Question

The radius of a circle is increased by \[1cm\] . Then ratio of new circumference to the new diameter is
A) \[\pi :3\]
B) \[\pi :2\]
C) \[\pi :1\]
D) \[\pi :\frac{1}{2}\]

Answer 1

Hint: From the given information we need to find the circumference and diameter of the new circle by considering the value of radius as any variable. To obtain the new radius we are required to add \[1cm\] to the variable considered. Consequently, we need to find the new circumference and diameter. And then the ratio can be found by dividing the new circumference with new diameter.

Complete step by step answer:
Let us consider the radius of the unchanged circle initially is \[r\]
The circumference of the circle with radius \[r\] can be taken as
\[C=2\pi r\]
The diameter of the circle with radius \[r\] is given by
\[d=2r\]
Given the radius is increased by \[1cm\]. The new radius will become
\[r+1\]
The circumference of new circle will be
\[{C}'=2\pi \left( r+1 \right)\]
The diameter of new circle will become
\[D=2\left( r+1 \right)\]
We need to find the ratio of new circumference and new diameter which is given by the expression
New Circumference: New Diameter
\[{C}':D\]
Substitute the values of new circumference and new diameter
\[2\pi \left( r+1 \right):2\left( r+1 \right)\]
Radius \[\left( r+1 \right)\] is common on both sides of the ratio hence we can cancel it. After that we get
  \[ 2\pi :2 \]
 \[\Rightarrow \pi :1\]
Hence the ratio of new circumference and new diameter which is given by Option (C)
 \[\pi :1\]

Note: The ratio is increased by \[1cm\] in the sense that the radius if present in other units can be changed to centimetres or the given quantity can be added as decimal for the radius. And the circumference of a circle is equal to \[\pi \] times that of diameter which remains unchanged.