Question

A map is drawn to the scale of $1:10000$. Find the actual length if the length on the map is 10cm.

Answer 1

Hint:
We can take the actual distance between the points as x. Then we can take the ratio of length on the map to the actual length and equate it with the scale of the map. Then we can solve for x and convert into kilometres to get the required answer.

Complete step by step solution:
We are given that the scale of the map is $1:10000$. This means that 1 unit in the graph is equivalent to 10000 units in actual.
We need to find the actual distance between two cities which is 10cm apart in the map.
Let x be the actual distance between the two cities. So, we can write it as a ratio of distance on the map to the actual distance
 $ \Rightarrow 10:x$
This ratio will be proportional to the given scale of the map. So we can write them as,
 $ \Rightarrow \dfrac{1}{{10000}} = \dfrac{{10}}{x}$
On cross multiplying, we get,
 \[ \Rightarrow x = 10 \times 10,000\]
On multiplication we get,
 $ \Rightarrow x = 100,000$
As the distance on the map is 5cm, the unit of x will be in cm.
 $ \Rightarrow x = 100,000cm$
We know that $1cm = \dfrac{1}{{100}}m$ . So, x will become,
 $ \Rightarrow x = \dfrac{{100,000}}{{100}}m$
We also know that $1m = \dfrac{1}{{1000}}km$ . So, x will become,
 $ \Rightarrow x = \dfrac{{100,000}}{{100 \times 1000}}km$
On cancelling the zeros, we get,
 $ \Rightarrow x = 1km$

Therefore, the actual length is 1km.

Note:
Alternate solution to this problem is given by,
We are given that the scale of the map is $1:10000$. This means that 1 unit in the graph is equivalent to 10000 units in actual.
So 10cm in the map will be equal to 10 times 10000 cm.
 $ \Rightarrow d = 10 \times 10,000cm$
We can write the zeros as powers of 10.
 $ \Rightarrow d = {10^5}cm$
We know that $1cm = {10^{ - 2}}m$ and $1m = {10^{ - 3}}km$ . So $1cm = {10^{ - 5}}km$
So, the distance will become,
 $ \Rightarrow d = {10^5} \times {10^{ - 5}}km$
On cancelling the powers, we get,
 $ \Rightarrow d = 1km$
 Therefore, the actual length is 1km.