assignment 11 writing equations from word problems

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and following it, you can be successful with word problems. So what should you do? Here are some recommended steps: to the quantity you are trying to find. you like. For example, if you are being asked to find a number, some students like to use the n. It is your choice. represents. will represent, you may think there is no need to write that down in words. However, by the time you read the problem several more times and solve the equation, it is easy to forget where you started. for the quantities given in the problem. does not necessarily you are finished with the problem. Many times you will need to take the answer you get from the and use it in some other way to answer the question originally given in the problem. true. If you are asked for a time value and end up with a negative number, this should indicate that you’ve made an error somewhere. If you are asked how fast a person is running and give an answer of 700 miles per hour, again you should be worried that there is an error. If you substitute these unreasonable answers into the you used in step 4 and it makes the true, then you should re-think the validity of your equation. Step 2: Assign a for the number. . Step 3: Write down what the represents. = a number Step 4: Write an equation. . If 6 is added to that, we get Step 5: Solve the equation. Step 6: Answer the question in the problem would be the number, so we have = 11. The number we are looking for is 11. Step 7: Check the answer. from Step 4. Step 2: Assign a for the number. . Step 3: Write down what the represents. = a number Step 4: Write an equation. + 9. We are then told to multiply that by -2, so we have Step 5: Solve the equation. Step 6: Answer the question in the problem would be the number, so we have = -5. The number we are looking for is -5. Step 7: Check the answer. from Step 4. Step 2: Assign a for the lowest test grade. . Step 3: Write down what the represents. = the lowest grade Step 4: Write an equation. Step 5: Solve the equation. Step 6: Answer the question in the problem would be the number, so we have = 48. The lowest grade on the algebra test was 48. Step 7: Check the answer. from Step 4. Step 2: Assign a for the number of tranquilizer prescriptions. . Step 3: Write down what the represents. = number of tranquilizer prescriptions Step 4: Write an equation. there were 4 prescriptions for antibiotics. Or if there were 30 tranquilizer prescriptions, then 4/3 as many for antibiotics, would there were 40 antibiotic prescriptions. In each case, we are taking the number of tranquilizers and multiplying by 4/3 to get the number of antibiotic prescriptions. So if is the number of tranquilizer prescriptions, then Step 5: Solve the equation. Step 6: Answer the question in the problem would be the number of prescriptions for tranquilizers, so we have = 36. There were 36 prescriptions for tranquilizers. Step 7: Check the answer. from Step 4. Step 2: Assign a variable. to. The number of miles driven by either Jamie or Rhonda will work. We need to just choose one and move to Step 3. Let’s assign a to represent the number of miles driving by Rhonda Let’s call it . Step 3: Write down what the represents. = the number of miles driven by Rhonda Step 4: Write an equation. , so the number of miles driven by Jamie is 2 . Together they drove a total of 90 miles. So we have (Rhonda) + (Jamie) = 90, or Step 5: Solve the equation. Step 6: Answer the question in the problem to the tells us = 30, which means Rhonda drove 30 miles. Now we have to find out how far Jamie drove. She drove twice as far as Rhonda, so the distance would be 20 miles. Step 7: Check the answer. from Step 4. Step 2: Assign a for the number of hours. . Step 3: Write down what the represents. = the number of hours Karen needs to work Step 4: Write an equation. . From that amount, we have to subtract the amount taken out for taxes and insurance. 25% of her salary is taken away. We need to write 25% as a decimal which gives 0.25. But we have to take 25% OF her salary or 25% of 6 . Karen’s goal is $450. We can now write an equation. (Salary) - 25%(Salary) = 450 Step 5: Solve the equation. Step 6: Answer the question in the problem = 100. Karen needs to work 100 hours to reach her goal of $450. Step 7: Check the answer. from Step 4. of a rectangular map is 15 inches and the is 50 inches. Find the width. Step 2: Assign a for the width. . Step 3: Write down what the represents. = the width of a Step 4: Write an equation. is 15 inches. We also know the is 50 inches. is the distance all the way around a figure. So to go all the way around a rectangle, you have = width + + width + length. is 15 inches, width is , and is 50, we get Step 5: Solve the equation. Step 6: Answer the question in the problem. = 10. The width of the is 10 inches. Don’t forget your units. Step 7: Check the answer. from Step 4. of a circular clock is 13.12 centimeters more than three times the radius. Find the of the face. of the of a circular clock. Step 2: Assign a for the radius. . Step 3: Write down what the represents. = the of the clock Step 4: Write an equation. and since those are two pieces of information in the problem. The formula for the is is 13.12 centimeters more than three times the radius. Three times the translates into 3 . Now we need to add 13.12 to that to get an for circumference. of a doesn’t change, these two expressions must be equal. Now we can set up the Step 5: Solve the equation. Step 6: Answer the question in the problem of the clock face. We decided that r would be the radius, so we have = 4. The of the clock is 4 centimeters. Don’t forget your units. Step 7: Check the answer. from Step 4.
		If 4 is subtracted from twice a number, the result is 10 less than the number. Find the number.
	What is your answer?

	Karin’s mom runs a dairy farm. Last year Betty the cow gave 375 gallons less than twice the amount from Bessie the cow. Together, Betty and Bessie produced 1464 gallons of milk. How many gallons did each cow give?
	What is your answer?

	Twice a number is added to the number and the answer is 90. Find the number.
	What is your answer?

	Jose has a board that is 44 inches long. He wishes to cut it into two pieces so that one piece will be 6 inches longer than the other. How long should the shorter piece be?
	What is your answer?

	Paula received a paycheck of $585. This amount reflects her weekly earnings less 10% of her earnings for deductions. How much was she paid before deductions were taken out?
	What is your answer?

	The of a triangular lot is 72 meters. One is 16 meters, and the other is twice the first side. Find the of the third side.
	What is your answer?

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Equation Word Problems Worksheets

This compilation of a meticulously drafted equation word problems worksheets is designed to get students to write and solve a variety of one-step, two-step and multi-step equations that involve integers, fractions, and decimals. These worksheets are best suited for students in grade 6 through high school. Click on the 'Free' icons to sample our handouts.

One Step Equation Word Problem Worksheets

Read and solve this series of word problems that involve one-step equations. Apply basic operations to find the value of unknowns.

(15 Worksheets)

Two-Step Equation Word Problems: Integers

Interpret this set of word problems that require two-step operations to solve the equations. Each printable worksheet has five word problems ideal for 6th grade, 7th grade, and 8th grade students.

• Download the set

Multi-Step Equation Word Problems: Integers

Read each multi-step word problem in these high school pdf worksheets and set up the equation. Solve and find the value of the unknown. More than two steps are required to solve the problems.

Two-Step Equation Word Problems: Fractions and Decimals

Read each word problem and set up the two-step equation. Solve the equation and find the solution. This selection of worksheets includes both fractions and decimals.

Multi-Step Equation Word Problems: Fractions and Decimals

Write multi-step equations that involve both fractions and decimals based on the word problems provided here. Validate your responses with our answer keys.

MCQ - Equation Word Problems

Pick the correct two-step equation that best matches word problems presented here. Evaluate the ability of students to solve two-step equations with this array of MCQ worksheets.

Related Worksheets

» One-Step Equation

» Two-Step Equation

» Multi-Step Equation

» Algebraic Identities

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Writing Equations From Word Problems – Differentiated Digital & Printable Worksheets

Updated:  28 Sep 2022

Practice writing equations from word problems with this set of differentiated worksheets.

Editable:  Google Slides

Non-Editable:  PDF

Pages:  8 Pages

• Curriculum Curriculum:  TEKS

Differentiated:  Yes

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Math 5.4(B)

Represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity;

Writing Equations from Word Problems

We all know that word problems are integral to a student’s math development. From a young age, students are exposed to word problems when learning about addition, subtraction, time, money, etc. Now that your students are working with higher-level math skills, there is an added twist to word problems that may come their way. Not only is it essential for students to know how to solve word problems, but they also have to know how to write equations with variables to represent a word problem. If you are looking for some worksheets to help with this skill, you’ve come to the right place!

This download includes two versions of the same resource, an on-grade level version and an advanced version. These worksheets aim to practice writing equations from word stories and visual models, using letters as the unknown quantity. A digital version is also available if you choose to assign this to your students in Google Classroom.  

An answer key is included with your download to make grading fast and easy!  

Tips for Differentiation + Scaffolding 

In addition to independent student work time, use this worksheet as an activity for:

• Guided math groups  
• Lesson warm-up
• Lesson wrap-up
• Fast finishers  
• Homework assignment
• Whole-class review (via smartboard)

If there are students who need a bit of a challenge, the resource has an advanced version with more complex numbers and multi-step equations for students who need a more challenging task on grade level. This version is marked with 2 stars in the upper right-hand corner.  

Consider assigning the on-grade level version of the resource for students who need additional support. This is marked with a single star in the upper right-hand corner of the worksheet. Students can complete this worksheet in a 1-on-1 or small group setting. Additionally, invite students to reference previous assignments, posters, or anchor charts while completing this activity.

🖨️ Easily Download & Print

Use the dropdown icon on the Download button to choose between the PDF or editable Google Slides version of this resource. 

Because this resource includes an answer sheet, we recommend you print one copy of the entire file. Then, make photocopies of the blank worksheet for students to complete. 

To save paper, we suggest printing this 2-page worksheet double-sided.

Turn this teaching resource into a sustainable activity by printing it on cardstock and slipping it into a dry-erase sleeve. Students can record their answers with a whiteboard marker, then erase and reuse them. 

Additionally, project the worksheet onto a screen and work through it as a class by having students record their answers in their notebooks.

This resource was created by Lorin Davies, a teacher in Texas and Teach Starter Collaborator. 

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Write and Solve Equations From Word Problems

Help learners understand and solve real-world problems using algebraic reasoning with these mixed operation word problems! In Write and Solve Equations From Word Problems, students will write and solve one-variable addition, subtraction, multiplication, and division equations from given word problems. But first they'll need to use their reading comprehension skills to decide on the important pieces of information needed to solve each problem. This sixth-grade math worksheet provides essential practice writing and solving one-step equations with multi-digit numbers, and even includes a few problems with decimals.

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Equations and Word Problems

These free  equations and word problems   worksheets  will help your students practice writing and solving equations that match real-world story problems.

These free algebra worksheets are printable and available in a variety of formats.  Of course, answer keys are provided with each free algebra worksheet.

Equations and Word Problems ( Two Step Equations ) Worksheets

Equations and Word Problems (Two Step Equations) Worksheet 1 – This 10 problem worksheet will help you practice writing and solving two step equations that match real world situations. Equations and Word Problems (Two Step Equations) Worksheet 1 RTF Equations and Word Problems (Two Step Equations) Worksheet 1 PDF Preview Equations and Word Problems Worksheet 1 In Your Web Browser View Answers

Equations and Word Problems (Two Step Equations) Worksheet 2 – This 10 problem worksheet will help you practice writing and solving two step equations that match real world situations. Equations and Word Problems (Two Step Equations) Worksheet 2 RTF Equations and Word Problems (Two Step Equations) Worksheet 2 PDF Preview Equations and Word Problems Worksheet 2 In Your Web Browser View Answers

Equations and Word Problems ( Combining Like Terms ) Worksheets

Equations and Word Problems (Combining Like Terms) Worksheet 1 – This 10 problem worksheet will help you practice writing and solving equations that match real world situations. You will have to combine like terms and then solve the equation. Equations and Word Problems (Combining Like Terms) Worksheet 1 RTF Equations and Word Problems (Combining Like Terms) Worksheet 1 PDF Preview Equations and Word Problems Worksheet 1 In Your Web Browser View Answers

Equations and Word Problems (Combining Like Terms) Worksheet 2 – This 10 problem worksheet will help you practice writing and solving equations that match real world situations. You will have to combine like terms and then solve the equation. Equations and Word Problems (Combining Like Terms) Worksheet 2 RTF Equations and Word Problems (Combining Like Terms) Worksheet 2 PDF Preview Equations and Word Problem Worksheet 2 In Your Web Browser View Answers

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REAL WORLD PROBLEMS: How to Write Equations Based on Algebra Word Problems

I know that you often sit in class and wonder, "Why am I forced to learn about equations, Algebra and variables?"

But... trust me, there are real situations where you will use your knowledge of Algebra and solving equations to solve a problem that is not school related. And... if you can't, you're going to wish that you remembered how.

It might be a time when you are trying to figure out how much you should get paid for a job, or even more important, if you were paid enough for a job that you've done. It could also be a time when you are trying to figure out if you were over charged for a bill.

This is important stuff - when it comes time to spend YOUR money - you are going to want to make sure that you are getting paid enough and not spending more than you have to.

Ok... let's put all this newly learned knowledge to work.

Click here if you need to review how to solve equations.

There are a few rules to remember when writing Algebra equations:

Writing Equations For Word Problems

• First, you want to identify the unknown, which is your variable. What are you trying to solve for? Identify the variable: Use the statement, Let x = _____. You can replace the x with whatever variable you are using.
• Look for key words that will help you write the equation. Highlight the key words and write an equation to match the problem.
• The following key words will help you write equations for Algebra word problems:

subtraction

Multiplication.

double (2x)

triple (3x)

quadruple (4x)

divided into

Let's look at an example of an algebra word problem.

Example 1: Algebra Word Problems

Linda was selling tickets for the school play.  She sold 10 more adult tickets than children tickets and she sold twice as many senior tickets as children tickets.

• Let x represent the number of children's tickets sold.
• Write an expression to represent the number of adult tickets sold.
• Write an expression to represent the number of senior tickets sold.
• Adult tickets cost $5, children's tickets cost $2, and senior tickets cost $3. Linda made $700. Write an equation to represent the total ticket sales.
• How many children's tickets were sold for the play? How many adult tickets were sold? How many senior tickets were sold?

As you can see, this problem is massive!  There are 5 questions to answer with many expressions to write. 

A few notes about this problem

1. In this problem, the variable was defined for you.  Let x represent the number of children’s tickets sold tells what x stands for in this problem.  If this had not been done for you, you might have written it like this:

        Let x = the number of children’s tickets sold

2. For the first expression, I knew that 10 more adult tickets were sold.  Since more means add, my expression was x +10 .  Since the direction asked for an expression, I don’t need an equal sign.  An equation is written with an equal sign and an expression is without an equal sign.  At this point we don’t know the total number of tickets.

3. For the second expression, I knew that my key words, twice as many meant two times as many.  So my expression was 2x .

4.  We know that to find the total price we have to multiply the price of each ticket by the number of tickets.  Take note that since x + 10 is the quantity of adult tickets, you must put it in parentheses!  So, when you multiply by the price of $5 you have to distribute the 5.

5.  Once I solve for x, I know the number of children’s tickets and I can take my expressions that I wrote for #1 and substitute 50 for x to figure out how many adult and senior tickets were sold.

Where Can You Find More Algebra Word Problems to Practice?

Word problems are the most difficult type of problem to solve in math. So, where can you find quality word problems WITH a detailed solution?

The Algebra Class E-course provides a lot of practice with solving word problems for every unit! The best part is.... if you have trouble with these types of problems, you can always find a step-by-step solution to guide you through the process!

Click here for more information.

The next example shows how to identify a constant within a word problem.

Example 2 - Identifying a Constant

A cell phone company charges a monthly rate of $12.95 and $0.25 a minute per call. The bill for m minutes is $21.20.

1. Write an equation that models this situation.

2. How many minutes were charged on this bill?

Notes For Example 2

• $12.95 is a monthly rate. Since this is a set fee for each month, I know that this is a constant. The rate does not change; therefore, it is not associated with a variable.
• $0.25 per minute per call requires a variable because the total amount will change based on the number of minutes. Therefore, we use the expression 0.25m
• You must solve the equation to determine the value for m, which is the number of minutes charged.

The last example is a word problem that requires an equation with variables on both sides.

Example 3 - Equations with Variables on Both Sides

You have $60 and your sister has $120. You are saving $7 per week and your sister is saving $5 per week. How long will it be before you and your sister have the same amount of money? Write an equation and solve.

Notes for Example 3

• $60 and $120 are constants because this is the amount of money that they each have to begin with. This amount does not change.
• $7 per week and $5 per week are rates. They key word "per" in this situation means to multiply.
• The key word "same" in this problem means that I am going to set my two expressions equal to each other.
• When we set the two expressions equal, we now have an equation with variables on both sides.
• After solving the equation, you find that x = 30, which means that after 30 weeks, you and your sister will have the same amount of money.

I'm hoping that these three examples will help you as you solve real world problems in Algebra!

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Last modified on April 17th, 2024

Inequality Word Problems

Inequalities are common in our everyday life. They help us express relationships between quantities that are unequal. Writing and solving word problems involving them helps develop our problem-solving approach, understanding, logical reasoning, and analytical skills.

Here are the 4 main keywords commonly used to write mathematical expressions involving inequalities.   

• At least →  ‘greater than or equal to’
• More than → ‘greater than’
• No more than or at most → ‘less than or equal to’
• Less than → ‘less than’

To participate in the annual sports day, Mr. Adams would like to have nine students in each group. But fewer than 54 students are in class today, so Mr. Adams is unable to make as many full groups as he wants. How many full groups can Mr. Adams make? Write the inequality that describes the situation.

Let ‘x’ be the total number of groups Mr. Adams can make. Since each group has 9 students, the total number of students in ‘x’ groups is 9x As we know, fewer than 54 students are in a class today. Thus, the inequality that represents the situation is: 9x < 54 On dividing both sides by 9, the maximum number of groups Mr. Adams can make is  x < 6 Thus, Mr. Adams can make a maximum of 6 full groups.

 Bruce needs at least \$561 to buy a new tablet. He has already saved \$121 and earns \$44 per month as a part-timer in a company. Write the inequality and determine how long he has to work to buy the tablet.

Let ‘x’ be the number of months Bruce needs to work. As we know,  The amount already saved by Bruce is \$121 He earns \$44 per month The cost of the tablet is at least \$561 After ‘x’ months of work, Bruce will have \$(121 + 44x) Now, the inequality representing the situation is: 121 + 44x ≥ 561 On subtracting 121 from both sides, 121 + 44x – 121 ≥ 561 – 121 ⇒ 44x ≥ 440 On dividing both sides by 44, x ≥ 10 Thus, Bruce needs to work for at least 10 months to buy the new tablet.

A store is offering a \$26 discount on all women’s clothes. Ava is looking at clothes originally priced between \$199 and \$299. How much can she expect to spend after the discount?

Let ‘x’ be the original price of the clothes Ava chooses. As we know, the original price range is 199 ≤ x ≤ 299, and the discount is \$26 Now, Ava pays \$(x – 26) after the discount. Thus, the inequality is: 199 – 26 ≤ x – 26 ≤ 299 – 26 ⇒ 173 ≤ x – 26 ≤ 273 Thus, she can expect to spend between \$173 and \$273 after the discount.

A florist makes a profit of \$6.25 per plant. If the store wants to profit at least \$4225, how many plants does it need to sell?

Let ‘P’ be the profit, ‘p’ be the profit per plant, and ‘n’ be the number of plants.  As we know, the store wants a profit of at least \$4225, and the florist makes a profit of \$6.25 per plant. Here, P ≥ 4225 and p = 6.25 …..(i) Also, P = p × n Substituting the values of (i), we get 6.25 × n ≥ 4225 On dividing both sides by 6.25, we get ${n\geq \dfrac{4225}{6\cdot 25}}$ ⇒ ${n\geq 676}$ Thus, the store needs to sell at least 676 plants to make a profit of \$4225.

Daniel had \$1200 in his savings account at the start of the year, but he withdraws \$60 each month to spend on transportation. He wants to have at least \$300 in the account at the end of the year. How many months can Daniel withdraw money from the account?

As we know, Daniel had \$1200 in his savings account at the start of the year, but he withdrew \$60 for transportation each month.  Thus, after ‘n’ months, he will have \$(1200−60n) left in his account. Also, Daniel wants to have at least \$300 in the account at the end of the year.  Here, the inequality is: 1200 – 60n ≥ 300 ⇒ 1200 – 60n – 1200 ≥ 300 – 1200 (by subtraction property) ⇒ -60n ≥ -900 ⇒ 60n ≤ 900 (by inversion property) ⇒ n ≤ ${\dfrac{900}{60}}$ ⇒ n ≤ 15 Thus, Daniel can withdraw money from the account for at most 15 months.

Anne is a model trying to lose weight for an upcoming beauty pageant. She currently weighs 165 lb. If she cuts 2 lb per week, how long will it take her to weigh less than 155 lb?

Let ‘t’ be the number of weeks to weigh less than 155 lb. As we know, Anne initially weighs 165 lb After ‘t’ weeks of cutting 2 lb per week, her weight will be 165 – 2t Now, Anne’s weight will be less than 155 lb Here, the inequality from the given word problem is: 165 – 2t < 155 On subtracting 165 from both sides, we get 165 – 2t – 165 < 155 – 165 ⇒ – 2t < -10 On dividing by -2, the inequality sign is reversed. ${\dfrac{-2t}{-2} >\dfrac{-10}{-2}}$ ⇒ t > 5

Rory and Cinder are on the same debate team. In one topic, Rory scored 5 points more than Cinder, but they scored less than 19 together. What are the possible points Rory scored?

Let Rory’s score be ‘r,’ and Cinder’s score be ‘c.’ As we know, Rory scored 5 points more than Cinder. Thus, Rory’s score is r = c + 5 …..(i) Also, their scores sum up to less than 19 points. Thus, the inequality is: r + c < 19 …..(ii) Substituting (i) in (ii), we get (c + 5) + c < 19 ⇒ 2c + 5 < 19 On subtracting 5 from both sides, we get 2c + 5 – 5 < 19 – 5 ⇒ 2c < 14 On dividing both sides by 2, we get ${\dfrac{2c}{2} >\dfrac{14}{2}}$ ⇒ c < 7 means Cinder’s score is less than 7 points. Now, from (i), r = c + 5 ⇒ c = r – 5 Thus, c < 7 ⇒ r – 5 < 7 On adding 5 to both sides, we get r – 5 + 5 < 7 + 5 ⇒ r < 12 means Rory’s score is less than 12 points. Hence, Rory’s scores can be 6, 7, 8, 9, 10, or 11 points.

An average carton of juice cans contains 74 pieces, but the number can vary by 4. Find out the maximum and minimum number of cans that can be present in a carton.

Let ‘c’ be the number of juice cans in a carton. As we know, the average number of cans in a carton is 74, and it varies by 4 cans. Thus, the required inequality is |c – 74| ≤ 4 ⇒ -4 ≤ c – 74 ≤ 4 On adding 74 to each side, we get -4 + 74 ≤ c – 74 + 74 ≤ 4 + 74 ⇒ 70 ≤ c ≤ 78 Hence, the minimum number of cans in a carton is 70, and the maximum number is 78.

 Layla rehearses singing for at least 12 hours per week, for three-fourths of an hour each session. If she has already sung 3 hours this week, how many more sessions remain for her to exceed her weekly practice goal?

Let ‘p’ be Layla’s total hours of practice in a week, and ‘s’ be the number of sessions she needs to complete. As we know, Layla has already rehearsed 3 hours, then her remaining rehearsal time is (p – 3) Each session lasts for three-fourths of an hour. Thus, we have the inequality: ${\dfrac{3}{4}s >p-3}$ …..(i) As we know, Layla rehearses for at least 12 hours, which means p ≥ 12 …..(ii) From (i), ${\dfrac{3}{4}s >p-3}$ ⇒ ${s >\dfrac{4}{3}\left( p-3\right)}$ From (ii), substituting the value p = 12 in (i), we get ${s >\dfrac{4}{3}\left( 12-3\right)}$ ⇒ ${s >\dfrac{4}{3}\cdot 9}$ ⇒ s > 12 Thus, Layla must complete more than 12 sessions to exceed her weekly rehearsal goal.

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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:

When you see		Think
add, total, sum, increase, more, combined, together, plus, more than		+
minus, less, difference, fewer, decreased, reduced		−
multiplied, times, of, product, factor		×
divided, quotient, per, out of, ratio, percent, rate		÷
maximize or minimize		geometry formulas
rate, speed		distance formulas
how long, days, hours, minutes, seconds		time

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?