In a straight line, 37 trees are planted at equal spacing. A car takes 20 seconds to travel from the first tree to the 13th tree. How many additional seconds will the car take to reach the last (37th) tree from the first tree?

Introduction / Context:
This problem focuses on constant speed motion over equally spaced points. The key idea is that if distance between consecutive points is equal and the speed of the car is constant, the travel time is directly proportional to the number of gaps between those points. Counting gaps instead of points is a crucial skill for such questions.

Given Data / Assumptions:
There are 37 trees in a straight line.
The distance between any two consecutive trees is the same.
The car takes 20 seconds to go from the first tree to the 13th tree.
The car continues with the same constant speed and direction.
We need the additional time required to go from the 13th tree position to the last 37th tree, expressed as extra seconds beyond the initial 20 seconds.

Concept / Approach:
When consecutive points are equally spaced, the number of intervals (gaps) between them is one less than the number of points. Time taken is proportional to distance, and distance is proportional to the number of gaps. Therefore, we can compare the ratios of gaps to find the ratio of times. This removes the need to know the exact physical distance between trees or the actual speed of the car.

Step-by-Step Solution:
Step 1: From the first tree to the 13th tree, the car passes through 13 − 1 = 12 equal gaps.Step 2: Time for these 12 gaps is given as 20 seconds.Step 3: From the first tree to the 37th tree, the car passes through 37 − 1 = 36 equal gaps.Step 4: The ratio of total gaps is 36 / 12 = 3.Step 5: Since speed is constant, time ratio is the same as the gap ratio, so total time from first to 37th tree = 3 * 20 seconds.Step 6: Total time = 60 seconds.Step 7: The question asks how much more time is needed beyond the 20 seconds already taken to reach the 13th tree.Step 8: Extra time = 60 − 20 = 40 seconds.

Verification / Alternative check:
We can calculate the time per gap. Time for 12 gaps is 20 seconds, so time per gap = 20 / 12 seconds. For 36 gaps, time = 36 * (20 / 12) = 3 * 20 = 60 seconds. Subtracting the initial 20 seconds still gives 40 seconds. This confirms that the extra time calculation is consistent with the proportional reasoning method.

Why Other Options Are Wrong:
36 seconds corresponds to incorrectly assuming there are only 18 additional gaps or misusing the ratio of points instead of gaps.
57 seconds and 44 seconds come from arithmetic slips or from mixing up the total time with the extra time.
60 seconds would be the total time for the journey, not the additional time after the first 20 seconds, so it does not answer the question as stated.

Common Pitfalls:
A frequent mistake is to count the number of trees instead of the number of spaces between them. Another common error is to give the total time instead of the requested extra time. Carefully reading phrases like "how much more time" or "additional time" is essential. Also, some students try to assume a particular distance between trees, which is not needed and can cause confusion.

Final Answer:
The car takes an additional 40 seconds to reach the last tree from the 13th tree.