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The scale of a map is represented by representative fraction as 1 : 4000. The distance between two points A and B in the map is 10 cm. The distance AB on the ground is (in km units):
Correct Answer:
D. 0.4
📝 Detailed Explanation: Understanding Representative Fraction (R.F.)
The Representative Fraction (R.F.) is a unitless ratio that shows the relationship between a distance on a map and the corresponding distance on the ground. An R.F. of 1 : 4000 means that 1 unit of distance on the map represents 4000 of the *same units* on the ground.
The key to solving these problems is to ensure the units are consistent before performing the final conversion.
Step-by-Step Calculation
- Identify Given Data:
- Map Distance = 10 cm
- R.F. = 1 : 4000 or 1/4000 - Calculate Ground Distance in Same Units:
Using the R.F., the ground distance is 4000 times the map distance.
Ground Distance = Map Distance × 4000
Ground Distance = 10 cm × 4000 = 40,000 cm - Convert to Desired Units (km):
Now, we convert the ground distance from centimeters to kilometers.
- To convert cm to meters, divide by 100: 40,000 cm / 100 = 400 m
- To convert meters to kilometers, divide by 1000: 400 m / 1000 = 0.4 km
⚗️ Formulas & Practical Example
Core Formulas
R.F. = Map Distance / Ground Distance
Ground Distance = Map Distance / R.F.
Map Distance = Ground Distance × R.F.
Ground Distance = Map Distance / R.F.
Map Distance = Ground Distance × R.F.
Example Problem
A city plan has a scale of 1 : 10,000. A new metro line on the plan measures 15 cm. What is the actual length of the metro line in kilometers?
- Step 1: Calculate Ground Distance in cm
Ground Distance = 15 cm / (1/10,000) = 15 cm × 10,000 = 150,000 cm - Step 2: Convert to km
150,000 cm ÷ 100 = 1,500 m
1,500 m ÷ 1000 = 1.5 km
💡 Key Concepts for Students
- Unit Consistency: The R.F. is a ratio. The most common mistake is forgetting that the units must be the same on both sides of the ratio during the initial calculation.
- Unit Conversion: Be confident in your metric conversions: 1 km = 1000 m, and 1 m = 100 cm. Therefore, 1 km = 100,000 cm.
- Scale Interpretation: A larger number in the denominator of the R.F. (e.g., 1:50,000) means a smaller scale map (less detail, larger area), while a smaller number (e.g., 1:1,000) means a larger scale map (more detail, smaller area).
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