Algebra II – Word Problems | Mathematics

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Study GuideAlgebra II–Word Problems1.General StrategyWord problems can sometimes feel confusing at first. They are written in sentences instead ofnumbers and equations, so it may take a moment to understand what they are asking.The goodnews is that most word problems can be solved by following a clear, step-by-step strategy.If you practice these steps, solving word problems becomes much easier.1. Read the Problem CarefullyStart by reading the problem slowly and carefully. You may need to read it more than once.Your goal is to understandwhat information is givenandwhat the question is asking you to find.Pay attention to important details such as numbers, relationships, and keywords.2. Choose a VariableNext, decide on a variable to represent the unknown value in the problem.A variable is simply a letter (such asx) that stands for the number you want to find. If the problem hasmore than one unknown, you may need more than one variable.3. Organize the InformationSometimes it helps to organize the information visually.You can:•Draw a diagramto represent the situation.•Create a chart or tableto arrange the information clearly.This step can make it easier to see how the different pieces of information relate to each other.4. Write an Algebraic Equation

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Study GuideNow translate the words in the problem into analgebraic sentence (equation)using the variable(s)you chose.This equation should represent the relationship described in the problem.5. Solve the EquationUse algebraic methods to solve the equation you created.This step will give you the value of the unknown variable.6. Check Your ResultAfter solving, take a moment tocheck your work. Substitute your answer back into the equation orsituation to make sure it works correctly.7. Make Sure the Answer Fits the QuestionRead the problem again and confirm that your result actually answers the question asked.Ask yourself:•Does the answer make sense?•Is it reasonable based on the information given?8.State the Final AnswerFinally, write your answer clearly. If necessary, include the correctunits(such as meters, dollars, orminutes).SummaryWord problems may look complicated at first, but when you break them into clear steps—read,represent, organize, translate, solve, and check—they become much easier to manage. Withpractice, this strategy will help you approach any word problem with confidence.

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Study Guide2.Simple InterestInterest is the money earned from investing money (or the money paid when borrowing). Onecommon way to calculate interest is calledsimple interest.The formula used to find simple interest is:Where:•I= interest earned•P= principal (the amount of money invested or borrowed)•R= annual interest rate (usually written as a decimal)•T= time inyearsThis formula helps you calculate how much interest is earned over a certain period of time.Example 1ProblemJim has$10,000to invest. He plans to invest part of the money at9% annual interestand the rest at12% annual interest.Afterone year, he expects to earn$165 morefrom the9% investmentthan from the12%investment.Question:How much money will he invest at each interest rate?Step 1: Choose a VariableLet:So theinterest earned from this investment after one year is:Since Jim invests a total of$10,000, the remaining amount invested at12%is:

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Study GuideThe interest earned from this part is:Step 2: Write an EquationThe problem says:The interest from the9% investment is $165 more than the interest from the 12% investment.So we can write the equation:Step 3: Solve the EquationFirst expand the expression:Combine constants:Add (0.12x) to both sides:Now solve for (x):Step 4: Interpret the ResultThis means:•$6500 is invested at 9%•The remaining amount is:

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Study GuideSo:•$3500 is invested at 12%Final AnswerJim will invest:•$6500 at 9% interest•$3500 at 12% interestThis combination will give him$165 more interest from the 9% investment than from the 12%investment after one year.3.Quiz: Simple Interest1. QuestionSmith invests$3000for one year at a rate of6%. How much interest will he earn at the end of thatyear?Answer Choices•$180•$200•$220Correct Answer:$180Why This Is CorrectUse thesimple interest formula:I = PrtWhere:

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Study Guide•(P) = principal•(r) = rate (decimal)•(t) = time in yearsSubstitute the values:(P = 3000)(r = 6% = 0.06)(t = 1)(I = 3000 × 0.06 × 1)(I = 180)So, the interest earned is$180.2. QuestionCortez invests$2500at a rate of7½%. What will herbalancebe at the end ofthree years?Answer Choices•$562•$2687.50•$3062.50Correct Answer:$3062.50Why This Is CorrectFirst find theinterest using the simple interest formula.(P = 2500)(r = 7.5% = 0.075)(t = 3)(I = 2500 × 0.075 × 3)

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Study Guide(I = 562.50)Now add the interest to the principal:(2500 + 562.50 = 3062.50)So, the final balance is$3062.50.3. QuestionMyles deposited$5000for4 yearsat a rate of5½%. What will hisbalancebe at the end of that time?Answer Choices•$1100•$6100•$6500Correct Answer:$6100Why This Is CorrectConvert the rate to a decimal:(5.5% = 0.055)Use the formula:(I = Prt)(I = 5000 × 0.055 ×4)(I = 1100)Now add the interest to the principal:(5000 + 1100 = 6100)So, the balance after four years is$6100.4. Question

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Study GuideReese deposited$7500fortwo yearsinto a money market account. At the end of two years she had$8700. Whatrate of interestdid she receive?Answer Choices•7%•7½%•8%Correct Answer:8%Why This Is CorrectFirst find theinterest earned:(8700-7500 = 1200)Now use thesimple interest formula and solve for (r):(I = Prt)(1200 = 7500 × r × 2)(1200 = 15000r)Divide both sides by15000:(r = 0.08)Convert to percent:(0.08 = 8%)So, the interest rate is8%.5. QuestionA certain amount of money was invested forone yearat a rate of7½%. At the end of that year it hadearned$675. How much money was invested?Answer Choices•$8000

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Study Guide•$9000•$10,000Correct Answer:$9000Why This Is CorrectUse the simpleinterest formula.(I = Prt)Substitute the known values:(675 = P × 0.075 × 1)(675 = 0.075P)Now divide both sides by0.075:(P = 9000)So, the amount invested was$9000.4.Compound InterestCompound interest is a way of calculating interest whereinterest is added to the principal, andthen future interest is calculated on thenew total amount. This means you earninterest on boththe original money and the interest that has already been added.The formula used to solve most compound interest problems is:What the Variables Mean•A= future value of the investment (the total amount after interest)•P= principal (the original amount invested)

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Study Guide•r= annual interest rate (written as a decimal)•t= time inyears•n= number of times the interest is compounded each yearFor example, if interest iscompounded monthly, thenn = 12.Example 1ProblemHow long will it take for an investment of$3,500to grow to$4,200if it is invested at6% interestcompounded monthly?Step 1: Identify the Known ValuesFrom the problem:•(A = 4200)•(P = 3500)•(r = 0.06)•(n = 12)We need to findt, the number of years.Step 2: Substitute into the FormulaSubstitute the known values into the compound interest formula:Step 3: Simplify the Expression

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