Question

The difference between the circumference of a circle and the radius of the circle is 111 cm. The radius is :
1) 23 cm
2) 20 cm
3) 21 cm
4) None of these

Answer 1

Hint: Suppose \[r\]is the radius of the circle. It is known that circumference of the circle is the length of the boundary of the circle. We calculate the circumference of a circle by using the formula \[2\pi r\], where \[r\]is the radius. Form an equation corresponding to the given conditions. Substitute the value of \[\pi \]in the formed equation to find the value of \[r\].

Complete step-by-step answer:
In the question, we are given the relationship between the circumference and the radius of the circle.
Let the radius of the circle be \[r\] cm.
We know that the circumference of the circle is the length of the boundary of the circle.
We calculate the circumference of a circle by using the formula \[2\pi r\], where \[r\]is the radius.
It is given that the difference between the circumference of a circle and the radius of the circle is 111cm.
We can write it as, \[2\pi r - r = 111\]
By substituting the value \[\dfrac{{22}}{7}\] for \[\pi \]in the above equation, we get,
\[2\left( {\dfrac{{22}}{7}} \right)r - r = 111\]
Take \[r\]common from the given equation,
\[r\left( {2\left( {\dfrac{{22}}{7}} \right) - 1} \right) = 111\]
Now, we will solve the bracket by multiplying and taking the L.C.M
$
  r\left( {\dfrac{{44}}{7} - 1} \right) = 111 \\
  r\left( {\dfrac{{44 - 7}}{7}} \right) = 111 \\
  r\left( {\dfrac{{37}}{7}} \right) = 111 \\
$
We can now find the value of \[r\] by multiplying both sides by \[\dfrac{7}{{37}}\]
$
  r = 111\left( {\dfrac{7}{{37}}} \right) \\
  r = 3 \times 7 \\
  r = 21 \\
$
Hence, the value of radius is 21cm.
Hence, option (C) is the correct option.
Note: In this question, note that the formula used is \[2\pi r\] for the circumference of the circle. Use \[\pi = \dfrac{{22}}{7}\] to avoid tricky calculations. Also, the formation of the equation is important in this question. Students make mistakes by writing the equations as \[r - 2\pi r = 111\], which gives the wrong answer. After completing the calculation, it is important to mention the unit which is cm in this case.