Proportional Reasoning - 2 | IT

Question 4

Using the map above, can you find the geographical distance between Bengaluru and Chennai? Also, find the geographical distance between Mangaluru and Chennai.

[Hint: Use a ruler to find the distance between the cities on the map. Then, use the ratio given on the map to find the actual geographical distance.]

Solution

Understand the Question

The map uses a Representative Fraction (RF) scale of $1 : 60,00,000$ , meaning $1\text{ unit}$ on the map represents $60,00,000\text{ units}$ on the ground.
To find actual geographical distances:
1. Convert the scale from centimeters to kilometers ( $1\text{ m} = 100\text{ cm}$ , $1\text{ km} = 1000\text{ m}$ ).
2. Measure the map distance between two cities with a ruler.
3. Multiply the measured map distance by the scale factor in $\text{km/cm}$ .

(i) Find the geographical distance between Bengaluru and Chennai.

Step 1 · Convert the Map Scale to km/cm

Given scale ratio is $1 : 60,00,000$ . Diagram 1

\begin{aligned} 1 \text{ cm on map} &= 60,00,000 \text{ cm in reality} \\[0.6em] &= \dfrac{60,00,000}{100} \text{ meters in reality} \\[0.6em] &= 60,000 \text{ meters in reality} \\[0.6em] &= \dfrac{60,000}{1000} \text{ kilometers in reality} \\[0.6em] &= 600 \text{ km in reality} \end{aligned}

$\text{Scale factor} = 600 \text{ km/cm}$

Step 2 · Calculate Distance between Bengaluru and Chennai

Measured map distance between Bengaluru and Chennai: $d_{\text{map1}} = 0.6 \text{ cm}$

Actual geographical distance:

\begin{aligned} D_{\text{actual1}} &= d_{\text{map1}} \times 600 \text{ km/cm} \\[0.6em] &= 0.6 \text{ cm} \times 600 \text{ km/cm} \\[0.6em] &= 360 \text{ km} \end{aligned}

Answer

(i) $360\text{ km}$

(ii) Find the geographical distance between Mangaluru and Chennai.

Step 1 · Calculate Distance between Mangaluru and Chennai

Measured map distance between Mangaluru and Chennai: $d_{\text{map2}} = 1.1 \text{ cm}$

Actual geographical distance:

\begin{aligned} D_{\text{actual2}} &= d_{\text{map2}} \times 600 \text{ km/cm} \\[0.6em] &= 1.1 \text{ cm} \times 600 \text{ km/cm} \\[0.6em] &= 660 \text{ km} \end{aligned}

Answer

(ii) $660\text{ km}$

Common Mistakes

Unit Conversion Errors: Forgetting that $1\text{ km} = 100{,}000\text{ cm}$ ( $100\text{ cm} \times 1000\text{ m}$ ), leading to incorrect division by $1000$ or $100$ only.
Ruler Measurement Errors: Misreading centimeters/millimeters when measuring straight-line distance between the city points on the map.

Question 4

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