On a certain map of India the actual distance of 1450 km between two cities Delhi and Kolkata is shown as 5 cm. What scale is used to draw the map? (A.T.M.A., 2004)

### Concept & Map Scales A map scale is a ratio of the distance on the map to the corresponding actual distance on the ground, expressed in the same units. $$ \text{Scale} = \text{Map Distance} : \text{Actual Distance} $$ ### Step-by-Step Solution * Given map distance = 5 cm. * Given actual distance = 1450 km. * Convert actual distance to centimeters to match the map units: * $1 \text{ km} = 1000 \text{ meters} = 100,000 \text{ cm} = 10^5 \text{ cm}$. * Actual distance in cm = $1450 \times 10^5$ cm. * Formulate the scale ratio: $5 : 1450 \times 10^5$. * Simplify by dividing both sides by 5: $1 : 290 \times 10^5$. * Adjust to standard scientific notation format shown in options: $290 \times 10^5 = 29 \times 10^6$. * The scale is $1 : 29 \times 10^6$. ### Exam Strategy & Shortcut Focus on the significant digits first. You need the ratio of $5$ to $1450$. $1450 \div 5 = 290$. Look at the options; only one contains the digits "29". You can safely select option (d) without perfectly calculating all the zeroes. ### Common Pitfall Failing to convert kilometers to centimeters correctly, often missing a zero ($10^4$ instead of $10^5$), leading to an incorrect power of 10 in the final ratio. ### Final Answer Therefore, the correct answer is **$1 : 29 \times 10^6$**.