Question

A man takes 4 h 20 minutes in walking to a certain place and riding back. If he walk on both sides he loses \(1 \mathrm{~h}\). The time he would take by riding both ways is : (a) \(2 \mathrm{~h} 20 \mathrm{~min}\) (b) \(3 \mathrm{~h} 20 \mathrm{~min}\) (c) \(2 \mathrm{~h}\) (d) 4 h 40 min

Short answer

Answer: 3h 20min

Step-by-step solution

Define Variables

Let's use W to represent the time taken to walk one way and R for the time taken to ride one way.

Write equations based on the given information

The man takes 4 hours and 20 minutes in walking to a certain place and riding back: \(W + R = 4\frac{1}{3}\) hours (equation 1) If he walks both ways, he loses 1 hour: \(W + W = R + 1\) (equation 2)

Solve the two equations for R and W

From equation 2, we get \(2W = R + 1\) Substitute this into equation 1: \(W + (2W - 1) = 4\frac{1}{3}\) \(3W - 1 = 4\frac{1}{3}\) Now solve for W: \(3W = 4\frac{1}{3} + 1 = 5\frac{1}{3}\) \(W = \frac{5\frac{1}{3}}{3} = \frac{16}{9}\) hours Now that we have W, we can find R: \(2W = R + 1\) \(R = 2W - 1 = 2\frac{16}{9} - 1 = \frac{14}{9}\) hours

Calculate the time he would take riding both ways

If he rides both ways, the total time is: \(R + R = 2R = 2\frac{14}{9} = \frac{28}{9} = 3\frac{1}{9}\) hours So the time he would take by riding both ways is \(3\mathrm{~h} 20\mathrm{~min}\). The correct answer is (b) \(3 \mathrm{~h} 20 \mathrm{~min}\).

Key concepts

Quantitative Aptitude

Quantitative aptitude encompasses the ability to perform calculations and solve problems involving numbers and data. In the context of time and distance problems, it includes understanding units of time, converting between these units, and applying arithmetic operations to find solutions. To solidify this concept, consider the exercise provided:

When confronted with a question where different modes of transport result in different travel times, one must convert the given times into a consistent unit. For instance, 4 hours 20 minutes would convert to 4 + 20/60 hours, or \(4 \frac{1}{3}\) hours. Such skills in handling fractions and mixed numbers are crucial. Furthermore, losing 1 hour in the context implies a time difference that must be accounted for in our calculations, such as in the given equation 2 (\(W + W = R + 1\)).

Applying quantitative aptitude in this scenario allows for setting up and manipulating algebraic equations to find the unknown variables, W for walking and R for riding, ultimately leading to the solution.

Algebraic Equations

The crux of solving time and distance problems often lies in setting up and solving algebraic equations. In our example, two key equations were formulated based on the given information. Algebra involves representing unknown quantities with variables, as with W and R here, and finding these unknowns by manipulating the equations.

The process includes:

• Identifying variables to represent unknown quantities.
• Declaring equations that represent the relationships between these variables based on the problem statement.
• Using algebraic methods to isolate variables and solve the equations.

In this exercise, substituting expressions for one variable into another equation allowed us to find their values. Once W was found, it was inserted back into another equation to find R. This algebraic manipulation ultimately led to determining the total riding time, showcasing the key role of algebra in solving real-world problems.

Problem-Solving Strategy

A clear problem-solving strategy simplifies complex problems by breaking them down into manageable steps. In the given exercise, the strategy included:

• Understanding the problem and determining what is being asked.
• Choosing appropriate variables to represent the unknowns.
• Translating the text into mathematical language, i.e., writing down equations.
• Solving the equations step-by-step.
• Interpreting the solution in the context of the problem to answer the question.

This structured approach helps prevent common mistakes and oversights. Especially crucial is the correct translation of the problem into equations. For example, the step that involved understanding that walking both ways adds an extra hour led to the equation \(W + W = R + 1\), which was pivotal in finding the solution. Encouraging students to develop a step-by-step strategy when facing mathematical problems is essential for effective problem-solving.