Algebra I – Word Problems - Quick Guides | Mathematics

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Study GuideAlgebra I–Word Problems1. Solving TechniqueWord problems can feel tricky at first. They may involve arithmetic, algebra, geometry, or even a mixof all three. But don’t worry—if you follow a clear and organized method, you can solve them withconfidence.Here’s a step-by-step technique to help you stay on track.1. First, Understand the QuesƟonStart by asking yourself:What am I trying to find?Are you solving for:•The distance a car traveled?•The speed of a plane?•The number of items someone can buy?Once you identify the final answer you’re looking for,mark it clearly(for example, circle it). Thiskeeps you focused and helps prevent solving for the wrong thing.2. IdenƟfy the InformaƟon You’re GivenNext, go through the problem carefully.•Underline important facts and numbers.•Pull out key information.•Look for important words (liketotal,difference,rate,per,sum, etc.).•If it helps,draw a picture or diagram.This step helps you see what information you already have and how the pieces might connect.

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Study Guide3. Set Up an EquaƟon (If Possible)Once you understand what’s given and what’s being asked, try to:•Write an equation•Or set up a simple system using the information providedThis turns the word problem into a math problem you know how to solve.4. Use Only What You NeedSometimes, problems includeextra information. Not everything you read will always be necessary.Ask yourself:•Do I actually need this detail to solve the problem?Focus only on the information that helps you reach the answer. Don’t waste time on irrelevant details.5. Solve Carefully and Watch Your UnitsNow it’s time to do the math.•Solve your equation carefully.•Double-check your calculations.•Make sure all units match.For example:•Convert feet to inches if needed.•Convert pounds to ounces.•Keep everything consistent.Mixing units is a common source of mistakes, so be careful here.6. Make Sure You Answered the Right QuesƟonThis is one of the most common mistakes students make.Before finishing, ask:

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Study Guide•Did I actually answer what the question asked?Sometimes students solve correctly but stop at the wrong value. Always compare your final answer tothe original question.7. Check if Your Answer Makes SenseFinally, pause and think.•Is your answer reasonable?•Does it seem too big or too small?•Could there have been a calculation mistake?If something looks strange, go back and review your setup and arithmetic.2. Key Words and PhrasesWhen you solve word problems, certain words act likeclues. These clue words help you figure outwhich math operation to use—addition, subtraction, multiplication, or division.Learning to spot these words makes word problems much easier. Let’s break them down clearly.Words That MeanAddThese words usually tell you to combine amounts.•Sum–“the sum of 2, 3, and 6”•Total–“the total of the first six payments”•Addition–“the addition of five pints”•Plus–“three liters plus two liters”•Increase–“her pay was increased by $15”•More than–“eight more than last week”•Added to–“$3 added to the cost”Whenever you see these words, think:Put things together.

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Study GuideWords That MeanSubtractThese words usually tell you to take away or compare amounts.•Difference–“the difference between…”•Fewer–“fifteen fewer men than women”•Remainder–“how many are left?”•Less than–“five less than another number”•Reduced–“the budget was reduced by $5,000”•Decreased–“decreased the speed by ten miles per hour”•Minus–“a number minus 9”These words signal:Take away or find what’s left.Words That MeanMultiplyThese words often involve repeated amounts or rates.•Product–“the product of 8 and 5”•Of–“one-half of the group”•Times–“five times as many”•At–“10 yards at 70¢ a yard”•Total–“$15 a week for three weeks”•Twice–“twice the value”These usually mean:Multiply or repeat an amount.Notice that the wordtotalcan sometimes mean add and sometimes mean multiply. The contextmatters!Words That MeanDivideThese words tell you to split something into equal parts.

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Study Guide•Quotient–“the final quotient”•Divided by–“a number divided by 12”•Divided into–“the group was divided into…”•Ratio–“the ratio of…”•Half–“half the profits” (which means divide by 2)These words suggest:Split or share equally.3. Simple InterestLet’s break this down in a clear and simple way.Example 1Question:How much simple interest will an account earn in five years if $500 is invested at 8%interest per year?Step 1:Identify What You’re FindingFirst, ask yourself:What is the problem asking for?It is asking for theinterest, not the total amount in the account.So we are solving forinterest.Step 2:Use the Simple Interest FormulaThe formula for simple interest is:[I = prt]Where:•I= interest•p= principal (the amount invested)

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Study Guide•r= rate (as a decimal)•t= time (in years)Step 3:Plug in the Given ValuesFrom the problem:•Principal,(p = 500)•Rate,(r = 8% = 0.08)•Time,(t = 5)yearsNow substitute into the formula:Final AnswerThe account will earn$200 in simple interestafter five years.4. Compound InterestNow let’s look atcompound interest.Compound interest is different from simple interest because you earn intereston your originalmoney and on the interest already added. In other words, your money grows faster because eachyear’s interest is added to the principal.Example 1Question:What will be the final total amount of money after three years if $1,000 is invested at 12%annual interest, compounded yearly?Step 1:Identify What You Are FindingThe question asks for thefinal total amount of money, not just the interest.Also notice:•The interest is12% per year

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Study Guide•It iscompounded yearly•The investment lasts3 yearsSince it’s compounded yearly, we calculate the growthone year at a time.We’ll still use the interest formula:Where:•(p)= principal•(r)= rate (as a decimal)•(t)= time (in years)Year 1So after one year:Total after Year 1 = $1,120Year 2Now the principal is no longer $1,000.It is$1,120, because interest was added.

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Study GuideAdd this to the previous total:Total after Year 2 = $1,254.40Year 3Again, the principal increases.Now it is$1,254.40.Add this:Final AnswerAfter three years, the total amount is:Why Compound Interest Grows FasterNotice how:•Year 1 interest = $120•Year 2 interest = $134.40•Year 3 interest = $150.53Each year, the interest increases because it is calculated on alarger principal.That’s the power of compound interest—you earninterest on interest.

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Study Guide5. RaƟo and ProporƟonRatios and proportions are very useful when solving word problems, especially when two quantitieschange at thesame rate.Let’s look at a clear example and break it down step by step.Example 1Question:If Arnold can type 600 pages of a manuscript in 21 days, how many days will it take him totype 230 pages if he works at the same rate?Step 1:Identify What You’re FindingFirst, ask yourself:What is the question asking?It asks:How many days?So our unknown is the number of days. Let’s call itx.Step 2:Set Up a Proportion (Framework)Since Arnold works at the same rate, the ratio betweenpages and daysstays constant.We can set up the proportion like this:Or we could flip both sides:Both ways are correct—just keep the order consistent.Now plug in the numbers:

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Study GuideStep 3:Cross MultiplyTo solve a proportion, we cross multiply:Now divide both sides by 600:Or as a decimal:Final AnswerIt will take8 1/20 days, or about8.05 days, to type 230 pages.6. PercentPercent problems may look confusing at first, but they become much easier when you follow a clearpattern.Let’s work through an example step by step.Example 1Question:Thirty students are awarded doctoral degrees at the graduate school, and this numberrepresents 40% of the total graduate student body. How many graduate students were enrolled?

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