The image shows a problem being solved using the concept of "Ratio and Proportion," commonly applied in time, speed, and distance problems. Here's a breakdown of the approach used in solving the problem:

Problem Setup:

The problem involves a person who:

• Rests for 9 hours every day and walks for 15 hours.
• In another case, the person rests for 18 hours but walks for 6 hours.
• The total distance covered with slower walking is known, and the goal is to find how many days it would take to cover double that distance if the person walks faster.

Key Assumptions:

• The time for walking at a slower pace is set to $x$ days.
• There are different walking schedules: one where the person walks for fewer hours and one where they walk for more hours but rest more.

Step-by-Step Explanation:

1. Ratios of Walking Time and Speed:

   • They first calculate the working hours per day for both cases, using a ratio of walking to resting time.
   • For slower walking, the ratio of time is 6:15 (6 hours walking, 15 hours resting).
   • The ratio of walking speed is $2:1$, indicating that the person walks faster in the second case.

2. Setting Up Proportions:

   • The total distance is directly proportional to speed and time. In this case, they use the ratio of walking time and distance covered.
   • The equation $\frac{6}{15} : \frac{2}{1}$ is set up, relating the two scenarios.

3. Applying Proportions:

   • The goal is to determine how long it will take the person to cover twice the distance with faster walking.
   • They use the concept that distance is proportional to speed and time to create an equation that balances the ratios of walking time and speed.

4. Multiplying the Terms:

   • By setting up a direct proportional equation $50 : x$, they multiply both sides by the middle terms of the ratios. This leads to solving for $x$, which represents the number of days it will take at the faster pace to cover twice the distance.

5. Final Calculation:

   • The final equation solves for $x = 125$ days, meaning it will take 125 days for the person to cover double the distance when walking faster.

Methodology Used:

• Ratio and Proportion: The entire problem relies on setting up a proportion between time, speed, and distance and solving for the unknown quantity.
• Direct and Inverse Proportion: Depending on whether more distance or less speed is involved, they use direct and inverse proportionality rules.
• Cross Multiplication: The solution involves cross-multiplying terms in the proportion to find the unknown value.

Tip:

In such problems, always identify the variable you need to solve for (in this case, the number of days) and set up ratios carefully for time, speed, and distance.