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• problem-solving

adjective as in analytic

Strongest matches

analytical , investigative

Weak matches

inquiring , rational , sound , systematic

adjective as in analytical

analytic , cogent , detailed , diagnostic , interpretive , investigative , penetrating , rational , scientific , systematic , thorough

conclusive , discrete , dissecting , explanatory , expository , inquiring , inquisitive , judicious , logical , organized , perceptive , perspicuous , precise , questioning , ratiocinative , reasonably , searching , solid , sound , studious , subtle , testing , valid

adjective as in analytic/analytical

cogent , conclusive , detailed , diagnostic , discrete , dissecting , explanatory , expository , inquiring , inquisitive , interpretive , investigative , judicious , logical , organized , penetrating , perceptive , perspicuous , precise , questioning , ratiocinative , rational , reasonable , scientific , searching , solid , sound , studious , subtle , systematic , testing , thorough , valid , well-grounded

Example Sentences

“These are problem-solving products but that incorporate technology in a really subtle, unobtrusive way,” she says.

And it is a “problem-solving populism” that marries the twin impulses of populism and progressivism.

“We want a Republican Party that returns to problem-solving mode,” he said.

Problem-solving entails accepting realities, splitting differences, and moving forward.

It teaches female factory workers technical and life skills, such as literacy, communication and problem-solving.

Problem solving with class discussion is absolutely essential, and should occupy at least one third of the entire time.

In teaching by the problem-solving method Professor Lancelot 22 makes use of three types of problems.

Sequential Problem Solving is written for those with a whole brain thinking style.

Thus problem solving involves both the physical world and the interpersonal world.

Sequential Problem Solving begins with the mechanics of learning and the role of memorization in learning.

Related Words

Words related to problem-solving are not direct synonyms, but are associated with the word problem-solving . Browse related words to learn more about word associations.

adjective as in logical

• investigative

adjective as in examining and determining

• explanatory
• inquisitive
• interpretive
• penetrating
• perspicuous
• questioning
• ratiocinative
• well-grounded

adjective as in examining

Viewing 5 / 11 related words

From Roget's 21st Century Thesaurus, Third Edition Copyright © 2013 by the Philip Lief Group.

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Our math problem solver that lets you input a wide variety of math math problems and it will provide a step by step answer. This math solver excels at math word problems as well as a wide range of math subjects.

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Math word problems require interpreting what is being asked and simplifying that into a basic math equation. Once you have the equation you can then enter that into the problem solver as a basic math or algebra question to be correctly solved. Below are math word problem examples and their simplified forms.

Word Problem: Rachel has 17 apples. She gives some to Sarah. Sarah now has 8 apples. How many apples did Rachel give her?

Simplified Equation: 17 - x = 8

Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b)

Simplified Equation: {r = d + 12, d = b + 6, r = 2 �� b}

Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left?

Simplified: 40 - 10 - 5

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Algebra Topics  - Introduction to Word Problems

Algebra topics  -, introduction to word problems, algebra topics introduction to word problems.

Algebra Topics: Introduction to Word Problems

Lesson 9: introduction to word problems.

/en/algebra-topics/solving-equations/content/

What are word problems?

A word problem is a math problem written out as a short story or scenario. Basically, it describes a realistic problem and asks you to imagine how you would solve it using math. If you've ever taken a math class, you've probably solved a word problem. For instance, does this sound familiar?

Johnny has 12 apples. If he gives four to Susie, how many will he have left?

You could solve this problem by looking at the numbers and figuring out what the problem is asking you to do. In this case, you're supposed to find out how many apples Johnny has left at the end of the problem. By reading the problem, you know Johnny starts out with 12 apples. By the end, he has 4 less because he gave them away. You could write this as:

12 - 4 = 8 , so you know Johnny has 8 apples left.

Word problems in algebra

If you were able to solve this problem, you should also be able to solve algebra word problems. Yes, they involve more complicated math, but they use the same basic problem-solving skills as simpler word problems.

You can tackle any word problem by following these five steps:

• Read through the problem carefully, and figure out what it's about.
• Represent unknown numbers with variables.
• Translate the rest of the problem into a mathematical expression.
• Solve the problem.
• Check your work.

We'll work through an algebra word problem using these steps. Here's a typical problem:

The rate to rent a small moving van is $30 per day, plus $0.50 per mile. Jada rented a van to drive to her new home. It took two days, and the van cost $360. How many miles did she drive?

It might seem complicated at first glance, but we already have all of the information we need to solve it. Let's go through it step by step.

Step 1: Read through the problem carefully.

With any problem, start by reading through the problem. As you're reading, consider:

• What question is the problem asking?
• What information do you already have?

Let's take a look at our problem again. What question is the problem asking? In other words, what are you trying to find out?

The rate to rent a small moving van is $30 per day, plus $0.50 per mile. Jada rented a van to drive to her new home. It took 2 days, and the van cost $360. How many miles did she drive?

There's only one question here. We're trying to find out how many miles Jada drove . Now we need to locate any information that will help us answer this question.

There are a few important things we know that will help us figure out the total mileage Jada drove:

• The van cost $30 per day.
• In addition to paying a daily charge, Jada paid $0.50 per mile.
• Jada had the van for 2 days.
• The total cost was $360 .

Step 2: Represent unknown numbers with variables.

In algebra, you represent unknown numbers with letters called variables . (To learn more about variables, see our lesson on reading algebraic expressions .) You can use a variable in the place of any amount you don't know. Looking at our problem, do you see a quantity we should represent with a variable? It's often the number we're trying to find out.

Since we're trying to find the total number of miles Jada drove, we'll represent that amount with a variable—at least until we know it. We'll use the variable m for miles . Of course, we could use any variable, but m should be easy to remember.

Step 3: Translate the rest of the problem.

Let's take another look at the problem, with the facts we'll use to solve it highlighted.

The rate to rent a small moving van is $30 per day , plus $0.50 per mile . Jada rented a van to drive to her new home. It took 2 days , and the van cost $360 . How many miles did she drive?

We know the total cost of the van, and we know that it includes a fee for the number of days, plus another fee for the number of miles. It's $30 per day, and $0.50 per mile. A simpler way to say this would be:

$30 per day plus $0.50 per mile is $360.

If you look at this sentence and the original problem, you can see that they basically say the same thing: It cost Jada $30 per day and $0.50 per mile, and her total cost was $360 . The shorter version will be easier to translate into a mathematical expression.

Let's start by translating $30 per day . To calculate the cost of something that costs a certain amount per day, you'd multiply the per-day cost by the number of days—in other words, 30 per day could be written as 30 ⋅ days, or 30 times the number of days . (Not sure why you'd translate it this way? Check out our lesson on writing algebraic expressions .)

$30 per day and $.50 per mile is $360

$30 ⋅ day + $.50 ⋅ mile = $360

As you can see, there were a few other words we could translate into operators, so and $.50 became + $.50 , $.50 per mile became $.50 ⋅ mile , and is became = .

Next, we'll add in the numbers and variables we already know. We already know the number of days Jada drove, 2 , so we can replace that. We've also already said we'll use m to represent the number of miles, so we can replace that too. We should also take the dollar signs off of the money amounts to make them consistent with the other numbers.

30 ⋅ 2 + .5 ⋅ m = 360

Now we have our expression. All that's left to do is solve it.

Step 4: Solve the problem.

This problem will take a few steps to solve. (If you're not sure how to do the math in this section, you might want to review our lesson on simplifying expressions .) First, let's simplify the expression as much as possible. We can multiply 30 and 2, so let's go ahead and do that. We can also write .5 ⋅ m as 0.5 m .

60 + .5m = 360

Next, we need to do what we can to get the m alone on the left side of the equals sign. Once we do that, we'll know what m is equal to—in other words, it will let us know the number of miles in our word problem.

We can start by getting rid of the 60 on the left side by subtracting it from both sides .

The only thing left to get rid of is .5 . Since it's being multiplied with m , we'll do the reverse and divide both sides of the equation with it.

.5 m / .5 is m and 300 / 0.50 is 600 , so m = 600 . In other words, the answer to our problem is 600 —we now know Jada drove 600 miles.

Step 5: Check the problem.

To make sure we solved the problem correctly, we should check our work. To do this, we can use the answer we just got— 600 —and calculate backward to find another of the quantities in our problem. In other words, if our answer for Jada's distance is correct, we should be able to use it to work backward and find another value, like the total cost. Let's take another look at the problem.

According to the problem, the van costs $30 per day and $0.50 per mile. If Jada really did drive 600 miles in 2 days, she could calculate the cost like this:

$30 per day and $0.50 per mile

30 ⋅ day + .5 ⋅ mile

30 ⋅ 2 + .5 ⋅ 600

According to our math, the van would cost $360, which is exactly what the problem says. This means our solution was correct. We're done!

While some word problems will be more complicated than others, you can use these basic steps to approach any word problem. On the next page, you can try it for yourself.

Let's practice with a couple more problems. You can solve these problems the same way we solved the first one—just follow the problem-solving steps we covered earlier. For your reference, these steps are:

If you get stuck, you might want to review the problem on page 1. You can also take a look at our lesson on writing algebraic expressions for some tips on translating written words into math.

Try completing this problem on your own. When you're done, move on to the next page to check your answer and see an explanation of the steps.

A single ticket to the fair costs $8. A family pass costs $25 more than half of that. How much does a family pass cost?

Here's another problem to do on your own. As with the last problem, you can find the answer and explanation to this one on the next page.

Flor and Mo both donated money to the same charity. Flor gave three times as much as Mo. Between the two of them, they donated $280. How much money did Mo give?

Problem 1 Answer

Here's Problem 1:

A single ticket to the fair costs $8. A family pass costs $25 more than half that. How much does a family pass cost?

Answer: $29

Let's solve this problem step by step. We'll solve it the same way we solved the problem on page 1.

Step 1: Read through the problem carefully

The first in solving any word problem is to find out what question the problem is asking you to solve and identify the information that will help you solve it . Let's look at the problem again. The question is right there in plain sight:

So is the information we'll need to answer the question:

• A single ticket costs $8 .
• The family pass costs $25 more than half the price of the single ticket.

Step 2: Represent the unknown numbers with variables

The unknown number in this problem is the cost of the family pass . We'll represent it with the variable f .

Step 3: Translate the rest of the problem

Let's look at the problem again. This time, the important facts are highlighted.

A single ticket to the fair costs $8 . A family pass costs $25 more than half that . How much does a family pass cost?

In other words, we could say that the cost of a family pass equals half of $8, plus $25 . To turn this into a problem we can solve, we'll have to translate it into math. Here's how:

• First, replace the cost of a family pass with our variable f .

f equals half of $8 plus $25

• Next, take out the dollar signs and replace words like plus and equals with operators.

f = half of 8 + 25

• Finally, translate the rest of the problem. Half of can be written as 1/2 times , or 1/2 ⋅ :

f = 1/2 ⋅ 8 + 25

Step 4: Solve the problem

Now all we have to do is solve our problem. Like with any problem, we can solve this one by following the order of operations.

• f is already alone on the left side of the equation, so all we have to do is calculate the right side.
• First, multiply 1/2 by 8 . 1/2 ⋅ 8 is 4 .
• Next, add 4 and 25. 4 + 25 equals 29 .

That's it! f is equal to 29. In other words, the cost of a family pass is $29 .

Step 5: Check your work

Finally, let's check our work by working backward from our answer. In this case, we should be able to correctly calculate the cost of a single ticket by using the cost we calculated for the family pass. Let's look at the original problem again.

We calculated that a family pass costs $29. Our problem says the pass costs $25 more than half the cost of a single ticket. In other words, half the cost of a single ticket will be $25 less than $29.

• We could translate this into this equation, with s standing for the cost of a single ticket.

1/2s = 29 - 25

• Let's work on the right side first. 29 - 25 is 4 .
• To find the value of s , we have to get it alone on the left side of the equation. This means getting rid of 1/2 . To do this, we'll multiply each side by the inverse of 1/2: 2 .

According to our math, s = 8 . In other words, if the family pass costs $29, the single ticket will cost $8. Looking at our original problem, that's correct!

So now we're sure about the answer to our problem: The cost of a family pass is $29 .

Problem 2 Answer

Here's Problem 2:

Answer: $70

Let's go through this problem one step at a time.

Start by asking what question the problem is asking you to solve and identifying the information that will help you solve it . What's the question here?

To solve the problem, you'll have to find out how much money Mo gave to charity. All the important information you need is in the problem:

• The amount Flor donated is three times as much the amount Mo donated
• Flor and Mo's donations add up to $280 total

The unknown number we're trying to identify in this problem is Mo's donation . We'll represent it with the variable m .

Here's the problem again. This time, the important facts are highlighted.

Flor and Mo both donated money to the same charity. Flor gave three times as much as Mo . Between the two of them, they donated $280 . How much money did Mo give?

The important facts of the problem could also be expressed this way:

Mo's donation plus Flor's donation equals $280

Because we know that Flor's donation is three times as much as Mo's donation, we could go even further and say:

Mo's donation plus three times Mo's donation equals $280

We can translate this into a math problem in only a few steps. Here's how:

• Because we've already said we'll represent the amount of Mo's donation with the variable m , let's start by replacing Mo's donation with m .

m plus three times m equals $280

• Next, we can put in mathematical operators in place of certain words. We'll also take out the dollar sign.

m + three times m = 280

• Finally, let's write three times mathematically. Three times m can also be written as 3 ⋅ m , or just 3 m .

m + 3m = 280

It will only take a few steps to solve this problem.

• To get the correct answer, we'll have to get m alone on one side of the equation.
• To start, let's add m and 3 m . That's 4 m .
• We can get rid of the 4 next to the m by dividing both sides by 4. 4 m / 4 is m , and 280 / 4 is 70 .

We've got our answer: m = 70 . In other words, Mo donated $70 .

The answer to our problem is $70 , but we should check just to be sure. Let's look at our problem again.

If our answer is correct, $70 and three times $70 should add up to $280 .

• We can write our new equation like this:

70 + 3 ⋅ 70 = 280

• The order of operations calls for us to multiply first. 3 ⋅ 70 is 210.

70 + 210 = 280

• The last step is to add 70 and 210. 70 plus 210 equals 280 .

280 is the combined cost of the tickets in our original problem. Our answer is correct : Mo gave $70 to charity.

/en/algebra-topics/distance-word-problems/content/

How do you solve word problems in math?

Master word problems with eight simple steps from a math tutor!

Author Amber Watkins

Published April 2024

• Key takeaways
• Students who struggle with reading, tend to struggle with understanding and solving word problems. So the best way to solve word problems in math is to become a better reader!
• Mastery of word problems relies on your child’s knowledge of keywords for word problems in math and knowing what to do with them.
• There are 8 simple steps each child can use to solve word problems- let’s go over these together.

Table of contents

• How to solve word problems

Lesson credits

As a tutor who has seen countless math worksheets in almost every grade – I’ll tell you this: every child is going to encounter word problems in math. The key to mastery lies in how you solve them! So then, how do you solve word problems in math?

In this guide, I’ll share eight steps to solving word problems in math.

How to solve word problems in math in 8 steps

Step 1: read the word problem aloud.

For a child to understand a word problem, it needs to be read with accuracy and fluency! That is why, when I tutor children with word problems, I always emphasize the importance of reading properly.

Mastering step 1 looks like this:

• Allow your child to read the word problem aloud to you. 
• Don’t let your child skip over or mispronounce any words. 
• If necessary, model how to read the word problem, then allow your child to read it again. Only after the word problem is read accurately, should you move on to step 2.

Step 2: Highlight the keywords in the word problem

The keywords for word problems in math indicate what math action should be taken. Teach your child to highlight or underline the keywords in every word problem. 

Here are some of the most common keywords in math word problems: 

• Subtraction words – less than, minus, take away
• Addition words – more than, altogether, plus, perimeter
• Multiplication words – Each, per person, per item, times, area 
• Division words – divided by, into
• Total words – in all, total, altogether

Let’s practice. Read the following word problem with your child and help them highlight or underline the main keyword, then decide which math action should be taken.

Michael has ten baseball cards. James has four baseball cards less than Michael. How many total baseball cards does James have? 

The words “less than” are the keywords and they tell us to use subtraction .

Step 3: Make math symbols above keywords to decode the word problem

As I help students with word problems, I write math symbols and numbers above the keywords. This helps them to understand what the word problem is asking.

Let’s practice. Observe what I write over the keywords in the following word problem and think about how you would create a math sentence using them:

Step 4: Create a math sentence to represent the word problem

Using the previous example, let’s write a math sentence. Looking at the math symbols and numbers written above the word problem, our math sentence should be: 10 – 5 = 5 ! 

Each time you practice a word problem with your child, highlight keywords and write the math symbols above them. Then have your child create a math sentence to solve. 

Step 5: Draw a picture to help illustrate the word problem

Pictures can be very helpful for problems that are more difficult to understand. They also are extremely helpful when the word problem involves calculating time , comparing fractions , or measurements . 

Step 6: Always show your work

Help your child get into the habit of always showing their work. As a tutor, I’ve found many reasons why having students show their work is helpful:

• By showing their work, they are writing the math steps repeatedly, which aids in memory
• If they make any mistakes they can track where they happened
• Their teacher can assess how much they understand by reviewing their work
• They can participate in class discussions about their work

Step 7: When solving word problems, make sure there is always a word in your answer!

If the word problem asks: How many peaches did Lisa buy? Your child’s answer should be: Lisa bought 10 peaches .

If the word problem asks: How far did Kyle run? Your child’s answer should be: Kyle ran 20 miles .

So how do you solve a word problem in math?

Together we reviewed the eight simple steps to solve word problems. These steps included identifying keywords for word problems in math, drawing pictures, and learning to explain our answers. 

Is your child ready to put these new skills to the test? Check out the best math app for some fun math word problem practice.

Parents, sign up for a DoodleMath subscription and see your child become a math wizard!

Amber Watkins

Amber is an education specialist with a degree in Early Childhood Education. She has over 12 years of experience teaching and tutoring elementary through college level math. "Knowing that my work in math education makes such an impact leaves me with an indescribable feeling of pride and joy!"

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problem-solving

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Word problems

Here is a list of all of the skills that cover word problems! These skills are organized by grade, and you can move your mouse over any skill name to preview the skill. To start practicing, just click on any link. IXL will track your score, and the questions will automatically increase in difficulty as you improve!

Here is a list of all of the skills that cover word problems! To start practicing, just click on any link.

Pre-K skills

• V.8 Addition word problems with pictures - sums up to 5
• W.8 Addition word problems with pictures - sums up to 10
• X.7 Subtraction word problems with pictures - numbers up to 5
• Y.7 Subtraction word problems with pictures - numbers up to 10

Kindergarten skills

• Q.1 Build cube trains to solve addition word problems - sums up to 5
• Q.2 Addition word problems with pictures - sums up to 5
• Q.3 Write addition sentences for word problems with pictures - sums up to 5
• Q.4 Addition word problems - sums up to 5
• Q.5 Model and write addition sentences for word problems - sums up to 5
• U.1 Build cube trains to solve addition word problems - sums up to 10
• U.2 Addition word problems with pictures - sums up to 10
• U.3 Write addition sentences for word problems with pictures - sums up to 10
• U.4 Addition word problems - sums up to 10
• U.5 Model and write addition sentences for word problems - sums up to 10
• V.4 Subtraction sentences up to 5 - what does the cube train show?
• X.1 Subtraction word problems with pictures - numbers up to 5
• X.2 Write subtraction sentences for word problems with pictures - up to 5
• X.3 Use cube trains to solve subtraction word problems - up to 5
• X.4 Subtraction word problems - numbers up to 5
• X.5 Model and write subtraction sentences for word problems - up to 5
• Y.4 Subtraction sentences up to 10 - what does the cube train show?
• AA.1 Subtraction word problems with pictures - numbers up to 10
• AA.2 Write subtraction sentences for word problems with pictures - up to 10
• AA.3 Use cube trains to solve subtraction word problems - up to 10
• AA.4 Subtraction word problems - numbers up to 10
• AA.5 Model and write subtraction sentences for word problems - up to 10
• CC.1 Addition and subtraction word problems with pictures
• CC.2 Use cube trains to solve addition and subtraction word problems - up to 10
• CC.3 Addition and subtraction word problems
• CC.4 Model and write addition and subtraction sentences for word problems

First-grade skills

• C.6 Skip-counting patterns - with tables
• H.1 Addition word problems with pictures - sums up to 10
• H.2 Write addition sentences for word problems with pictures - sums up to 10
• H.3 Build cube trains to solve addition word problems - sums up to 10
• H.4 Addition word problems - sums up to 10
• H.5 Model and write addition sentences for word problems - sums up to 10
• H.6 Addition sentences for word problems - sums up to 10
• I.5 Subtraction sentences up to 10: what does the cube train show?
• L.1 Subtraction word problems with pictures - up to 10
• L.2 Write subtraction sentences for word problems with pictures - up to 10
• L.3 Use cube trains to solve subtraction word problems - up to 10
• L.4 Subtraction word problems - up to 10
• L.5 Model and write subtraction sentences for word problems - up to 10
• L.6 Subtraction sentences for "take apart" word problems - up to 10
• L.7 Subtraction sentences for word problems - up to 10
• N.1 Comparison word problems up to 10: how many more?
• N.2 Subtraction sentences for comparison word problems up to 10: how many more?
• N.3 Comparison word problems up to 10: how many fewer?
• N.4 Subtraction sentences for comparison word problems up to 10: how many fewer?
• N.5 Comparison word problems up to 10: how many more or fewer?
• N.6 Subtraction sentences for comparison word problems up to 10: how many more or fewer?
• N.7 Comparison word problems up to 10: what is the larger amount?
• N.8 Comparison word problems up to 10: what is the smaller amount?
• N.9 Comparison word problems up to 10
• O.1 Addition and subtraction word problems with pictures - up to 10
• O.2 Use cube trains to solve addition and subtraction word problems - up to 10
• O.3 Word problems with unknown sums and differences - up to 10
• O.4 Addition and subtraction sentences for word problems - up to 10
• O.5 Word problems with change unknown - up to 10
• O.6 Word problems with start unknown - up to 10
• O.7 Word problems with one addend unknown - up to 10
• O.8 Word problems with both addends unknown - up to 10
• O.9 Word problems involving addition and subtraction - up to 10
• O.10 Match word problems to addition and subtraction sentences - up to 10
• R.1 Addition word problems with models - sums up to 20
• R.2 Addition word problems - sums up to 20
• R.3 Addition sentences for word problems - sums up to 20
• R.4 Add three numbers - word problems
• U. New! Subtraction word problems with models - up to 20
• U.1 Subtraction word problems - up to 20
• U.2 Subtraction sentences for word problems - up to 20
• W.1 Comparison word problems up to 20: how many more or fewer?
• W.2 Comparison word problems up to 20: what is the larger amount?
• W.3 Comparison word problems up to 20: what is the smaller amount?
• W.4 Comparison word problems up to 20: part 1
• W.5 Comparison word problems up to 20: part 2
• X. New! Word problems with sum or difference unknown - up to 20
• X. New! Word problems with change unknown - up to 20
• X. New! Word problems with start unknown - up to 20
• X.1 Word problems with one addend unknown - up to 20
• X.2 Word problems with both addends unknown - up to 20
• X.3 Use models to solve word problems involving addition and subtraction - up to 20
• X.4 Word problems involving addition and subtraction without comparisons - up to 20
• X. New! Word problems involving addition and subtraction - up to 20
• X.5 Addition and subtraction sentences for word problems - up to 20
• X.6 Match word problems to addition and subtraction sentences - up to 20
• BB.4 Compare numbers up to 100: word problems
• DD.13 Addition word problems - one-digit plus two-digit numbers
• DD.14 Addition sentences for word problems - one-digit plus two-digit numbers
• EE.11 Customary units of length: word problems
• EE.13 Metric units of length: word problems
• FF.7 Time and clocks: word problems
• HH.8 Money - word problems

Second-grade skills

• B.5 Greatest and least - word problems - up to 100
• B.6 Greatest and least - word problems - up to 1,000
• C.6 Skip-counting stories
• C.10 Skip-counting puzzles
• G.4 Addition word problems - sums to 20
• G.5 Addition sentences for word problems - sums to 20
• G.10 Addition word problems - three one-digit numbers
• G.12 Addition word problems - four or more one-digit numbers
• I.3 Subtraction word problems - up to 20
• I.4 Subtraction sentences for word problems - up to 20
• K.1 Comparison word problems - up to 20
• K.2 Use models to solve addition and subtraction word problems - up to 20
• K.3 Addition and subtraction word problems - up to 20
• K.4 Match word problems to addition and subtraction sentences - up to 20
• K.5 Two-step addition and subtraction word problems - up to 20
• K.6 Solve word problems using guess-and-check - up to 20
• L.16 Guess the number
• N.8 Addition word problems - up to two digits
• N.16 Addition word problems - three numbers up to two digits each
• N.19 Addition word problems - four numbers up to two digits each
• P.10 Subtraction word problems - up to two digits
• R. New! Use models to solve addition and subtraction word problems - up to 100
• R.1 Addition and subtraction word problems - up to 100
• R.2 Two-step addition and subtraction word problems - up to 100
• T.5 Addition word problems - up to three digits
• V.6 Subtraction word problems - up to three digits
• W.4 Addition and subtraction word problems - up to 1,000
• X.6 Solve word problems using repeated addition - sums to 25
• AA.14 Making change
• BB.2 Add money up to $1: word problems
• BB.4 Subtract money up to $1: word problems
• BB.6 Add and subtract money up to $1: word problems
• GG.6 Compare lengths: customary units
• GG.7 Customary units of length: word problems
• HH.4 Compare lengths: metric units
• HH.5 Metric units of length: word problems

Third-grade skills

• A.6 Place value word problems
• A.7 Guess the number
• B.5 Ordering puzzles
• D.3 Estimate sums by rounding: word problems
• E.3 Estimate differences by rounding: word problems
• F.3 Estimate sums and differences: word problems
• G.7 Add two numbers up to three digits: word problems
• G.12 Add three numbers up to three digits each: word problems
• H.7 Subtract numbers up to three digits: word problems
• I.3 Add two numbers up to four digits: word problems
• I.7 Add three numbers up to four digits each: word problems
• J.3 Subtract two numbers up to four digits: word problems
• K.6 Addition and subtraction word problems
• K.7 Age puzzles
• K.8 Find two numbers based on sum and difference
• M.3 Skip-counting puzzles
• S.1 Use equal groups and arrays to solve multiplication word problems
• S.2 Multiplication word problems with factors up to 5
• S.3 Use strip models to solve multiplication word problems
• S.4 Multiplication word problems with factors up to 10
• S.5 Multiplication word problems with factors up to 5: find the missing number
• S.6 Multiplication word problems with factors up to 10: find the missing number
• S.7 Compare numbers using multiplication: word problems
• T.7 Multiply one-digit numbers by two-digit numbers: word problems
• T.9 Multiply three numbers: word problems
• Y.1 Use equal groups to solve division word problems
• Y.2 Use arrays to solve division word problems
• Y.3 Use equal groups and arrays to solve division word problems
• Y.4 Division word problems
• Z.6 Multiplication and division word problems
• AA.4 Addition, subtraction, multiplication, and division word problems
• AA.5 Find two numbers based on sum, difference, product, and quotient
• BB.1 Two-step addition and subtraction word problems
• BB.2 Two-step multiplication and division word problems
• BB.3 Two-step mixed operation word problems
• BB.4 Two-step word problems: identify reasonable answers
• CC.5 Write equations with unknown numbers to represent word problems: multiplication and division only
• CC.6 Write equations with unknown numbers to represent word problems
• FF.1 Unit fractions: modeling word problems
• FF.2 Unit fractions: word problems
• FF.3 Fractions of a whole: modeling word problems
• FF.4 Fractions of a whole: word problems
• FF.5 Fractions of a group: word problems
• KK.5 Compare fractions in recipes
• MM.12 Find the area of rectangles: word problems
• MM.13 Find the missing side length of a rectangle: word problems
• NN.6 Perimeter: word problems
• OO.1 Find the area, perimeter, or side length: word problems
• SS.3 Find the end time: word problems
• SS.4 Find the elapsed time: word problems
• SS.5 Find start and end times: two-step word problems
• UU.6 Measurement word problems
• WW.6 Making change
• WW.10 Add money amounts - word problems

Fourth-grade skills

• A.9 Place value word problems
• B.5 Find the order
• C.5 Rounding puzzles
• D.2 Estimate sums: word problems
• D.4 Add two multi-digit numbers: word problems
• E.2 Estimate differences: word problems
• E.4 Subtract two multi-digit numbers: word problems
• F.8 Compare numbers using multiplication: word problems
• F.9 Comparison word problems: addition or multiplication?
• G.2 Divisibility rules: word problems
• H.4 Estimate products word problems: identify reasonable answers
• H.10 Multiply 1-digit numbers by 2-digit numbers: word problems
• H.11 Multiply 1-digit numbers by 2-digit numbers: multi-step word problems
• H.17 Multiply 1-digit numbers by 3-digit or 4-digit numbers: word problems
• H.18 Multiply 1-digit numbers by 3-digit or 4-digit numbers: multi-step word problems
• I.3 Multiply two multiples of ten: word problems
• I.5 Estimate products: word problems
• I.11 Multiply a 2-digit number by a 2-digit number: word problems
• I.12 Multiply a 2-digit number by a 2-digit number: multi-step word problems
• J.2 Division facts to 10: word problems
• J.4 Division facts to 12: word problems
• L.1 Divide numbers ending in zeros by 1-digit numbers: word problems
• L.2 Divide 2-digit numbers by 1-digit numbers: interpret remainders
• L.3 Divide 2-digit numbers by 1-digit numbers: word problems
• L.4 Divide larger numbers by 1-digit numbers: interpret remainders
• L.5 Divide larger numbers by 1-digit numbers: word problems
• M.2 Estimate sums, differences, products, and quotients: word problems
• M.5 Addition, subtraction, multiplication, and division word problems
• M.7 Find two numbers based on sum and difference
• M.9 Find two numbers based on sum, difference, product, and quotient
• M.11 Write equations to represent word problems
• M.13 Use equations to solve addition and subtraction word problems
• N.1 Multi-step addition and subtraction word problems
• N.2 Multi-step word problems with strip diagrams
• N.3 Use strip diagrams to represent and solve multi-step word problems
• N.4 Multi-step word problems
• N.5 Multi-step word problems involving remainders
• N.6 Multi-step word problems: identify reasonable answers
• N.7 Word problems with extra or missing information
• N.8 Solve word problems using guess-and-check
• O.7 Number patterns: word problems
• P.1 Fractions of a whole: word problems
• P.2 Fractions of a group: word problems
• T.4 Add and subtract fractions with like denominators: word problems
• T.5 Add and subtract fractions with like denominators in recipes
• T.12 Add and subtract mixed numbers with like denominators in recipes
• T.13 Add and subtract mixed numbers with like denominators: word problems
• U.7 Add and subtract fractions with unlike denominators: word problems
• V.6 Multiply unit fractions by whole numbers: word problems
• W.7 Multiply fractions by whole numbers: word problems
• W.10 Multiply fractions and mixed numbers by whole numbers in recipes
• W.13 Fractions of a number: word problems
• Z.7 Add and subtract decimals: word problems
• Z.10 Add 3 or more decimals: word problems
• Z.13 Solve decimal problems using diagrams
• AA.5 Find the change, price, or amount paid
• AA.8 Multi-step word problems with money: addition and subtraction only
• AA.9 Multi-step word problems with money
• CC.5 Elapsed time: word problems
• CC.6 Find start and end times: multi-step word problems
• FF.1 Measurement word problems
• FF.2 Measurement word problems with fractions
• HH.8 Relationship between area and perimeter
• HH.9 Area and perimeter: word problems
• HH.10 Rectangles: relationship between perimeter and area word problems

Fifth-grade skills

• B.2 Estimate sums and differences: word problems
• B.4 Add and subtract whole numbers: word problems
• D.3 Multiply numbers ending in zeros: word problems
• D.6 Estimate products: word problems
• D.8 Multiply by 1-digit numbers: word problems
• D.13 Multiply by 2-digit numbers: word problems
• E.3 Divide numbers ending in zeros: word problems
• E.7 Divide by 1-digit numbers: interpret remainders
• E.8 Divide multi-digit numbers by 1-digit numbers: word problems
• E.12 Divide 2-digit and 3-digit numbers by 2-digit numbers: word problems
• E.14 Divide 4-digit numbers by 2-digit numbers: word problems
• F.5 Divisibility rules: word problems
• G.2 Add, subtract, multiply, and divide whole numbers: word problems
• I.1 Write numerical expressions for word problems
• I.2 Multi-step word problems
• I.3 Multi-step word problems involving remainders
• I.4 Multi-step word problems: identify reasonable answers
• L.7 Add and subtract fractions with unlike denominators: word problems
• L.9 Add 3 or more fractions: word problems
• M.6 Add and subtract mixed numbers: word problems
• M.7 Add and subtract fractions and mixed numbers in recipes
• O.3 Multiply fractions by whole numbers: word problems
• O.6 Fractions of a number: word problems
• P.2 Multiply two fractions: word problems
• R.7 Multiplication with mixed numbers: word problems
• R.8 Multiply fractions and mixed numbers in recipes
• V.2 Add, subtract, multiply, and divide fractions and mixed numbers: word problems
• X.6 Compare, order, and round decimals: word problems
• AA.6 Add and subtract decimals: word problems
• CC.8 Multiply decimals and whole numbers: word problems
• FF.7 Division with decimal quotients: word problems
• GG.2 Add, subtract, multiply, and divide decimals: word problems
• HH.2 Add and subtract money: word problems
• HH.3 Add and subtract money: multi-step word problems
• HH.5 Multiply money amounts: word problems
• HH.6 Multiply money amounts: multi-step word problems
• HH.8 Divide money amounts: word problems
• HH.11 Find the number of each type of coin
• II.10 Multi-step problems with customary unit conversions
• JJ.8 Multi-step problems with metric unit conversions
• JJ.9 Multi-step problems with customary or metric unit conversions
• KK.5 Number patterns: word problems
• MM.2 Write variable expressions: word problems
• MM.4 Write variable equations: word problems
• TT.7 Area and perimeter: word problems
• UU.4 Volume of rectangular prisms made of unit cubes: word problems
• UU.6 Volume of cubes and rectangular prisms: word problems
• UU.7 Compare volumes and dimensions of rectangular prisms: word problems
• VV.1 Income and payroll taxes: understanding pay stubs
• VV.2 Income and payroll taxes: word problems
• VV.3 Sales and property taxes: word problems
• VV.9 Reading financial records
• VV.10 Keeping financial records

Sixth-grade skills

• A.2 Add and subtract whole numbers: word problems
• B.2 Multiply whole numbers: word problems
• B.4 Multiply numbers ending in zeros: word problems
• C.3 Divide numbers ending in zeros: word problems
• E.2 Add, subtract, multiply, or divide two whole numbers: word problems
• E.3 Estimate to solve word problems
• E.4 Multi-step word problems
• E.5 Multi-step word problems: identify reasonable answers
• F.10 GCF and LCM: word problems
• H.2 Add and subtract decimals: word problems
• H.3 Add and subtract money amounts: word problems
• I.6 Divide decimals by whole numbers: word problems
• I.11 Multiply and divide decimals: word problems
• J.2 Add, subtract, multiply, or divide two decimals: word problems
• K.2 Add and subtract fractions with like denominators: word problems
• K.4 Add and subtract fractions with unlike denominators: word problems
• K.7 Add and subtract mixed numbers: word problems
• L.3 Multiply fractions by whole numbers: word problems
• L.7 Multiply fractions: word problems
• L.14 Multiply mixed numbers: word problems
• M.5 Divide fractions by whole numbers in recipes
• M.12 Divide fractions and mixed numbers: word problems
• N.2 Add, subtract, multiply, or divide two fractions: word problems
• O.10 Absolute value and integers: word problems
• P.8 Add and subtract integers: word problems
• Q.6 Compare and order rational numbers: word problems
• S.3 Write a ratio: word problems
• S.8 Equivalent ratios: word problems
• S.11 Calculate speed, distance, or time: word problems
• S.12 Ratios and rates: complete a table and make a graph
• S.13 Use tape diagrams to solve ratio word problems
• S.14 Compare ratios: word problems
• S.15 Compare rates: word problems
• S.16 Ratios and rates: word problems
• S.19 Scale drawings: word problems
• T.3 Identify proportional relationships by graphing
• T.4 Interpret graphs of proportional relationships
• U.5 Convert between percents, fractions, and decimals: word problems
• U.7 Compare percents and fractions: word problems
• V.5 Percents of numbers: word problems
• V.8 Find what percent one number is of another: word problems
• V.11 Solve percent word problems
• W.10 Compare temperatures above and below zero
• X.7 Percents - calculate tax, tip, mark-up, and more
• Y.3 Write variable expressions: word problems
• Y.7 Evaluate variable expressions: word problems
• AA.13 Solve one-step addition and subtraction equations: word problems
• AA.14 Solve one-step multiplication and division equations: word problems
• AA.15 Write a one-step equation: word problems
• AA.16 Solve one-step equations: word problems
• AA.17 Which word problem matches the one-step equation?
• BB.4 Write and graph inequalities: word problems
• CC.2 Identify independent and dependent variables in tables and graphs
• CC.4 Identify independent and dependent variables: word problems
• CC.6 Find a value using two-variable equations: word problems
• CC.7 Solve word problems by finding two-variable equations
• CC.13 Graph a two-variable equation
• CC.14 Interpret a graph: word problems
• GG.17 Area of quadrilaterals and triangles: word problems
• HH.3 Volume of cubes and rectangular prisms: word problems
• JJ.10 Interpret measures of center and variability
• KK.1 Counting principle
• LL.1 Compare checking accounts

Seventh-grade skills

• A.6 Quantities that combine to zero: word problems
• B.14 Add and subtract integers: word problems
• D.2 Add and subtract decimals: word problems
• D.4 Multiply decimals and whole numbers: word problems
• D.6 Divide decimals by whole numbers: word problems
• D.9 Add, subtract, multiply, and divide decimals: word problems
• E.4 GCF and LCM: word problems
• F.1 Understanding fractions: word problems
• F.4 Fractions: word problems with graphs and tables
• F.7 Compare fractions: word problems
• G.2 Add and subtract fractions: word problems
• G.4 Add and subtract mixed numbers: word problems
• G.10 Multiply fractions and mixed numbers: word problems
• G.14 Divide fractions and mixed numbers: word problems
• G.16 Add, subtract, multiply, and divide fractions and mixed numbers: word problems
• I.6 Identify quotients of rational numbers: word problems
• I.11 Multi-step word problems with positive rational numbers
• L.4 Equivalent ratios: word problems
• L.7 Compare ratios: word problems
• L.8 Compare rates: word problems
• L.10 Do the ratios form a proportion: word problems
• L.12 Solve proportions: word problems
• L.13 Estimate population size using proportions
• N.1 Find the constant of proportionality from a table
• N.2 Write equations for proportional relationships from tables
• N.3 Identify proportional relationships by graphing
• N.4 Find the constant of proportionality from a graph
• N.5 Write equations for proportional relationships from graphs
• N.10 Interpret graphs of proportional relationships
• N.11 Write and solve equations for proportional relationships
• O.7 Percents of numbers: word problems
• O.9 Solve percent equations: word problems
• O.11 Percent of change: word problems
• O.12 Percent of change: find the original amount word problems
• O.13 Percent error: word problems
• P.1 Add, subtract, multiply, and divide money amounts: word problems
• P.8 Find the percent: tax, discount, and more
• P.10 Multi-step problems with percents
• R.3 Write variable expressions: word problems
• S.14 Identify equivalent linear expressions: word problems
• T.11 Choose two-step equations: word problems
• T.12 Solve two-step equations: word problems
• U.6 One-step inequalities: word problems
• V.5 Sequences: word problems
• X.1 Identify independent and dependent variables
• X.8 Interpret a graph: word problems
• BB.4 Area and perimeter: word problems
• BB.7 Circles: word problems
• CC.6 Volume of cubes and rectangular prisms: word problems
• DD.2 Scale drawings: word problems
• DD.3 Scale drawings: scale factor word problems
• HH. New! Populations and samples
• HH.9 Make inferences from multiple samples
• HH.10 Compare populations using measures of center and spread
• II.4 Experimental probability
• II.10 Find the number of outcomes: word problems

Eighth-grade skills

• A.6 Add and subtract integers: word problems
• B.7 Add and subtract rational numbers: word problems
• B.10 Multiply and divide rational numbers: word problems
• B.14 Multi-step word problems
• G.2 Solve proportions: word problems
• G.3 Estimate population size using proportions
• G.4 Scale drawings: word problems
• G.5 Scale drawings: scale factor word problems
• H.4 Find what percent one number is of another: word problems
• H.7 Percents of numbers: word problems
• H.11 Percent of change: word problems
• H.12 Percent of change: find the original amount word problems
• I.6 Find the percent: tax, discount, and more
• I.8 Multi-step problems with percents
• K.4 Write variable expressions: word problems
• L.9 Identify equivalent linear expressions: word problems
• M.10 Solve one-step and two-step equations: word problems
• M.14 Solve equations with variables on both sides: word problems
• T.5 Pythagorean theorem: word problems
• V.3 Area and perimeter: word problems
• V.5 Circles: word problems
• X.1 Find the constant of proportionality from a table
• X.2 Write equations for proportional relationships from tables
• X.3 Identify proportional relationships by graphing
• X.4 Find the constant of proportionality from a graph
• X.5 Write equations for proportional relationships from graphs
• X.8 Identify proportional relationships: word problems
• X.9 Graph proportional relationships and find the slope
• X.10 Interpret graphs of proportional relationships
• X.11 Write and solve equations for proportional relationships
• X.12 Compare proportional relationships represented in different ways
• BB.3 Identify independent and dependent variables
• CC.4 Interpret points on the graph of a linear function
• CC.6 Interpret the slope and y-intercept of a linear function
• CC.10 Write linear functions: word problems
• FF.5 Sequences: word problems
• GG.3 Solve a system of equations by graphing: word problems
• GG.9 Solve a system of equations using substitution: word problems
• GG.11 Solve a system of equations using elimination: word problems
• GG.13 Solve a system of equations using any method: word problems
• II.10 Interpret lines of best fit: word problems
• JJ.3 Experimental probability
• JJ.10 Counting principle

Algebra 1 skills

• C.9 Solve one-step and two-step linear equations: word problems
• C.11 Consecutive integer problems
• C.16 Solve linear equations with variables on both sides: word problems
• D.1 Area and perimeter: word problems
• E.1 Scale drawings: word problems
• E.5 Multi-step problems with unit conversions
• E.6 Rate of travel: word problems
• E.7 Weighted averages: word problems
• M.3 Identify independent and dependent variables
• N.4 Evaluate a linear function from its graph: word problems
• N.5 Interpret the slope and y-intercept of a linear function
• N.8 Domain and range of linear functions: word problems
• O.3 Solve a system of equations by graphing: word problems
• O.9 Solve a system of equations using substitution: word problems
• O.11 Solve a system of equations using elimination: word problems
• O.13 Solve a system of equations using augmented matrices: word problems
• O.15 Solve a system of equations using any method: word problems
• P.5 Write two-variable inequalities: word problems
• V.7 Write exponential functions: word problems
• V.8 Exponential growth and decay: word problems
• Z.7 Solve quadratic equations: word problems
• AA.5 Write linear and exponential functions: word problems
• JJ.6 Interpret lines of best fit: word problems
• JJ.8 Interpret regression lines
• JJ.9 Analyze a regression line of a data set
• KK.6 Identify independent and dependent events
• KK.8 Counting principle
• KK.9 Permutations

Geometry skills

• A.1 Identify hypotheses and conclusions
• A.2 Counterexamples
• P.1 Pythagorean theorem
• V.9 Calculate density, mass, and volume
• AA.5 Counting principle
• AA.6 Permutations
• AA.15 Find probabilities using the addition rule

Algebra 2 skills

• B.3 Solve linear equations: word problems
• E.3 Solve a system of equations by graphing: word problems
• E.7 Solve a system of equations using substitution: word problems
• E.9 Solve a system of equations using elimination: word problems
• E.11 Solve a system of equations using any method: word problems
• F.1 Write two-variable linear inequalities: word problems
• O.7 Solve quadratic equations: word problems
• Z.3 Write exponential functions: word problems
• Z.9 Exponential growth and decay: word problems
• CC.4 Compound interest: word problems
• CC.5 Continuously compounded interest: word problems
• OO.2 Counting principle
• OO.4 Find probabilities using combinations and permutations
• OO.5 Find probabilities using two-way frequency tables
• OO.10 Find conditional probabilities using two-way frequency tables
• OO.11 Find probabilities using the addition rule
• PP.8 Write the probability distribution for a game of chance
• PP.9 Expected values for a game of chance
• PP.10 Choose the better bet
• QQ.1 Find probabilities using the binomial distribution
• RR.5 Find confidence intervals for population means
• RR.6 Find confidence intervals for population proportions
• RR.7 Interpret confidence intervals for population means
• RR.8 Experiment design
• RR.9 Analyze the results of an experiment using simulations
• SS.5 Interpret regression lines
• SS.6 Analyze a regression line of a data set
• TT.19 Solve a system of equations using augmented matrices: word problems

Precalculus skills

• D.9 Solve quadratic equations: word problems
• K.5 Exponential growth and decay: word problems
• K.6 Compound interest: word problems
• N.2 Solve a system of equations by graphing: word problems
• N.5 Solve a system of equations using substitution: word problems
• N.7 Solve a system of equations using elimination: word problems
• N.9 Solve a system of equations using augmented matrices: word problems
• CC.3 Find probabilities using combinations and permutations
• CC.4 Find probabilities using two-way frequency tables
• CC.8 Find conditional probabilities using two-way frequency tables
• CC.9 Find probabilities using the addition rule
• DD.8 Write the probability distribution for a game of chance
• DD.9 Expected values for a game of chance
• DD.10 Choose the better bet
• EE.1 Find probabilities using the binomial distribution
• EE.2 Mean, variance, and standard deviation of binomial distributions
• EE.9 Use normal distributions to approximate binomial distributions
• FF.5 Find confidence intervals for population means
• FF.6 Find confidence intervals for population proportions
• FF.7 Interpret confidence intervals for population means
• FF.8 Experiment design
• FF.9 Analyze the results of an experiment using simulations
• GG.5 Interpret regression lines
• GG.6 Analyze a regression line of a data set
• GG.7 Analyze a regression line using statistics of a data set

Solving Word Questions

With LOTS of examples!

In Algebra we often have word questions like:

Example: Sam and Alex play tennis.

On the weekend Sam played 4 more games than Alex did, and together they played 12 games.

How many games did Alex play?

How do we solve them?

The trick is to break the solution into two parts:

Turn the English into Algebra.

Then use Algebra to solve.

Turning English into Algebra

To turn the English into Algebra it helps to:

• Read the whole thing first
• Do a sketch if possible
• Assign letters for the values
• Find or work out formulas

You should also write down what is actually being asked for , so you know where you are going and when you have arrived!

Also look for key words:

Thinking Clearly

Some wording can be tricky, making it hard to think "the right way around", such as:

Example: Sam has 2 dollars less than Alex. How do we write this as an equation?

• Let S = dollars Sam has
• Let A = dollars Alex has

Now ... is that: S − 2 = A

or should it be: S = A − 2

or should it be: S = 2 − A

The correct answer is S = A − 2

( S − 2 = A is a common mistake, as the question is written "Sam ... 2 less ... Alex")

Example: on our street there are twice as many dogs as cats. How do we write this as an equation?

• Let D = number of dogs
• Let C = number of cats

Now ... is that: 2D = C

or should it be: D = 2C

Think carefully now!

The correct answer is D = 2C

( 2D = C is a common mistake, as the question is written "twice ... dogs ... cats")

Let's start with a really simple example so we see how it's done:

Example: A rectangular garden is 12m by 5m, what is its area ?

Turn the English into Algebra:

• Use w for width of rectangle: w = 12m
• Use h for height of rectangle: h = 5m

Formula for Area of a Rectangle : A = w × h

We are being asked for the Area.

A = w × h = 12 × 5 = 60 m 2

The area is 60 square meters .

Now let's try the example from the top of the page:

Example: Sam and Alex play Tennis. On the weekend Sam played 4 more games than Alex did, and together they played 12 games. How many games did Alex play?

• Use S for how many games Sam played
• Use A for how many games Alex played

We know that Sam played 4 more games than Alex, so: S = A + 4

And we know that together they played 12 games: S + A = 12

We are being asked for how many games Alex played: A

Which means that Alex played 4 games of tennis.

Check: Sam played 4 more games than Alex, so Sam played 8 games. Together they played 8 + 4 = 12 games. Yes!

A slightly harder example:

Example: Alex and Sam also build tables. Together they make 10 tables in 12 days. Alex working alone can make 10 in 30 days. How long would it take Sam working alone to make 10 tables?

• Use a for Alex's work rate
• Use s for Sam's work rate

12 days of Alex and Sam is 10 tables, so: 12a + 12s = 10

30 days of Alex alone is also 10 tables: 30a = 10

We are being asked how long it would take Sam to make 10 tables.

30a = 10 , so Alex's rate (tables per day) is: a = 10/30 = 1/3

Which means that Sam's rate is half a table a day (faster than Alex!)

So 10 tables would take Sam just 20 days.

Should Sam be paid more I wonder?

And another "substitution" example:

Example: Jenna is training hard to qualify for the National Games. She has a regular weekly routine, training for five hours a day on some days and 3 hours a day on the other days. She trains altogether 27 hours in a seven day week. On how many days does she train for five hours?

• The number of "5 hour" days: d
• The number of "3 hour" days: e

We know there are seven days in the week, so: d + e = 7

And she trains 27 hours in a week, with d 5 hour days and e 3 hour days: 5d + 3e = 27

We are being asked for how many days she trains for 5 hours: d

The number of "5 hour" days is 3

Check : She trains for 5 hours on 3 days a week, so she must train for 3 hours a day on the other 4 days of the week.

3 × 5 hours = 15 hours, plus 4 × 3 hours = 12 hours gives a total of 27 hours

Some examples from Geometry:

Example: A circle has an area of 12 mm 2 , what is its radius?

• Use A for Area: A = 12 mm 2
• Use r for radius

And the formula for Area is: A = π r 2

We are being asked for the radius.

We need to rearrange the formula to find the area

Example: A cube has a volume of 125 mm 3 , what is its surface area?

Make a quick sketch:

• Use V for Volume
• Use A for Area
• Use s for side length of cube
• Volume of a cube: V = s 3
• Surface area of a cube: A = 6s 2

We are being asked for the surface area.

First work out s using the volume formula:

Now we can calculate surface area:

An example about Money:

Example: Joel works at the local pizza parlor. When he works overtime he earns 1¼ times the normal rate. One week Joel worked for 40 hours at the normal rate of pay and also worked 12 hours overtime. If Joel earned $660 altogether in that week, what is his normal rate of pay?

• Joel's normal rate of pay: $N per hour
• Joel works for 40 hours at $N per hour = $40N
• When Joel does overtime he earns 1¼ times the normal rate = $1.25N per hour
• Joel works for 12 hours at $1.25N per hour = $(12 × 1¼N) = $15N
• And together he earned $660, so:

$40N + $(12 × 1¼N) = $660

We are being asked for Joel's normal rate of pay $N.

So Joel’s normal rate of pay is $12 per hour

Joel’s normal rate of pay is $12 per hour, so his overtime rate is 1¼ × $12 per hour = $15 per hour. So his normal pay of 40 × $12 = $480, plus his overtime pay of 12 × $15 = $180 gives us a total of $660

More about Money, with these two examples involving Compound Interest

Example: Alex puts $2000 in the bank at an annual compound interest of 11%. How much will it be worth in 3 years?

This is the compound interest formula:

So we will use these letters:

• Present Value PV = $2,000
• Interest Rate (as a decimal): r = 0.11
• Number of Periods: n = 3
• Future Value (the value we want): FV

We are being asked for the Future Value: FV

Example: Roger deposited $1,000 into a savings account. The money earned interest compounded annually at the same rate. After nine years Roger's deposit has grown to $1,551.33 What was the annual rate of interest for the savings account?

The compound interest formula:

• Present Value PV = $1,000
• Interest Rate (the value we want): r
• Number of Periods: n = 9
• Future Value: FV = $1,551.33

We are being asked for the Interest Rate: r

So the annual rate of interest is 5%

Check : $1,000 × (1.05) 9 = $1,000 × 1.55133 = $1,551.33

And an example of a Ratio question:

Example: At the start of the year the ratio of boys to girls in a class is 2 : 1 But now, half a year later, four boys have left the class and there are two new girls. The ratio of boys to girls is now 4 : 3 How many students are there altogether now?

• Number of boys now: b
• Number of girls now: g

The current ratio is 4 : 3

Which can be rearranged to 3b = 4g

At the start of the year there was (b + 4) boys and (g − 2) girls, and the ratio was 2 : 1

b + 4 g − 2 = 2 1

Which can be rearranged to b + 4 = 2(g − 2)

We are being asked for how many students there are altogether now: b + g

There are 12 girls !

And 3b = 4g , so b = 4g/3 = 4 × 12 / 3 = 16 , so there are 16 boys

So there are now 12 girls and 16 boys in the class, making 28 students altogether .

There are now 16 boys and 12 girls, so the ratio of boys to girls is 16 : 12 = 4 : 3 At the start of the year there were 20 boys and 10 girls, so the ratio was 20 : 10 = 2 : 1

And now for some Quadratic Equations :

Example: The product of two consecutive even integers is 168. What are the integers?

Consecutive means one after the other. And they are even , so they could be 2 and 4, or 4 and 6, etc.

We will call the smaller integer n , and so the larger integer must be n+2

And we are told the product (what we get after multiplying) is 168, so we know:

n(n + 2) = 168

We are being asked for the integers

That is a Quadratic Equation , and there are many ways to solve it. Using the Quadratic Equation Solver we get −14 and 12.

Check −14: −14(−14 + 2) = (−14)×(−12) = 168 YES

Check 12: 12(12 + 2) = 12×14 = 168 YES

So there are two solutions: −14 and −12 is one, 12 and 14 is the other.

Note: we could have also tried "guess and check":

• We could try, say, n=10: 10(12) = 120 NO (too small)
• Next we could try n=12: 12(14) = 168 YES

But unless we remember that multiplying two negatives make a positive we might overlook the other solution of (−14)×(−12).

Example: You are an Architect. Your client wants a room twice as long as it is wide. They also want a 3m wide veranda along the long side. Your client has 56 square meters of beautiful marble tiles to cover the whole area. What should the length of the room be?

Let's first make a sketch so we get things right!:

• the length of the room: L
• the width of the room: W
• the total Area including veranda: A
• the width of the room is half its length: W = ½L
• the total area is the (room width + 3) times the length: A = (W+3) × L = 56

We are being asked for the length of the room: L

This is a quadratic equation , there are many ways to solve it, this time let's use factoring :

And so L = 8 or −14

There are two solutions to the quadratic equation, but only one of them is possible since the length of the room cannot be negative!

So the length of the room is 8 m

L = 8, so W = ½L = 4

So the area of the rectangle = (W+3) × L = 7 × 8 = 56

There we are ...

... I hope these examples will help you get the idea of how to handle word questions. Now how about some practice?

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