. How many trays of each type of
muffins should you make to maximize profit?

60. A
company found that -2p + 1000 models the number of TVs sold per month
where p can be set as low as 200 or as high as 300. How can the
company maximize the revenue?

More hard word problems in algebra

61. Your
friends say that he has $2.40 in equal numbers of quarters, dimes,
and nickels. How many of each coin does he have?

62. I
am a two-digit number whose digit in the tenth place is 1 less than
twice the digit in the ones place. When the digit in the tenth place
is divided by the digit in the ones place, The quotient is 1 and the
remainder is 4. What number am I?

63. A
two-digit number is formed by randomly selecting from the digits 2,
4, 5,
and 7 without replacement. What is the probability that a two-digit number
contains a 2 or a 7?

64. Suppose
you interview 30 females and 20 males at your school to find out who
among them are using an electric toothbrush. Your survey revealed
that only 2 males use an electric toothbrush while 6 females use it.
What is the probability that a respondent did not use an electric
toothbrush given that the respondent is a female?

65. An
employer pays 15 dollars per hour plus an extra 5 dollars per hour
for every hour worked beyond 8 hours up to a maximum daily wage of
220 dollars. Find
a piecewise function that models this situation.

66. Divide
me by 7, the remainder is 5. Divide me by 3, the remainder is 1 and
my quotient is 2 less than 3 times my previous quotient. What number
am I?

67. A
company making luggage have these requirements to follow. The length is
15 inches greater than the depth and the sum of length, width, and
depth may not exceed 50 inches. What is the maximum value for the
depth if the manufacturer will only use whole numbers?

68. To
make an open box, a man cuts equal squares from each corner of a
sheet of metal that is 12 inches wide and 16 inches long. Find an
expression for the volume in terms of x.

69. Ten
candidates are running for president, vice-president, and secretary
in the students government. You may vote for as many as 3 candidates.
In how many ways can you vote for 3 or fewer candidates?

70. The
half-life of a medication prescribed by a doctor is 6 hours. How
many mg of this medication is left after 78 hours if the doctor prescribed
100 mg?

71. Suppose
you roll a red number cube and a yellow number cube. Find
P(red 2, yellow 2) and the probability to get any matching pairs of
numbers.

72. A
movie theater in a small town usually open its doors 3 days in a row
and then closes the next day for maintenance. Another movie theater 3
miles away open 4 days in a row and then closes the next day for the
same reason. Suppose both movie theaters are closed today and today
is Wednesday, when is the next time they will both be closed again on
the same day?

73. An
investor invests 5000 dollars at 10% and the rest at 5%. How much was
invested at 5% if the yield is one-fifth of the amount invested at at
10%?

74. 20000
students took a standardized math test. The scores on the test are
normally distributed, with a mean score of 85 and a standard
deviation of 5. About how many students scored between 90 and 95?

75. A
satellite, located 2400 km above Earth’s surface, is in circular
orbit around the earth. If it takes the satellite 3 hours to complete
1 orbit, how far is the satellite after 1 hour?

76. In
a group of 10 people, what is the probability that at least two
people in
the group have the same birthday?

77. During
a fundraising for cancer at a gala, everybody shakes hands with
everyone else in the room before the event and after the event is
finished. If
n people attended the gala, how many different handshakes occur?

78. Two
cubes have side lengths that are equal to 2x and 4x. How many times greater
than the surface of the small cube is the surface area of the large
cube?

79. Suppose
you have a job in a restaurant that pays $8 per hour. You also have
a job at Walmart that pays $10 per hour. You want to earn at least
200 per week. However, you want to work no more than 25 hours per
week . Show 3 different ways you could work at each job.

80. Two
company offer tutoring services. Company A realizes that when they
tutor for 3 hours, they make 45 dollars. When they tutor for 7 hours
they make 105 hours. Company B realizes that when they tutor for 2
hours, they make 34 dollars. When they tutor for 6 hours they make
102 hours. Assuming that the number of hours students sign for
tutoring is the same for both company, which company will generate
more revenue?

81. You
want to fence a rectangular area for kids in the backyard. To save on
fences, you will use the back of your house as one of the four sides.
Find the possible dimensions if the house is 60 feet wide and you
want to use at least 160 feet of fencing.

82. When
a number in increased by 20%, the result is the same when it is
decreased by 10% plus 12. What is the number?

83. The
average of three numbers is 47. The biggest number is five more than
twice the smallest. The range is 35. What are the three numbers?

84. The
percent of increase of a number from its original amount to 36 is
80%. What
is the original amount of the number?

85. When
Peter drives to work, he averages 45 miles per hour because of
traffic. On the way back home, he averages 60 miles per hour because
traffic is not as bad. The total travel time is 2 hours. How far is
Peter’s house from work?

86. An
advertising company takes 20% from all revenue that it generates for
its affiliates. If the affiliates were paid 15200 dollars this month,
how much revenue
did the advertising company generate this month?

87. A
company’s revenue can be modeled with a quadratic equation. The
company noticed that when they sell either 2 or or 12 items, the
revenue is 0. How much is the revenue when they sell 20 items?

88. A
ball bounced 4 times, reaching three-fourths of its previous height
with each bounce. After the fourth bounce, the ball reached a height
of 25 cm. How high was the ball when it was dropped?

89. A
rental company charges 40 dollars per day plus $0.30 per mile. You
rent a car and drop it off 4 days later. How many miles did you drive
the car if you paid 325.5 dollars which included a 5% sales tax?

90. Two
students leave school at the same time and travel in opposite
directions along the same road. One walk at a rate of 3 mi/h. The
other bikes at a rate of 8.5 mi/h. After how long will they be 23
miles part?

91. Brown
has the same number of brothers as sisters. His sister Sylvia as
twice as many brothers as sisters. How many children are in the
family?

92. Ethan
has the same number of male classmates as female classmates. His
classmate Olivia has three-fourths as many female classmates as male
classmates. How many students are in the class?

93. Noah
wants to share a certain amount of money with 10 people. However, at
the last minute, he is thinking about decreasing the amount by 20 so
he can keep 20 for himself and share the money with only 5 people.
How much money is Noah trying to share if each person still gets the
same amount?

94. The
square root of me plus the square root of me is me. Who am I?

95. A
cash drawer contains 160 bills, all 10s and 50s. If the total value
of the 10s and 50s is $1,760. How many of each type of bill are in
the drawer?

96. You
want to make 28 grams of protein snack mix made with peanuts and
granola. Peanuts contain 7 grams of protein per ounce and granola
contain 3 grams of protein per ounce. How many ounces of granola
should you use for 1 ounce of peanuts?

97. The
length of a rectangular prism is quadrupled, the width is doubled,
and the height is cut in half. If V is the volume of the rectangular
prism before the modification, express the volume after the
modification in terms of V.

98. A
car rental has CD players in 85% of its cars. The CD players are
randomly distributed throughout the fleet of cars. If a person rents
4 cars, what is the probability that at least 3 of them will have CD
players?

99. Jacob’s
hourly wage is 4 times as much as Noah. When Jacob got a raise of 2
dollars, Noah accepted a new position that pays him 2 dollars less
per hour. Jacob now earns 5 times as much money as Noah. How much
money do they make per hour after Jacob got the raise?

100.

You
own a catering business that makes specialty cakes. Your company has
decided to create three types of cakes. To create these cakes, it
takes a team that consists of a decorator, a baker, and a design
consultant. Cake A takes the decorator 9 hours, the baker 6 hours,
and the design consultant 1 hour to complete. Cake B takes the
decorator 10 hours, the baker 4 hours, and the design consultant 2
hours. Cake C takes the decorator 12 hours, the baker 4 hour, and the
design consultant 1 hour. Without hiring additional employees, there
are 398 decorator hours available, 164 baker hours available, and 58
design consultant hours available. How many of each type of cake can
be created?

Want a solution to these tough algebra problems? Add to your shopping cart and purchase a Detailed 150+ PAGES SOLUTION and TOP-NOTCH EXPLANATIONS with PayPal.