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Proportional Reasoning-2 - Ratios in Maps

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A map scale is a ratio that compares the distance on a map to the actual distance on the ground.

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Map scales are typically expressed as 1:n1 : n, where 11 unit on the map represents nn units in real life.

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Maps are drawn in direct proportion. This means the ratio Map DistanceActual Distance\frac{\text{Map Distance}}{\text{Actual Distance}} remains constant for any two points on the map.

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When calculating scales, ensure that both quantities are in the same units. Use the conversion 1 km=1,00,000 cm1 \text{ km} = 1,00,000 \text{ cm} or 1 m=100 cm1 \text{ m} = 100 \text{ cm}.

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A 'large scale' map (e.g., 1:1001:100) shows a small area in great detail, while a 'small scale' map (e.g., 1:1,000,0001:1,000,000) shows a large area with less detail.

📐Formulae

Scale=Distance on MapActual Distance\text{Scale} = \frac{\text{Distance on Map}}{\text{Actual Distance}}

Actual Distance=Map Distance×Scale Factor\text{Actual Distance} = \text{Map Distance} \times \text{Scale Factor}

Map Distance=Actual DistanceScale Factor\text{Map Distance} = \frac{\text{Actual Distance}}{\text{Scale Factor}}

1 km=105 cm1 \text{ km} = 10^5 \text{ cm}

💡Examples

Problem 1:

On a map, the scale is given as 1:3,00,0001 : 3,00,000. If the distance between two cities on the map is 5 cm5 \text{ cm}, find the actual distance between them in kilometers.

Solution:

Given: Scale =1:3,00,000= 1 : 3,00,000 Map distance =5 cm= 5 \text{ cm}

Actual distance in cm: 5×3,00,000=15,00,000 cm5 \times 3,00,000 = 15,00,000 \text{ cm}

Convert cm to km: Actual Distance=15,00,0001,00,000 km=15 km\text{Actual Distance} = \frac{15,00,000}{1,00,000} \text{ km} = 15 \text{ km}

Explanation:

Since 1 cm1 \text{ cm} on the map represents 3,00,000 cm3,00,000 \text{ cm} in reality, we multiply the map distance by the scale factor. Finally, we divide by 1,00,0001,00,000 to convert the answer from centimeters to kilometers.

Problem 2:

The actual distance between two points is 120 km120 \text{ km}. If these points are represented 4 cm4 \text{ cm} apart on a map, find the scale of the map.

Solution:

Map Distance =4 cm= 4 \text{ cm} Actual Distance =120 km=120×1,00,000 cm=1,20,00,000 cm= 120 \text{ km} = 120 \times 1,00,000 \text{ cm} = 1,20,00,000 \text{ cm}

Scale=Map DistanceActual Distance\text{Scale} = \frac{\text{Map Distance}}{\text{Actual Distance}} Scale=41,20,00,000=130,00,000\text{Scale} = \frac{4}{1,20,00,000} = \frac{1}{30,00,000}

The scale is 1:30,00,0001 : 30,00,000.

Explanation:

To find the scale, we first convert the actual distance into the same units as the map distance (cm). Then, we simplify the ratio so that the numerator (map part) is 11.

Problem 3:

A blueprint of a building uses a scale of 1:2001 : 200. If the actual length of a room is 10 m10 \text{ m}, what is its length on the blueprint in centimeters?

Solution:

Actual Length =10 m=10×100 cm=1000 cm= 10 \text{ m} = 10 \times 100 \text{ cm} = 1000 \text{ cm} Scale =1:200= 1 : 200

Blueprint Length=Actual LengthScale Factor\text{Blueprint Length} = \frac{\text{Actual Length}}{\text{Scale Factor}} Blueprint Length=1000200=5 cm\text{Blueprint Length} = \frac{1000}{200} = 5 \text{ cm}

Explanation:

We first convert the actual length to centimeters. Using the scale, we divide the actual length by the scale factor (200200) to find the corresponding length on the drawing.