Word Problems With Multi Step Equations

Let's tackle the challenge of multi-step equation word problems. These problems require not just algebraic skills, but also the ability to translate real-world scenarios into mathematical models. Successfully navigating these problems involves careful reading, strategic planning, and consistent execution.

Understanding the Basics

Before diving into complex scenarios, it's essential to master the fundamentals. Multi-step equations are algebraic equations that require more than one operation to solve. These operations can include addition, subtraction, multiplication, division, and distribution. The key is to isolate the variable on one side of the equation by performing inverse operations in the correct order, often following the reverse of the order of operations (PEMDAS/BODMAS).

Translating Words into Equations: The Foundation

The biggest hurdle in solving word problems is converting the given information into a mathematical equation. This involves:

• Identifying the unknown: Determine what the problem is asking you to find. This unknown will be your variable (usually represented as x, but you can choose any letter).
• Defining variables: Assign variables to represent the unknown quantities. If there are multiple unknowns, try to express them in terms of a single variable.
• Recognizing keywords: Certain words and phrases indicate specific mathematical operations:
  • "Sum," "total," "more than," "increased by" suggest addition (+).
  • "Difference," "less than," "decreased by," "subtracted from" suggest subtraction (-).
  • "Product," "times," "multiplied by" suggest multiplication (*).
  • "Quotient," "divided by," "ratio" suggest division (/).
  • "Is," "equals," "results in" suggest equality (=).
• Formulating the equation: Piece together the information, variables, and operations to create a balanced equation that accurately represents the problem.

A Step-by-Step Approach to Solving Word Problems

Here’s a structured method to tackle word problems involving multi-step equations:

1. Read and Understand: Carefully read the entire problem. Identify the question being asked and the information provided. Highlight or underline key phrases and numbers.
2. Define Variables: Assign variables to represent the unknown quantities. If there are multiple unknowns, express them in terms of a single variable whenever possible.
3. Translate into an Equation: Convert the word problem into a mathematical equation using the identified variables, keywords, and relationships.
4. Solve the Equation: Use algebraic techniques to solve for the variable. Simplify the equation by combining like terms, distributing, and isolating the variable on one side.
5. Check Your Solution: Substitute the solution back into the original equation to verify that it satisfies the equation. Also, check if the answer makes sense in the context of the problem.
6. Answer the Question: Make sure you answer the specific question that was asked in the problem. Sometimes, solving for the variable is just the first step, and you might need to perform additional calculations to get the final answer.

Example Problems with Detailed Solutions

Let's walk through several examples to illustrate the process:

Example 1: The Concert Tickets

Problem: John and Mary want to go to a concert. John buys the tickets, and they cost him $55. He also pays a convenience fee of $3 per ticket. Mary promises to pay him back half of the total cost. How much money does Mary owe John?

Solution:

1. Understand: We need to find out how much Mary owes John, which is half the total cost of the tickets and fees.
2. Define Variables:
   • Let t be the number of tickets.
   • Let c be the total cost.
   • Let m be the amount Mary owes.
3. Translate into an Equation:
   • Cost of tickets: $55
   • Convenience fee per ticket: $3
   • Number of tickets: 2
   • Total convenience fee: 2 * $3 = $6
   • Total cost (c): $55 + $6 = $61
   • Amount Mary owes (m): $61 / 2
4. Solve the Equation:
   • m = $61 / 2 = $30.50
5. Check Your Solution:
   • Half of $61 is indeed $30.50. The answer makes sense.
6. Answer the Question: Mary owes John $30.50.

Example 2: The Rectangular Garden

Problem: The length of a rectangular garden is 3 feet longer than twice its width. If the perimeter of the garden is 54 feet, what are the dimensions of the garden?

Solution:

1. Understand: We need to find the length and width of the rectangular garden given the perimeter and the relationship between the length and width.
2. Define Variables:
   • Let w be the width of the garden (in feet).
   • Let l be the length of the garden (in feet).
3. Translate into an Equation:
   • Length in terms of width: l = 2w + 3
   • Perimeter of a rectangle: 2l + 2w = 54
4. Solve the Equation:
   • Substitute the expression for l into the perimeter equation: 2*(2w + 3) + 2w = 54
   • Distribute: 4w + 6 + 2w = 54
   • Combine like terms: 6w + 6 = 54
   • Subtract 6 from both sides: 6w = 48
   • Divide by 6: w = 8
   • Now find the length: l = 28 + 3 = 16 + 3 = 19
5. Check Your Solution:
   • Perimeter = 219 + 28 = 38 + 16 = 54. This checks out.
6. Answer the Question: The width of the garden is 8 feet, and the length is 19 feet.

Example 3: The Age Problem

Problem: Sarah is three times as old as her younger brother, Michael. In five years, Sarah will be twice as old as Michael. How old are Sarah and Michael now?

Solution:

1. Understand: We need to find Sarah and Michael's current ages based on the given relationships.
2. Define Variables:
   • Let m be Michael's current age.
   • Let s be Sarah's current age.
3. Translate into an Equation:
   • Sarah's age in terms of Michael's age: s = 3m
   • In five years:
     • Sarah's age: s + 5
     • Michael's age: m + 5
   • Relationship in five years: s + 5 = 2(m + 5)
4. Solve the Equation:
   • Substitute s = 3m into the second equation: 3m + 5 = 2(m + 5)
   • Distribute: 3m + 5 = 2m + 10
   • Subtract 2m from both sides: m + 5 = 10
   • Subtract 5 from both sides: m = 5
   • Now find Sarah's age: s = 35 = 15
5. Check Your Solution:
   • Currently, Sarah is 15 and Michael is 5.
   • In five years, Sarah will be 20 and Michael will be 10.
   • 20 is indeed twice 10. This checks out.
6. Answer the Question: Sarah is currently 15 years old, and Michael is currently 5 years old.

Example 4: The Investment Problem

Problem: A person invests $10,000, part at 6% annual interest and the rest at 8% annual interest. If the total interest earned for the year is $720, how much was invested at each rate?

Solution:

1. Understand: We need to find out how much money was invested at each interest rate.
2. Define Variables:
   • Let x be the amount invested at 6%.
   • Then, 10000 - x is the amount invested at 8%.
3. Translate into an Equation:
   • Interest from 6% investment: 0.06x
   • Interest from 8% investment: 0.08(10000 - x)
   • Total interest: 0.06x + 0.08(10000 - x) = 720
4. Solve the Equation:
   • Distribute: 0.06x + 800 - 0.08x = 720
   • Combine like terms: -0.02x + 800 = 720
   • Subtract 800 from both sides: -0.02x = -80
   • Divide by -0.02: x = 4000
   • Amount invested at 8%: 10000 - 4000 = 6000
5. Check Your Solution:
   • Interest from $4000 at 6%: 0.06 * 4000 = $240
   • Interest from $6000 at 8%: 0.08 * 6000 = $480
   • Total interest: $240 + $480 = $720. This checks out.
6. Answer the Question: $4000 was invested at 6%, and $6000 was invested at 8%.

Example 5: Consecutive Integers

Problem: The sum of three consecutive even integers is 72. What are the integers?

Solution:

1. Understand: We need to find three consecutive even integers that add up to 72.
2. Define Variables:
   • Let x be the first even integer.
   • Then, x + 2 is the second even integer.
   • And x + 4 is the third even integer.
3. Translate into an Equation:
   • Sum of the integers: x + (x + 2) + (x + 4) = 72
4. Solve the Equation:
   • Combine like terms: 3x + 6 = 72
   • Subtract 6 from both sides: 3x = 66
   • Divide by 3: x = 22
   • The integers are: 22, 24, 26
5. Check Your Solution:
   • 22 + 24 + 26 = 72. This checks out.
6. Answer the Question: The three consecutive even integers are 22, 24, and 26.

Example 6: Mixture Problem

Problem: How many liters of a 20% acid solution should be mixed with 10 liters of a 60% acid solution to obtain a 30% acid solution?

Solution:

1. Understand: We need to find the amount of 20% acid solution to mix with 10 liters of 60% solution to get a 30% solution.
2. Define Variables:
   • Let x be the number of liters of the 20% solution.
3. Translate into an Equation:
   • Amount of acid in the 20% solution: 0.20x
   • Amount of acid in the 60% solution: 0.60 * 10 = 6
   • Total amount of solution: x + 10
   • Amount of acid in the 30% solution: 0.30(x + 10)
   • Equation: 0.20x + 6 = 0.30(x + 10)
4. Solve the Equation:
   • Distribute: 0.20x + 6 = 0.30x + 3
   • Subtract 0.20x from both sides: 6 = 0.10x + 3
   • Subtract 3 from both sides: 3 = 0.10x
   • Divide by 0.10: x = 30
5. Check Your Solution:
   • Acid in 30 liters of 20% solution: 0.20 * 30 = 6 liters
   • Acid in 10 liters of 60% solution: 0.60 * 10 = 6 liters
   • Total acid: 6 + 6 = 12 liters
   • Total solution: 30 + 10 = 40 liters
   • Concentration: 12 / 40 = 0.30 or 30%. This checks out.
6. Answer the Question: You need 30 liters of the 20% acid solution.

Advanced Strategies and Tips

• Draw Diagrams: Visual aids can be incredibly helpful, especially for geometry-related problems. Draw diagrams to represent the given information and relationships.
• Create Tables: For problems involving rates, time, and distance, organizing the information in a table can clarify the relationships and help formulate the equation.
• Simplify Complex Fractions: If your equation involves fractions, eliminate them by multiplying both sides by the least common denominator (LCD).
• Practice Regularly: The more word problems you solve, the better you become at recognizing patterns, translating words into equations, and applying algebraic techniques.
• Break Down Complex Problems: If a problem seems overwhelming, break it down into smaller, more manageable parts. Solve each part separately and then combine the results.
• Use Estimation: Before solving the equation, make an estimate of the answer. This can help you identify if your solution is reasonable.
• Consider Units: Always pay attention to the units of measurement. Ensure that your units are consistent throughout the problem and that your final answer is expressed in the correct units.

Common Mistakes to Avoid

• Misinterpreting the Problem: Rushing through the problem without fully understanding the information can lead to incorrect equations and solutions.
• Incorrectly Defining Variables: Assigning the wrong variables or not defining them clearly can cause confusion and errors.
• Making Arithmetic Errors: Simple arithmetic mistakes can derail your solution, so double-check your calculations.
• Not Checking the Solution: Failing to check your solution can result in incorrect answers, as you might not catch arithmetic errors or misinterpretations of the problem.
• Forgetting Units: Omitting units in your final answer can make it unclear or even incorrect.

The Importance of Practice

Solving word problems involving multi-step equations is a skill that improves with practice. Regularly working through a variety of problems will help you develop your problem-solving abilities, enhance your understanding of algebraic concepts, and build confidence in your mathematical skills.

By mastering the art of translating words into equations and consistently applying a systematic approach, you can conquer even the most challenging word problems and unlock the power of algebra in real-world scenarios.