Lesson 4.1: Translating Word Problems

Introduction

In this lesson, we will explore how to effectively translate word problems into mathematical equations, a crucial skill for tackling GMAT quantitative reasoning questions. Many students find word problems intimidating; however, with a systematic approach, they can be broken down into manageable pieces. By the end of this lesson, students will be able to articulate a clear process for turning words into equations, define variables accurately, and verify that their answers correspond to the questions being asked.

Learning Objectives

• A repeatable process for turning words into equations.
• Defining variables and identifying the question asked.
• Checking answers against the problem's meaning.
• Translate a word problem into solvable equations systematically.
• Verify that a numerical answer addresses the question asked.

Section 1: Understanding Word Problems

Word problems can often seem complex at first glance. They blend real-life scenarios with mathematical concepts, requiring keen reading and analytical skills to decipher. Let's begin by establishing a fundamental understanding of word problems and the consistency in their structure.

Key Concepts

1. Identify the Components of the Problem: Most word problems will present various elements, including:

• Variables: These are the unknowns we need to find.
• Relationships: How the variables relate to each other (e.g., rates, totals).
• Units: The measurements (e.g., hours, miles, dollars).

1. What is Being Asked: Understand what the problem is requesting. Look for keywords like "how many", "what is the total", or "find the rate" that indicate the core question of the problem.

Example 1: Simple Addition Problem

Problem: A farmer has 50 apples. He buys 30 more apples. How many apples does he have now?

Step-by-Step Solution:

• Identify variables: Let $x$ represent the total number of apples.
• Formulate the equation: We can express the problem as:

$$ x = 50 + 30 $$

• Calculate the result: To find $x$, calculate:

$$ x = 80 $$

• Verify the context: The farmer now has 80 apples, addressing the original question.

Section 2: Translating Verbal Context to Mathematical Relationships

Once we grasp the components of word problems, the next step is to establish the mathematical relationship between them. This often requires converting phrases into symbols. Let's discuss some common phrases and their mathematical representations.

Common Phrases and Their Translations

• Sum: The term indicates addition. For example, "the sum of" translates to $+$.
• Difference: Indicates subtraction, e.g., "the difference between" translates to $-$.
• Product: Means multiplication, e.g., "twice a number" translates to $2x$ if the number is $x$.
• Quotient: Indicates division, e.g., "a number divided by" translates to $\frac{x}{y}$.

Example 2: Mixture Problem

Problem: A solution contains 5 liters of water and 2 liters of salt. What is the concentration of salt in the solution?

Step-by-Step Solution:

• Identify the variables: Let $s$ represent the total amount of salt.
• Formulate the relationship:
• Total solution volume = water + salt = $5 + 2 = 7$ liters.
• Concentration = \frac{Amount \, of \, Salt}{Total \, Volume}$ = \frac{2}{7}$.
• Calculate:

$$ \text{Concentration} = \frac{2}{7} \approx 0.2857 = 28.57\% $$

• Context Verification: Thus, the salt concentration is approximately 28.57%.

Section 3: Establishing the Framework for Solutions

With a clear understanding of translating phrases into mathematical representations, we can create a structured approach to solve word problems. This section will highlight a systematic process that students can use to ensure clean setups.

A Systematic Process

1. Read the Problem Carefully: Do not rush. Understand each part of the problem.
2. Identify What is Given: Note down numbers, units, and initial conditions.
3. Define Variables Clearly: Assign letters to unknown values but ensure it's intuitive.
4. Translate into Equations: Using the phrases mentioned, convert the problem into mathematical expressions.
5. Solve the Equation: Utilize algebraic techniques to solve the generated equations.
6. Check Your Work: Confirm that the solution meets the original problem's criteria.

Example 3: Rate Problems

Problem: Sarah can paint a fence in 5 hours, while her brother, Tom, can do it in 3 hours. How long will it take for them to paint the fence together?

Step-by-Step Solution:

• Variables: Let $t$ be the time it takes for both to complete the task together.
• Set up the equation:
• Sarah's rate = $\frac{1}{5}$ fences per hour.
• Tom's rate = $\frac{1}{3}$ fences per hour.
• Combined rate = $\frac{1}{t}$ of a fence per hour.
• The equation becomes:

$$ \frac{1}{5} + \frac{1}{3} = \frac{1}{t} $$

• Solve the equation: Finding a common denominator:

$$ \frac{3}{15} + \frac{5}{15} = \frac{1}{t} $$

$$ \frac{8}{15} = \frac{1}{t} $$

• Cross-multiply to solve for $t$:

$$ 8t = 15 $$

$$ t = \frac{15}{8} = 1.875 \text{ hours} $$

• Verify: Together, Sarah and Tom will finish painting the fence in approximately 1.875 hours.

Conclusion

In this lesson, students has learned how to systematically approach word problems by understanding their structure, translating verbal instructions into mathematical equations, and establishing a clear solution framework. With this knowledge in hand, solving word problems becomes a manageable and methodical process.

Study Notes

• Understand the components of word problems: variables, relationships, and units.
• Familiarize yourself with common mathematical phrases and their symbolic representations.
• Follow a systematic process for translating problems into equations.
• Always verify that your numerical solutions address the problem's question.
• Practice regularly with diverse word problems to enhance your skills.