Question 5

Map scale helps us convert distances on a map to real-world distances.

Step 1 — Understanding Map Scales

We use maps to represent large areas on a small piece of paper. A map scale tells us the ratio between a distance on the map and the actual distance on the ground. For example, a scale of 1:2,000,000 means 1 unit on the map represents 2,000,000 units in reality.

Let us consider two different maps, Map A and Map B. Both maps show the same region. Map A has a scale of 1:2,000,000. This means 1 cm on Map A represents 2,000,000 cm in reality. Map B has a scale of 1:4,000,000. This means 1 cm on Map B represents 4,000,000 cm in reality.

Step 2 — Measuring Distances on Maps

Let us choose two cities, Delhi and Jaipur, for our measurement. We use a ruler to measure the straight-line distance between Delhi and Jaipur on each map.

On Map A, we measure the distance to be 13.5 cm. On Map B, we measure the distance to be 6.75 cm.

Notice that the map distance is smaller on Map B. This is because Map B has a larger scale factor, showing a larger area.

Step 3 — Calculating Actual Geographical Distances

We will now use the scale of each map to find the actual distance between Delhi and Jaipur. The formula for actual distance is: Actual Distance = Map Distance × Scale Factor.

For Map A: Let $D_A$ be the map distance on Map A. Let $S_A$ be the scale factor for Map A. We have $D_A = 13.5 \text{ cm}$. The scale is $1:2,000,000$, so $S_A = 2,000,000$.

The actual distance from Map A, let us call it $A_A$, is: $A_A = D_A \times S_A$ $A_A = 13.5 \text{ cm} \times 2,000,000$ $A_A = 27,000,000 \text{ cm}$ We convert centimeters to kilometers. We know that $1 \text{ km} = 100,000 \text{ cm}$. $A_A = \frac{27,000,000}{100,000} \text{ km}$

$\boxed{270 \text{ km}}$

For Map B: Let $D_B$ be the map distance on Map B. Let $S_B$ be the scale factor for Map B. We have $D_B = 6.75 \text{ cm}$. The scale is $1:4,000,000$, so $S_B = 4,000,000$.

The actual distance from Map B, let us call it $A_B$, is: $A_B = D_B \times S_B$ $A_B = 6.75 \text{ cm} \times 4,000,000$ $A_B = 27,000,000 \text{ cm}$ We convert centimeters to kilometers. $A_B = \frac{27,000,000}{100,000} \text{ km}$

$\boxed{270 \text{ km}}$

Step 4 — Comparing the Geographical Distances

We calculated the actual distance between Delhi and Jaipur using two different maps. From Map A, the actual distance is 270 km. From Map B, the actual distance is 270 km.

Answer

Yes, both maps give the same geographical distance, approximately. The map scale correctly adjusts for how distances are represented on each map. Any minor differences in real-world measurements would be due to small measurement errors on the map, not the scale itself.