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Algebra II – Word Problems

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    Study Guide Algebra II – Word Problems 1. General Strategy Word problems can sometimes feel confusing at first. They are written in sentences instead of numbers and equations, so it may take a moment to understand what they are asking. The good news 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 Carefully Start by reading the problem slowly and carefully. You may need to read it more than once. Your goal is to understand what information is given and what the question is asking you to find . Pay attention to important details such as numbers, relationships, and keywords. 2. Choose a Variable Next, decide on a variable to represent the unknown value in the problem. A variable is simply a letter (such as x ) that stands for the number you want to find. If the problem has more than one unknown, you may need more than one variable. 3. Organize the Information Sometimes it helps to organize the information visually. You can: • Draw a diagram to represent the situation. • Create a chart or table to 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 Guide Now translate the words in the problem into an algebraic sentence (equation) using the variable(s) you chose. This equation should represent the relationship described in the problem. 5. Solve the Equation Use algebraic methods to solve the equation you created. This step will give you the value of the unknown variable. 6. Check Your Result After solving, take a moment to check your work . Substitute your answer back into the equation or situation to make sure it works correctly. 7. Make Sure the Answer Fits the Question Read 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 Answer Finally, write your answer clearly. If necessary, include the correct units (such as meters, dollars, or minutes). Summary Word 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. With practice, this strategy will help you approach any word problem with confidence.

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    Study Guide 2. Simple Interest Interest is the money earned from investing money (or the money paid when borrowing). One common way to calculate interest is called simple 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 in years This formula helps you calculate how much interest is earned over a certain period of time. Example 1 Problem Jim has $10,000 to invest. He plans to invest part of the money at 9% annual interest and the rest at 12% annual interest . After one year , he expects to earn $165 more from the 9% investment than from the 12% investment . Question: How much money will he invest at each interest rate? Step 1: Choose a Variable Let: So the interest earned from this investment after one year is: Since Jim invests a total of $10,000 , the remaining amount invested at 12% is:

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

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    Study Guide So: • $3500 is invested at 12% Final Answer Jim will invest: • $6500 at 9% interest • $3500 at 12% interest This combination will give him $165 more interest from the 9% investment than from the 12% investment after one year . 3. Quiz: Simple Interest 1. Question Smith invests $3000 for one year at a rate of 6% . How much interest will he earn at the end of that year? Answer Choices • $180 • $200 • $220 Correct Answer : $180 Why This Is Correct Use the simple interest formula : I = Prt Where:

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    Study Guide • (P) = principal • (r) = rate (decimal) • (t) = time in years Substitute the values: (P = 3000) (r = 6% = 0.06) (t = 1) (I = 3000 × 0.06 × 1) (I = 180) So, the interest earned is $180 . 2. Question Cortez invests $2500 at a rate of 7½% . What will her balance be at the end of three years ? Answer Choices • $562 • $2687.50 • $3062.50 Correct Answer : $3062.50 Why This Is Correct First find the interest 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. Question Myles deposited $5000 for 4 years at a rate of 5½% . What will his balance be at the end of that time? Answer Choices • $1100 • $6100 • $6500 Correct Answer : $6100 Why This Is Correct Convert 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 Guide Reese deposited $7500 for two years into a money market account. At the end of two years she had $8700 . What rate of interest did she receive? Answer Choices • 7% • 7½% • 8% Correct Answer : 8% Why This Is Correct First find the interest earned : (8700 - 7500 = 1200) Now use the simple interest formula and solve for (r): (I = Prt) (1200 = 7500 × r × 2) (1200 = 15000r) Divide both sides by 15000 : (r = 0.08) Convert to percent: (0.08 = 8%) So, the interest rate is 8% . 5. Question A certain amount of money was invested for one year at a rate of 7½% . At the end of that year it had earned $675 . How much money was invested? Answer Choices • $8000

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    Study Guide • $9000 • $10,000 Correct Answer : $9000 Why This Is Correct Use the simple interest formula. (I = Prt) Substitute the known values: (675 = P × 0.075 × 1) (675 = 0.075P) Now divide both sides by 0.075 : (P = 9000) So, the amount invested was $9000 . 4. Compound Interest Compound interest is a way of calculating interest where interest is added to the principal , and then future interest is calculated on the new total amount . This means you earn interest on both the 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 in years • n = number of times the interest is compounded each year For example, if interest is compounded monthly , then n = 12 . Example 1 Problem How long will it take for an investment of $3,500 to grow to $4,200 if it is invested at 6% interest compounded monthly ? Step 1: Identify the Known Values From the problem: • (A = 4200) • (P = 3500) • (r = 0.06) • (n = 12) We need to find t , the number of years. Step 2: Substitute into the Formula Substitute the known values into the compound interest formula: Step 3: Simplify the Expression

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