A surveyor measures a distance between two points on a map of representative fraction of 1:100 is 60 m. But later he found that he used a wrong representative fraction of 1:50. What is the correct distance between the two points?
📝 Detailed Explanation: Correcting Measurements with Wrong Scales
This problem deals with a common error in surveying where a measurement is calculated using an incorrect scale or Representative Fraction (R.F.). Understanding the relationship between map distance, ground distance, and R.F. is key to solving it.
Principle 1: Understanding Representative Fraction (R.F.)
The R.F. is the ratio of the distance on the map to the corresponding distance on the ground. It's a unitless fraction.
R.F. = Distance on Map / Distance on Ground
For example, an R.F. of 1:100 means 1 unit on the map represents 100 units on the ground (e.g., 1 cm = 100 cm, or 1 m = 100 m).
Principle 2: The Constant Factor - Actual Map Distance
The most important concept here is that the physical distance measured on the paper map is a fixed value. The error is not in the physical measurement on the paper, but in the *calculation* that converted it to a ground distance. Our goal is to find this fixed map distance and then re-calculate the ground distance with the correct scale.
⚙️ Solution Method 1: The Step-by-Step Approach
This method involves finding the actual distance on the map first and then using the correct scale to find the true ground distance.
- Step 1: Find the actual distance measured on the map.
- Step 2: Calculate the correct ground distance using the correct R.F.
We use the incorrect information to work backward. The surveyor used the wrong R.F. (1:50) to get a ground distance of 60 m.
Map Distance = Ground Distance × R.F.
Map Distance = 60 m × (1/50) = 1.2 m
This means the actual physical line drawn between the two points on the map sheet was 1.2 meters long.
Now, we use this actual map distance (1.2 m) with the R.F. that *should* have been used (1:100).
Correct Ground Distance = Map Distance / R.F.
Correct Ground Distance = 1.2 m / (1/100)
Correct Ground Distance = 1.2 m × 100 = 120 m
⚙️ Solution Method 2: The Proportionality Formula
A quicker way to solve this is by using a direct relationship. Since the map distance is constant, we can set up a proportion.
Correct Ground Length × Correct R.F. = Incorrect Ground Length × Incorrect R.F.
Let 'C' be the Correct Ground Length.
C × (1/100) = 60 m × (1/50)
C / 100 = 60 / 50
C = (60 / 50) × 100
C = 1.2 × 100
C = 120 m
Logical Check: Understanding the Scales
A scale of 1:50 is a larger scale than 1:100 (it shows features as twice as large). Because the surveyor used a larger scale than intended, the initial calculation of 60 m was an underestimate. The correct answer must be larger than 60 m, which makes 120 m the logical choice.