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How to Write Expressions

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Basics on the topic How to Write Expressions

How to Write Expressions

Writing mathematical expressions is a fundamental skill that helps translate real-world situations into manageable mathematical problems. It involves identifying key quantities and their relationships and then using symbols and numbers to express these relationships succinctly. This guide will explore the basics of constructing mathematical expressions, equipping you with the tools to solve problems efficiently and accurately.

How to Write Expressions – Definition

An algebraic expression is a mathematical phrase containing numbers, variables, and operation symbols. In business planning, algebraic expressions can model financial scenarios, calculate costs, and predict growth.

Here's a table summarizing common keywords associated with each of the basic mathematical operations, which can help in identifying what operation to use when writing mathematical expressions:

Operation Keywords
Addition add, plus, increase, total, sum, together, more than
Subtraction subtract, minus, less, decrease, fewer, less than
Multiplication multiply, times, product, doubled, tripled
Division divide, divided by, quotient, per, ratio

How to Write Expressions – Step-by-Step Process

Here's a concise step-by-step guide on how to write an algebraic expression, followed by two simple examples to demonstrate the process:

Step-by-Step Process of Writing an Algebraic Expression

Step Description
Step 1: Identify the Variables Determine what the unknowns are in the problem and assign a variable (like x x , y y , etc.) to each.
Step 2: Determine the Operation Based on the keywords or problem context, decide which mathematical operations (addition, subtraction, multiplication, division) are required.
Step 3: Formulate the Expression Combine the variables and numbers using the operations identified to create an algebraic expression that represents the problem.

Let’s look at some examples.

Example 1:: Suppose you have 5 dimes and you receive x x additional dimes from a friend.

  • Identify the Variables: Let x x represent the number of dimes received.
  • Determine the Operation: The problem involves adding the dimes you received to the ones you already have.
  • Formulate the Expression: The algebraic expression for this situation is 5+x 5 + x .

Example 2:: You buy y y pencils, and each pencil costs 3 dollars.

  • Identify the Variables: Let y y represent the number of pencils.
  • Determine the Operation: Since each pencil costs a certain amount, you multiply the number of pencils by the cost per pencil.
  • Formulate the Expression: The algebraic expression for this situation is 3y 3y .

How to Write Expressions – Real-World Application

Example: For Mateo's taco catering business, he has a simple pricing model. He sells each taco for $\2.50 and charges an additional service fee of $\$200. An algebraic expression to represent Mateo's total charge for 't' tacos is:

2.50t+2002.50t + 200

With 'tt' representing the number of tacos sold, we can calculate Mateo's revenue by substituting the value of 'tt' into the expression.

Mateo serves three tacos per person and needs to account for kitchen accidents that usually cost him 10 tacos. To calculate the number of people Mateo can serve, we use the following expression:

(t10)÷3(t - 10) \div 3

This expression takes the total number of tacos 'tt', subtracts the 1010 accident-prone tacos, and divides by 33 since each person gets three tacos.

Mateo projects that his company's value will double each week. To model this exponential growth, we use the following expression:

10,000×2w10,000 × 2^{w}

Here, 'w' represents the number of weeks, and the entire expression predicts the company's value after 'w' weeks.

How to Write Expressions – Guided Practice

1. Hannah makes and sells handmade necklaces. She charges $\15 for each necklace and a fixed shipping fee of $\$5 per order.

Algebraic Expression: The total charge for 'n' necklaces would be: 15n+5 15n + 5

With 'nn' representing the number of necklaces sold, we can calculate Hannah's revenue by substituting the value of 'nn' into the expression.

2. Alex tutors students and charges $\20 per hour. Additionally, he charges a one-time material fee of $\$10 per student.

Algebraic Expression: The total charge for 'h' hours of tutoring would be: 20h+10 20h + 10

With 'hh' representing the number of tutoring hours, we can calculate Alex's total earnings by substituting the value of 'hh' into the expression.

How to Write Expressions – Practice

How to Write Expressions – Summary

Key Learnings from this Text:

  • Algebraic expressions can simplify financial calculations in business planning.
  • Translating business scenarios into algebraic expressions is essential for strategic planning and forecasting.

Engage further in exploration of the interplay between algebra and real-world applications like business planning, through our engaging content, interactive practice problems, and additional learning resources on our platform.

How to Write Expressions – Frequently Asked Questions

Transcript How to Write Expressions

Loco Mateo is crazy for tacos, so he decides to start a catering service which will feature his signature creation: Tacos el Fuego. Because he knows his tacos are so insanely delicious, he’s absolutely certain that his new business is going to be a success. So, he needs to write up a business plan to make sure he’s ready for all the HUGE orders he’s definitely going to get once he launches. To help him prepare for his inevitable financial success, Mateo is going to have to enlist the help of some basic algebraic expressions. Mateo’s plan is to cater to weddings, funerals, family reunions..anywhere there is a big group of people fiending for tacos. Since each group is going to be different, he needs to come up with an expression to represent how much he'll charge for each catering order, based on the number of tacos sold. Mateo decides to charge 2.50foreachTacoelFuegoplus,hesgoingtochargea2.50 for each Taco el Fuego plus, he’s going to charge a 200 service fee for his impeccable taco servicing skills. By focusing in on the keywords, we can turn Mateo's business plan into an algebraic expression. Since each taco costs 2.50,weshouldmultiply2.50byt,orthenumberoftacossold,togetthetotalamountofmoneyMateocanearnbysellingtacos.ThevariabletrepresentsthenumberoftacosMateosells.Whenmultiplyingbyavariable,youcaneitheruseamultiplicationsignorjustputthenumberrightnexttothevariable.Finally,weknowthatthekeyword"plus"indicatesaddition,soweaddaplus200tocompletetheexpression.Next,Mateoisgoingtoneedtofigureouthowmanypeoplehecanfeedwitheachbatchoftacoshemakes.Becausehewantstokeephisbusinessexclusive,hedecidestolimitthenumberoftacostothreeperperson.But,Mateoisalsonotthemostefficientchef.Everytimehepreparesabatchoftacos,healwaysburnsorbreakssomeofthem,soheendsupwith10fewerthanthetotalnumberoftacosinthebatch.Letsusethisinformationtowriteanexpression.WeknowsthatMateowillendupwith10fewertacosthanthetotalnumberoftacoshemakes.Wedontknowhowmanythetotalwillbe,sowecanusethevariablettostandforthetotalnumberoftacos.Thewords"fewerthan"tellusthatwearegoingtoneedtosubtract10fromthetotal,t.Wealsoknowthathewillservethreetacosperperson.Thekeyword"per"tellsthatweneedtosetupadivisionproblem.Sowetakethetotalnumberoftacosremainingafterthe10misfires...Thendividethatallbythree.There!Nowwhenweknowthenumberoftacos,t,wecanusethisexpressiontofigureouthowmanypeopleMateocanfeed!Nowthathesgotabasicbusinessmodelinplace,MateojustKNOWSthatstocksinhiscompanyaregoingtosoar!Rightnow,hefiguresthatitsworthabout2.50, we should multiply 2.50 by 't', or the number of tacos sold, to get the total amount of money Mateo can earn by selling tacos. The variable 't' represents the number of tacos Mateo sells. When multiplying by a variable, you can either use a multiplication sign or just put the number right next to the variable. Finally, we know that the keyword "plus" indicates addition, so we add a plus 200 to complete the expression. Next, Mateo is going to need to figure out how many people he can feed with each batch of tacos he makes. Because he wants to keep his business exclusive, he decides to limit the number of tacos to three per person. But, Mateo is also not the most efficient chef. Every time he prepares a batch of tacos, he always burns or breaks some of them, so he ends up with 10 fewer than the total number of tacos in the batch. Let's use this information to write an expression. We knows that Mateo will end up with 10 fewer tacos than the total number of tacos he makes. We don't know how many the total will be, so we can use the variable 't' to stand for the total number of tacos. The words "fewer than" tell us that we are going to need to subtract 10 from the total, 't'. We also know that he will serve three tacos per person. The keyword "per" tells that we need to set up a division problem. So we take the total number of tacos remaining after the 10 misfires... Then divide that all by three. There! Now when we know the number of tacos, 't', we can use this expression to figure out how many people Mateo can feed! Now that he’s got a basic business model in place, Mateo just KNOWS that stocks in his company are going to soar! Right now, he figures that it’s worth about 10,000 but if everything goes according to plan, his company will double in value every week. How can we use this information to come up with an expression? Let's start with the initial value of the company, 10,000.Next,wehavethekeywords"doubleeachweek",whichmeansweregoingtodorepeatedmultiplicationby2everyweek.Letsdrawupaquicktable.Attheendofweekone,weknowthecompanywillbeworth10,000. Next, we have the keywords "double each week", which means we're going to do repeated multiplication by 2 every week. Let's draw up a quick table. At the end of week one, we know the company will be worth 10,000 x 2, or 20,000.Attheendofweektwo,itwillbeworth20,000. At the end of week two, it will be worth 10,000 x 2 x 2, or 40,000.Attheendofweek3,itwillbeworth40,000. At the end of week 3, it will be worth 10,000 x 2 x 2 x 2, or 80,000.Doyounoticeapatternhere?Eachoftheseexpressionscanberewrittenusingexponents.80,000. Do you notice a pattern here? Each of these expressions can be rewritten using exponents. 10,000 x 2 can be written as 10,000x2tothefirstpower.10,000 x 2 to the first power. 10,000 x 2 x 2 can be written as 10,000x2tothesecondpower.Similarly,10,000 x 2 to the second power. Similarly, 10,000 x 2 x 2 x 2 can be written as 10x2tothethirdpower.Usingthispattern,wecancalculatehowcrazyrichMateosgoingtobeonweekw,becauseweknowthatwellhavetomultiply10 x 2 to the third power. Using this pattern, we can calculate how crazy rich Mateo’s going to be on week w, because we know that we'll have to multiply 10,000 by 2, "w" times! We can write this as the expression, $10,000 x 2 to the 'w' power! It looks like Loco Mateo has just finished up a batch of Tacos el Fuego in time to fill his first order! But where are all of his customers? I guess that’s why they call him Loco!

How to Write Expressions exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video How to Write Expressions.