Word Problems With One Step Equations

One-step equations form the bedrock of algebra, representing simple yet fundamental relationships between variables and constants. Mastering the art of translating word problems into these equations is crucial for developing problem-solving skills that extend far beyond the classroom.

Decoding the Language: From Words to Math

Word problems often seem daunting due to their narrative format, but with a systematic approach, they can be demystified. The key lies in identifying the unknown variable, recognizing the mathematical operation implied by the words, and constructing the equation accordingly.

Identifying the Unknown

The first step is to pinpoint what the problem is asking you to find. This unknown quantity will be represented by a variable, typically x, y, or z. Look for keywords like "what," "how many," "find," or "determine" to guide you.

Example: "Sarah has some apples. After giving 5 apples to her friend, she has 12 left. How many apples did Sarah have initially?"

Here, the unknown is the initial number of apples Sarah had. We can represent this with the variable x.

Recognizing Mathematical Operations

Certain words and phrases consistently indicate specific mathematical operations. Understanding these keywords is crucial for accurately translating the word problem into an equation.

• Addition: Sum, plus, more than, increased by, total
• Subtraction: Difference, minus, less than, decreased by, fewer than
• Multiplication: Product, times, multiplied by, of
• Division: Quotient, divided by, per, ratio, shared equally

Example: "The price of a shirt is increased by $7. The new price is $25. What was the original price?"

The phrase "increased by" indicates addition. If the original price is x, the equation will involve adding 7 to x.

Constructing the Equation

Once you've identified the unknown and the operation, you can construct the one-step equation. This involves expressing the relationship described in the word problem using mathematical symbols.

Example: "A number multiplied by 3 equals 21. What is the number?"

Here, the unknown number is x. "Multiplied by 3" translates to 3 * x, or 3x. "Equals 21" means the expression is set equal to 21. The equation is therefore 3x = 21.

Mastering the Four Operations: One-Step Equation Examples

Let's explore examples of word problems involving each of the four basic mathematical operations: addition, subtraction, multiplication, and division.

Addition

Word Problem: John earned some money mowing lawns. After receiving $15 from his aunt, he had a total of $42. How much money did John earn mowing lawns?

Solution:

• Unknown: The amount John earned mowing lawns (let's call it x)
• Operation: Addition ("total" indicates addition)
• Equation: x + 15 = 42
• Solving the Equation: To isolate x, subtract 15 from both sides of the equation:
  • x + 15 - 15 = 42 - 15
  • x = 27

Answer: John earned $27 mowing lawns.

Subtraction

Word Problem: Maria had some stickers. She gave 8 stickers to her friend and now has 23 stickers left. How many stickers did Maria have originally?

Solution:

• Unknown: The original number of stickers Maria had (x)
• Operation: Subtraction ("gave away" indicates subtraction)
• Equation: x - 8 = 23
• Solving the Equation: To isolate x, add 8 to both sides of the equation:
  • x - 8 + 8 = 23 + 8
  • x = 31

Answer: Maria originally had 31 stickers.

Multiplication

Word Problem: A certain number of boxes each contain 6 pencils. If there are a total of 54 pencils, how many boxes are there?

Solution:

• Unknown: The number of boxes (x)
• Operation: Multiplication ("each contain" implies multiplication)
• Equation: 6x = 54
• Solving the Equation: To isolate x, divide both sides of the equation by 6:
  • 6x / 6 = 54 / 6
  • x = 9

Answer: There are 9 boxes.

Division

Word Problem: A roll of ribbon is cut into 12 equal pieces, each measuring 5 inches long. What was the original length of the ribbon?

Solution:

• Unknown: The original length of the ribbon (x)
• Operation: Division (the reverse of "cut into equal pieces" is multiplication, so to find the original, we need division)
• Equation: x / 12 = 5
• Solving the Equation: To isolate x, multiply both sides of the equation by 12:
  • (x / 12) * 12 = 5 * 12
  • x = 60

Answer: The original length of the ribbon was 60 inches.

Advanced Scenarios: Multi-Step Thinking within One-Step Equations

While the equations themselves remain one-step, some word problems require a bit more thought to set up the equation correctly. This often involves understanding the context of the problem and applying some basic reasoning.

Dealing with "Less Than"

The phrase "less than" can be tricky because it reverses the order of the terms.

Word Problem: A number is 5 less than 18. What is the number?

Solution:

• Unknown: The number (x)
• Operation: Subtraction
• Equation: x = 18 - 5
• Solving the Equation: x = 13

Answer: The number is 13. Note that we write 18 - 5, not 5 - 18, because the number is less than 18.

Working with "Of"

The word "of" often indicates multiplication, especially when dealing with fractions or percentages.

Word Problem: One-third of a number is 7. What is the number?

Solution:

• Unknown: The number (x)
• Operation: Multiplication (specifically, multiplying by a fraction)
• Equation: (1/3) * x = 7 (This can also be written as x/3 = 7)
• Solving the Equation: Multiply both sides by 3:
  • ((1/3) * x) * 3 = 7 * 3
  • x = 21

Answer: The number is 21.

Problems Involving Consecutive Integers

These problems involve finding integers that follow each other in sequence.

Word Problem: One more than a number is 6. What is the number?

Solution:

• Unknown: The number (x)
• Operation: Addition
• Equation: x + 1 = 6
• Solving the Equation: Subtract 1 from both sides:
  • x + 1 - 1 = 6 - 1
  • x = 5

Answer: The number is 5.

Tips and Tricks for Solving Word Problems

• Read Carefully: Read the problem multiple times to ensure you understand what it's asking.
• Highlight Key Information: Underline or highlight the important numbers and keywords.
• Define the Variable: Clearly state what your variable represents.
• Check Your Answer: After solving, plug your answer back into the original word problem to see if it makes sense.
• Practice Regularly: The more you practice, the better you'll become at recognizing patterns and translating word problems into equations.

Why Are Word Problems Important?

While they might seem like abstract exercises, word problems are crucial for developing essential skills:

• Critical Thinking: They force you to analyze information and identify relevant details.
• Problem-Solving: They teach you how to break down complex problems into smaller, manageable steps.
• Mathematical Reasoning: They help you understand the relationship between numbers and the real world.
• Application of Knowledge: They allow you to apply mathematical concepts to practical situations.

Real-World Applications

One-step equations, and the ability to solve word problems related to them, are foundational for many real-world applications:

• Budgeting: Calculating expenses, savings, and income.
• Cooking: Adjusting recipes and measuring ingredients.
• Shopping: Calculating discounts, sales tax, and total costs.
• Travel: Determining distances, travel times, and fuel consumption.
• Construction: Measuring materials and calculating dimensions.

Common Mistakes to Avoid

• Misinterpreting Keywords: Pay close attention to the meaning of words like "less than" or "of."
• Incorrectly Identifying the Operation: Make sure you choose the correct mathematical operation based on the wording of the problem.
• Not Defining the Variable: Always clearly state what your variable represents to avoid confusion.
• Forgetting to Check Your Answer: Plug your answer back into the original problem to ensure it makes sense.
• Giving Up Too Easily: Word problems can be challenging, but don't get discouraged. Break the problem down into smaller steps and keep trying.

Examples with Detailed Explanations

Let's dive into some more complex examples with detailed explanations of each step.

Example 1: Discounted Price

Word Problem: A store is having a sale where all items are 20% off. If a shirt costs $24 after the discount, what was the original price of the shirt?

Solution:

1. Understand the Problem: The discounted price ($24) represents 80% of the original price (100% - 20% = 80%).

2. Define the Variable: Let x represent the original price of the shirt.

3. Set up the Equation: 0.80x = $24 (80% of the original price equals $24)

4. Solve the Equation: Divide both sides by 0.80:

   • (0.80x) / 0.80 = $24 / 0.80
   • x = $30

5. Check the Answer: 20% of $30 is $6. $30 - $6 = $24. The answer checks out.

Answer: The original price of the shirt was $30.

Example 2: Sharing Candy

Word Problem: Three friends, Alice, Bob, and Carol, decide to share a bag of candy equally. After sharing, each friend has 15 pieces of candy. How many pieces of candy were originally in the bag?

Solution:

1. Understand the Problem: Each friend receiving 15 pieces represents one-third of the original amount of candy.

2. Define the Variable: Let x represent the original number of pieces of candy in the bag.

3. Set up the Equation: x / 3 = 15 (The original amount divided by 3 equals 15)

4. Solve the Equation: Multiply both sides by 3:

   • (x / 3) * 3 = 15 * 3
   • x = 45

5. Check the Answer: If there were 45 pieces originally, each friend would receive 45 / 3 = 15 pieces. The answer checks out.

Answer: There were originally 45 pieces of candy in the bag.

Example 3: Distance and Speed

Word Problem: A train travels at a constant speed for 2 hours and covers a distance of 180 miles. What is the speed of the train?

Solution:

1. Understand the Problem: Distance equals speed multiplied by time (Distance = Speed * Time).

2. Define the Variable: Let x represent the speed of the train in miles per hour (mph).

3. Set up the Equation: 2x = 180 (Time * Speed = Distance)

4. Solve the Equation: Divide both sides by 2:

   • (2x) / 2 = 180 / 2
   • x = 90

5. Check the Answer: If the train travels at 90 mph for 2 hours, it will cover a distance of 90 * 2 = 180 miles. The answer checks out.

Answer: The speed of the train is 90 mph.

Conclusion: Building a Foundation for Algebraic Success

Mastering one-step equations and their corresponding word problems is an essential step in building a strong foundation in algebra. By understanding the keywords, identifying the unknown, and practicing regularly, you can develop the problem-solving skills necessary to tackle more complex mathematical challenges. Don't be afraid to break down problems into smaller steps, and remember that persistence and practice are key to success. The ability to translate real-world scenarios into mathematical equations is a valuable skill that will serve you well in various aspects of life.