Question

Prachi starts from Barabanki at \(6 \mathrm{am}\) at constant speed of \(60 \mathrm{~km} / \mathrm{h}\). She halts at Lucknow for half an hour and then drives at \(40 \mathrm{~km} / \mathrm{h}\). If she reaches Kanpur at \(9: 30 \mathrm{am}\), which is \(160 \mathrm{~km}\) from Barabanki, how far is Barabanki from Lucknow? (a) \(75 \mathrm{~km}\) (b) \(80 \mathrm{~km}\) (c) \(100 \mathrm{~km}\) (d) \(120 \mathrm{~km}\)

Short answer

Question: Find the distance between Barabanki and Lucknow based on Prachi's journey. Solution: The distance between Barabanki and Lucknow is 120 km.

Step-by-step solution

Calculate the total time duration

The duration between \(6 \mathrm{am}\) (when Prachi starts from Barabanki) and \(9:30 \mathrm{am}\) (when she reaches Kanpur) is equal to \(3.5\) hours.

Exclude the halting time

Since Prachi halts for half an hour (0.5 hours) at Lucknow and then continues her journey, we need to remove this from the total time duration to calculate the actual time taken to travel. So, the total time spent driving is \(3.5 - 0.5 = 3\) hours.

Calculate the distance equation

Prachi drives the first leg of her journey at a constant speed of 60 km/h until Lucknow and the second leg at a constant speed of 40 km/h till Kanpur. Let's denote the distance between Barabanki and Lucknow as x km. Then, the distance between Lucknow and Kanpur would be 160-x km.

Use the formula of distance, speed and time

As we know: distance = speed × time For the first leg of the journey, let's express the time that Prachi spends driving in terms of distance and speed: \(\frac{x}{60}\) For the second leg, the equation would be: \(\frac{160-x}{40}\) Since the total time spent driving is 3 hours, we can write the equation as follows: \(\frac{x}{60}+\frac{160-x}{40}=3\)

Solve the equation

First, we need to find the least common multiple (LCM) of 60 and 40, which is 120. Then we will multiply each term in the equation by 120 to remove the fractions: \(x\cdot2+(160-x)\cdot3=120\cdot3\) Now we simplify the equation and solve for x: \(2x+480-3x=360\) Move the x terms to the left and the constants to the right, then solve for x: \(x=480-360\) \(x=120\)

Answer

The distance between Barabanki and Lucknow is 120 km (option d).

Key concepts

Quantitative Aptitude

Quantitative aptitude entails the ability to handle numerical data and use mathematical tools to make sense of it. In questions like Prachi's journey from Barabanki to Kanpur, this core concept is tested by presenting a real-world context where numbers represent speed, time, and distances. To excel in such problems, the student must:

• Understand the units used (in this case, km for distance and hours for time).
• Grasp the concept of speed as distance per unit of time.
• Apply arithmetic operations such as addition, subtraction, and working with fractions effectively.

One improvement tip for educators is to emphasize how to translate a word problem into numerical expressions and focus on unit conversions if necessary. Encourage practice with different scenarios to ensure students can adapt the techniques to various problems.

Problem Solving

Problem-solving skills come into play when students face a multi-step challenge like the journey problem in our example. It involves several stages: understanding the problem, formulating equations, and then solving them. A good problem solver will:

• Carefully read the problem to identify the key pieces of information.
• Break the problem into smaller, manageable parts (as done with the separate legs of Prachi's journey).
• Create equations based on the information provided and the relationship between distance, speed, and time.
• Apply logical reasoning to solve for unknown variables, maintaining patience and accuracy.

An effective teaching strategy is to guide students through the stages using clear examples, allowing them to gain confidence as they learn to tackle similar problems independently.

Mathematical Equations

Understanding and forming mathematical equations are vital in solving distance, speed, and time problems. The equation \(\frac{d}{s} = t\) connects the three variables, where distance (\(d\)) equals speed (\(s\)) multiplied by time (\(t\)). In the given exercise, this relationship helps us set up two key equations based on Prachi's speed during different parts of her journey. Here's how:

• Identify the variables and constants from the problem: \(x\) is the unknown distance between Barabanki and Lucknow, while speeds and total time are given.
• Set up equations that reflect the scenario, as seen in steps 3 and 4 of the solution.
• Manipulate and solve the resulting equations methodically (step 5 of the solution).

Teachers should emphasize the importance of clearly labeling variables and constants, and show the step-by-step process of forming and solving equations. Practice is beneficial, as is the encouragement to double-check work to prevent errors.