What is the representative fraction for a scale of 10 cm = 20 km ?
📝 Detailed Explanation: Calculating Representative Fraction (R.F.)
A Representative Fraction (R.F.) is a way of expressing the scale of a map as a ratio. The most important rule for R.F. is that it must be a unitless ratio. This means the numerator (map distance) and the denominator (ground distance) must be in the same units before the fraction is simplified.
⚙️ Step-by-Step Calculation
Step 1: Convert Both Sides to the Same Unit
The given scale is 10 cm = 20 km. To create a unitless ratio, we must convert kilometers (km) to centimeters (cm). We will use the smallest unit (cm) for the conversion.
- 1 km = 1,000 meters (m)
- 1 m = 100 centimeters (cm)
First, convert 20 km to meters:
20 km × 1,000 m/km = 20,000 m
Next, convert 20,000 m to centimeters:
20,000 m × 100 cm/m = 2,000,000 cm
So, the scale is now: 10 cm = 2,000,000 cm
Step 2: Create and Simplify the Fraction
Now that both values are in centimeters, we can write the fraction.
R.F. = Map Distance / Ground Distance
R.F. = 10 cm / 2,000,000 cm
To simplify, we divide both the numerator and the denominator by 10:
R.F. = 1 / 200,000
Conclusion
The Representative Fraction is 1 / 200,000, which can also be written as 1 : 200,000. This means that 1 unit of length on the map represents 200,000 of the same units on the ground.