Question

Out of the following statements, the incorrect one is:
 $ \left( a \right) $ Doubling the length of a given rectangle doubles the area.
 $ \left( b \right) $ Doubling the altitude of a triangle doubles the area.
 $ \left( c \right) $ Doubling the radius of a given circle doubles the area.
 $ \left( d \right) $ Doubling the divisor of a fraction and dividing its numerator by $ 2 $ changes the quotient.

Answer 1

Hint: Taking one option after the other and verify with formula
Here, we are going to take the formula of area of rectangle and substitute the length as double and get the resultant area, then for the area of triangle substitute the height as double and get the resultant area and similarly we have to do for the two options and find the incorrect statement.

Complete step-by-step answer:
As we know that the area of the rectangle is given by $ Area = L \times B $ and here, $ L $ is the length and $ B $ is the width. So in the question, it is saying that the length becomes, $ 2L $ so the area will be equal to $ Are{a_{new}} = 2L \times B $ . So from this, we can see that the new area becomes double of the original area.
So, the option $ \left( a \right) $ is correct.

The area of the triangle is given by $ \dfrac{1}{2} \times Base \times Height $ . So, by doubling the altitude of the triangle we will get the new area as two times the original area.
 $
  Area = \dfrac{1}{2} \times Base \times \left( {2height} \right) \\
  Area = 2\left( {\dfrac{1}{2} \times Base \times height} \right) \;
 $
So, the option $ \left( b \right) $ is correct.

As we know the area of the circle is given by $ \pi {r^2} $ , since we can see that the area of the circle is directly proportional to the square of the radius. It gives us four times the original area
 $
  Area = \pi {\left( {2r} \right)^2} \\
  Area = 4\pi {r^2} \;
 $
So, the option $ \left( c \right) $ is incorrect.

The last option we have is given as doubling the divisor of a fraction and dividing its numerator by $ 2 $ changes the quotient. So, let’s check this one. So by substituting the values, and checking we will get the value of the quotient changed.
Let’s take our fraction as \[\dfrac{a}{b}\]
Now doing as mentioned in option D
 \[
  \dfrac{a}{b} \to \dfrac{{\dfrac{a}{2}}}{{2 \times b}} \\
   = \dfrac{1}{4} \times \dfrac{a}{b} \\
   = \dfrac{a}{{4b}} \;
\]
We saw that multiply and dividing 2 both on top and bottom gave us a different result.
So, the option $ \left( d \right) $ is correct.
Hence, the option $ \left( c \right) $ is incorrect.
So, the correct answer is “Option C”.

Note: For solving this type of question we should always go through the options one by one and check it. To make it easy and answer firstly we can also check by substituting the values according to the question and then comparing it. We will get the result. So in this way, we can solve this type of question.