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Algebra I – Word Problems

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    Study Guide Algebra I – Word Problems 1. Solving Technique Word problems can feel tricky at first. They may involve arithmetic, algebra, geometry, or even a mix of all three. But don’t worry — if you follow a clear and organized method, you can solve them with confidence. Here’s a step - by - step technique to help you stay on track. 1. First, Understand the Ques Ɵ on Start 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). This keeps you focused and helps prevent solving for the wrong thing. 2. Iden Ɵ fy the Informa Ɵ on You’re Given Next, go through the problem carefully. • Underline important facts and numbers. • Pull out key information. • Look for important words (like total , 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 Guide 3. Set Up an Equa Ɵ on (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 provided This turns the word problem into a math problem you know how to solve. 4. Use Only What You Need Sometimes, problems include extra 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 Units Now 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 Ques Ɵ on This 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 to the original question. 7. Check if Your Answer Makes Sense Finally, 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 Phrases When you solve word problems, certain words act like clues . These clue words help you figure out which 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 Mean Add These 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 Guide Words That Mean Subtract These 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 Mean Multiply These 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 word total can sometimes mean add and sometimes mean multiply. The context matters! Words That Mean Divide These 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 Interest Let’s break this down in a clear and simple way. Example 1 Question: 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 Finding First, ask yourself: What is the problem asking for? It is asking for the interest , not the total amount in the account. So we are solving for interest . Step 2: Use the Simple Interest Formula The 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 Values From the problem: • Principal, ( p = 500 ) • Rate, ( r = 8% = 0.08 ) • Time, ( t = 5 ) years Now substitute into the formula: Final Answer The account will earn $200 in simple interest after five years. 4. Compound Interest Now let’s look at compound interest . Compound interest is different from simple interest because you earn interest on your original money and on the interest already added . In other words, your money grows faster because each year’s interest is added to the principal. Example 1 Question: 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 Finding The question asks for the final total amount of money , not just the interest. Also notice: • The interest is 12% per year

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    Study Guide • It is compounded yearly • The investment lasts 3 years Since it’s compounded yearly, we calculate the growth one year at a time . We’ll still use the interest formula: Where: • ( p ) = principal • ( r ) = rate (as a decimal) • ( t ) = time (in years) Year 1 So after one year: Total after Year 1 = $1,120 Year 2 Now the principal is no longer $1,000. It is $1,120 , because interest was added.

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

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    Study Guide 5. Ra Ɵ o and Propor Ɵ on Ratios and proportions are very useful when solving word problems, especially when two quantities change at the same rate . Let’s look at a clear example and break it down step by step. Example 1 Question: If Arnold can type 600 pages of a manuscript in 21 days, how many days will it take him to type 230 pages if he works at the same rate? Step 1: Identify What You’re Finding First, ask yourself: What is the question asking? It asks: How many days? So our unknown is the number of days. Let’s call it x . Step 2: Set Up a Proportion (Framework) Since Arnold works at the same rate, the ratio between pages and days stays 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 Guide Step 3: Cross Multiply To solve a proportion, we cross multiply: Now divide both sides by 600: Or as a decimal: Final Answer It will take 8 1/20 days , or about 8.05 days , to type 230 pages. 6. Percent Percent problems may look confusing at first, but they become much easier when you follow a clear pattern. Let’s work through an example step by step. Example 1 Question: Thirty students are awarded doctoral degrees at the graduate school, and this number represents 40% of the total graduate student body. How many graduate students were enrolled?

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