For a scale of 1 cm = 2.5 km the representative fraction is:
Correct Answer:
D. 1: 250000
📝 Detailed Explanation: Converting Engineer's Scale to R.F.
This question involves converting a stated scale, often called an "Engineer's Scale," into a Representative Fraction (R.F.). An Engineer's Scale relates a map unit to a ground unit, often in different units (e.g., cm to km). An R.F. is a pure, unitless ratio where both sides represent the same unit.
The cardinal rule for conversion is: Convert both sides of the scale to the same unit. The simplest approach is usually to convert the larger unit (km) to the smaller unit (cm).
Step-by-Step Calculation
- Start with the given scale:
1 cm = 2.5 km - Convert kilometers (km) to centimeters (cm):
- We know 1 km = 1000 meters.
- We know 1 meter = 100 centimeters.
- Therefore, 1 km = 1000 × 100 = 100,000 cm. - Apply the conversion to the scale:
2.5 km = 2.5 × 100,000 cm = 250,000 cm. - Rewrite the scale with the same units:
1 cm = 250,000 cm - Express as a Representative Fraction (R.F.):
Since the units are now the same, we can write it as a ratio: 1 : 250,000
⚗️ Formulas & Applications
The core principle is unit consistency. The formula is simply the ratio of the map distance to the ground distance in the same units.
R.F. = (Map Distance) / (Ground Distance in same units)
Practical Example
Convert the scale "1 inch = 2 miles" to R.F.
- Step 1: Convert miles to inches.
- 1 mile = 5280 feet
- 1 foot = 12 inches
- 1 mile = 5280 × 12 = 63,360 inches
- Step 2: Apply to the scale.
- 2 miles = 2 × 63,360 = 126,720 inches
- Step 3: State the R.F.
- The scale is 1 inch = 126,720 inches, so the R.F. is 1 : 126,720.
💡 Key Concepts for Students
- R.F. is Unitless: Always remember that a Representative Fraction is a pure ratio and has no units. This is why the conversion to a common unit is the most critical step.
- Memorize Key Conversions: Knowing common metric (1 km = 100,000 cm) and imperial (1 mile = 63,360 inches) conversions is essential for speed and accuracy in exams.
- Scale Direction: An Engineer's Scale (1 cm = X km) and an R.F. (1 : Y) are just different ways of expressing the same relationship between the map and the real world.