Question

In a race, the man who came two places ahead of the last man finished one place ahead of the man who came three places behind the man just ahead of the one who stood second. How many men finished the race? (a) 6 (b) 5 (c) 4 (d) 8

Short answer

Answer: 6 men finished the race.

Step-by-step solution

Understand the given conditions

Let's break down the given conditions: 1. The man who came two places ahead of the last man finished one place ahead of... 2. The man who came three places behind the man just ahead of the one who stood second. Let's denote the positions of the men involved in these conditions: - X: Two places ahead of the last man - Y: One place ahead of the man who came three places behind the man just ahead of the one who stood second - Z: The man just ahead of the one who stood second

Set up equations using the conditions

Based on the given conditions, we have: 1. X = L - 2 (Where L represents the last man's position) 2. Y = X + 1 3. Y = Z + 3 (Y is three places behind Z) Now we have three equations with three variables.

Test the multiple-choice options

Test each option and see which one satisfies all the given conditions: Option (a): 6 men - L = 6 - X = L - 2 = 4 - Y = X + 1 = 5 - Z = Y - 3 = 2 In this case, the given conditions are satisfied. Option (b): 5 men - L = 5 - X = L - 2 = 3 - Y = X + 1 = 4 - Z = Y - 3 = 1 In this case, the third condition is not satisfied, as the man just ahead of the one who stood second should have been 2. Option (c): 4 men - L = 4 - X = L - 2 = 2 - Y = X + 1 = 3 - Z = Y - 3 = 0 In this case, the third condition is not satisfied, as there should be at least one man ahead of the one who stood second. Option (d): 8 men - L = 8 - X = L - 2 = 6 - Y = X + 1 = 7 - Z = Y - 3 = 4 In this case, the third condition is not satisfied, as the man just ahead of the one who stood second should have been 2.

Choose the correct option

Out of all the options, only (a) 6 men satisfies all the given conditions. Therefore, the correct answer is (a) 6 men finished the race.

Key concepts

Problem Solving

Approaching a problem like this requires a clear understanding of the conditions and variables involved. Effective problem solving often starts with breaking down the problem into smaller, manageable pieces. You have to carefully read through the problem to extract all necessary information and identify relationships between the elements provided.

In this particular issue, you had to determine how many men finished the race based on layered clues. By defining variables for each specific condition, you could set up a framework to work within. This step is crucial as it aids in visualizing the solution path. Once you outline these points, you can proceed to solve tied logical equations which reveal hidden details that clarify the problem statement.

When faced with a complex problem:

• Break it down into smaller parts.
• Identify and define any variables.
• Use logical reasoning to understand relationships between different parts.
• Test potential solutions against the given conditions.

Mathematical Equations

Mathematical equations play a key role in translating word problems into manageable numeric expressions. In this race problem, setting equations based on the positions helps to systematically determine the correct answer. When you turn conditions into mathematical expressions, you're making the problem more structured and easy to solve step by step.

Here, you derived three equations from the relations: 1. X, representing the position of the man two places ahead of the last man, is formulated as \( X = L - 2 \). 2. Y is positioned as one place ahead of X, thus \( Y = X + 1 \). 3. Y is also three places behind Z, illustrated with \( Y = Z + 3 \).

You then sequentially tested each option against these equations to check which configuration of positions satisfies all assumptions. This systematic approach ensures that you consider all possibilities and identify the correct answer without assumptions or guesswork.

Critical Thinking

Critical thinking is essential in dissecting and solving complex word problems. It's not about making assumptions but about asking the right questions and evaluating every detail logically. In this race problem, critical thinking helps you look beyond the surface to identify how all the conditions interlock to determine the sequence of finishers.

Employing critical thinking means you actively question each part of the problem:

• What does each condition imply about the positions?
• How do these conditions connect to each other logically?
• Is there a specific order or sequence that emerges from combining the conditions?
• Do multiple scenarios exist, or is one arrangement consistent with all clues?

By critically analyzing the elements given, you avoid rushing to conclusions and instead work towards a coherent, informed solution. It's a careful and deliberate process to follow clues and ensure alignment with the stipulations.