Understanding the Problem: Calculating Scale's Representative Fraction (R.F.)

This question asks us to find the Representative Fraction (R.F.) of a scale. We are given the actual area of a field and its corresponding dimensions (length and breadth) as shown on a map. The R.F. represents the ratio of a distance on the map to the corresponding distance on the ground.

Given Information

• Actual Area of the field = 33600 m²
• Map Length = 12 cm
• Map Breadth = 7 cm

Calculating Map Area

First, we calculate the area of the field as depicted on the map using the given map dimensions:

Map Area = Map Length $\times$ Map Breadth

Map Area = 12 cm $\times$ 7 cm = 84 cm²

Converting Ground Area Units

The actual area is given in square meters (m²), while the map dimensions are in centimeters (cm). To find the R.F., both distances or areas must be in the same units. We need to convert the actual ground area from square meters to square centimeters.

We know that 1 meter = 100 centimeters.

Therefore, 1 square meter (m²) = (100 cm)² = 10000 cm².

Now, we convert the actual area:

Actual Area = 33600 m² $\times$ 10000 cm²/m²

Actual Area = 336,000,000 cm²

Calculating the Representative Fraction (R.F.)

The R.F. is defined as the ratio of the area on the map to the corresponding area on the ground. Mathematically, this is represented as:

R.F.² = Area on Map / Area on Ground

R.F.² = (Map Area) / (Actual Area)

Substitute the calculated values:

R.F.² = 84 cm² / 336,000,000 cm²

R.F.² = 84 / 336,000,000

To simplify the fraction, we can divide both the numerator and the denominator by 84:

R.F.² = 1 / (336,000,000 / 84)

R.F.² = 1 / 4,000,000

Now, to find the R.F., we take the square root of both sides:

R.F. = $\sqrt{1 / 4,000,000}$

R.F. = 1 / $\sqrt{4,000,000}$

R.F. = 1 / 2000

Conclusion

The Representative Fraction (R.F.) of the scale is 1:2000. This means that 1 unit of measurement on the map represents 2000 of the same units on the ground.