Problem Statement
A surveyor measured the distance between two points on a plan and calculated the length to be equal to 650 m assuming the scale of the plan was 1 cm = 50 m. Later, it was discovered that the actual scale of the plan was 1 cm = 40 m. Find the true distance between the points.
Step-by-Step Solution
Key Information
- Measured Length (based on wrong scale) = 650 m
- Wrong Scale (assumed) = 1 cm = 50 m
- Correct Scale (actual) = 1 cm = 40 m
- Goal: Find the True Distance.
Step 1: Calculate Representative Fractions (R.F.)
The Representative Fraction (R.F.) is the ratio of map distance to ground distance in the same units.
Wrong Scale R.F.:
1 cm on map = 50 m on ground
1 cm on map = 50 * 100 cm on ground = 5000 cm
R.F. (Wrong) = 1 / 5000
Correct Scale R.F.:
1 cm on map = 40 m on ground
1 cm on map = 40 * 100 cm on ground = 4000 cm
R.F. (Correct) = 1 / 4000
Step 2: Apply the Correction Formula
The relationship between true length, measured length, and the scales is:
True Length / Correct Scale R.F. = Measured Length / Wrong Scale R.F.
Rearranging to find the True Length:
True Length = Measured Length * (Wrong Scale R.F. / Correct Scale R.F.)
Alternatively, using the scale factors directly:
True Length = Measured Length * (Wrong Scale Factor / Correct Scale Factor)
True Length = 650 m * (50 m/cm / 40 m/cm)
Or using the R.F. values:
True Length = 650 m * [(1/5000) / (1/4000)]
Step 3: Calculate the True Distance
Using the R.F. ratio:
True Length = 650 m * (4000 / 5000)
True Length = 650 m * (4 / 5)
True Length = (650 * 4) / 5 m
True Length = 2600 / 5 m
True Length = 520 m
Final Result
Conceptual Explanation & Applications
Core Concepts:
- Map Scale: The ratio between a distance on a map or plan and the corresponding distance on the ground.
- Representative Fraction (R.F.): A way of expressing scale as a ratio (e.g., 1:5000 or 1/5000), where both numerator and denominator are in the same units. A smaller denominator means a larger scale (more detail).
- Scale Correction: When a measurement is made using an incorrect scale assumption, the true distance can be found by adjusting the measured length based on the ratio of the correct scale to the incorrect scale (or the inverse ratio of their R.F.s).
- Proportional Reasoning: The measured length on the plan corresponds to a certain physical distance on the paper. This physical paper distance represents different ground distances depending on the scale used.
Real-World Applications:
- Surveying and Cartography: Essential for accurately interpreting distances and areas from maps and plans.
- Engineering and Architecture: Reading blueprints and scaled drawings correctly.
- Land Management and Planning: Calculating property dimensions or project areas from plans.
- Navigation: Using maps with stated scales to estimate travel distances.
Why It Works:
The calculation works because the physical measurement on the plan paper is constant. Let this physical length be ‘L_paper’.
Using the wrong scale (1 cm = 50 m): Measured Ground Length = L_paper * 50 m/cm = 650 m.
This means L_paper = 650 m / (50 m/cm) = 13 cm.
Now, using the correct scale (1 cm = 40 m) with the actual paper length:
True Ground Length = L_paper * 40 m/cm = 13 cm * 40 m/cm = 520 m.
The formula True Length = Measured Length * (Wrong Scale Factor / Correct Scale Factor) encapsulates this logic: 520 m = 650 m * (50 / 40). Since the correct scale (1 cm = 40 m) represents a shorter ground distance per cm compared to the wrong scale (1 cm = 50 m), the true distance is less than the incorrectly calculated distance.
