Question

What is the distance between two cities, 4.5 cm apart, on a map drawn to scale 1cm = 45km?

Answer 1

Hint: The question is based on the concept of proportionality. Use the property that length on the map is directly proportional to actual distance.

Complete step-by-step answer:

To start with the question, we let the length on the map be l cm, and the actual distance represented by l be d km.

Let us first try to interpret the question in mathematical terms. We know that the length on the map is directly proportional to the actual distance.

$length\text{ }on\text{ }map\text{ }\alpha \text{ }actual\text{ }dis\tan ce\text{.}$

$l\text{ }\alpha \text{ d}$

Now according to the definition of proportionality, we can represent the above statement as:
$l=kd$

In the above equation, k is the constant of proportionality.

Now according to the question 1cm on the map represents 45km of actual distance. Representing it mathematically using the above equation, we get

$l=kd$

$\Rightarrow 1=45k$

$\Rightarrow k=\dfrac{1}{45}\text{ cm/km}.............(i)$

Therefore, the value of the constant of proportionality is $\dfrac{1}{45}\text{ cm/km}$ .

We will now find the distance between the two cities using the given data that the distance on the map between the two cities is 4.5cm.

$l=kd$

On substituting the value of k from equation (i), we get

$l=\dfrac{d}{45}$

$\Rightarrow 4.5=\dfrac{d}{45}$

$\Rightarrow d=202.5Km$

Therefore, the actual distance between the two cities is equal to 202.5 km.

Note: In the question related to proportionality, the key thing is to find the constant of proportionality. Also, the unit of the constant of proportionality is very important and needs to be used wisely.