Question

If the length of a rectangle is increased by one- third and its width is decreased by one third, then find the percentage decrease in its area.

Answer 1

Hint: For this we first let the length and breadth of a rectangle as ‘x’ and ‘y’ and then calculate its area. Then we find new length and breadth and so new areas. Then to find the percentage we divide the difference of areas so obtained by area of given rectangle and multiplying this ratio by $ 100 $ to get required percentage decrease in area.
Formulas used: area of rectangle = $ Length \times Breadth $

Complete step-by-step answer:
Let length of a rectangle = x
Breadth of a rectangle = y
Then area of a rectangle = xy
New length of a rectangle = $ x + \dfrac{1}{3}x $
Therefore, new length of a rectangle is = $ \dfrac{4}{3}x $
New, breadth of a rectangle = $ y - \dfrac{1}{3}y $
Therefore, new breadth of a rectangle = $ \dfrac{2}{3}y $
Area of new rectangle formed by new length and breadth is = $ \left( {\dfrac{4}{3}x} \right) \times \left( {\dfrac{2}{3}y} \right) $
Therefore, new area of a rectangle = $ \dfrac{8}{9}xy $
Now, to see whether an area of a rectangle is increased or decreased by changing its length and breadth can be calculated by finding the difference of two areas obtained above. We have,
Change in area = $ New\,\,area\, - \,given\,\,area $
 $
   \Rightarrow \,change\,\,in\,\,area\, = \dfrac{8}{9}xy - xy \\
   \Rightarrow \,change\,\,in\,\,area\, = \dfrac{{8xy - 9xy}}{9} \\
   \Rightarrow \,change\,\,in\,\,area\, = - \dfrac{{xy}}{9} \;
  $
Therefore, from above we see that there is a decrease in area.
Now, to find a percentage decrease in area. We divide change in area by given area to find ratio and then multiplying this ratio with $ 100 $ to get its percentage.
 $
   \Rightarrow \,Percentage\,\,decrease\,\,in\,\,area\,\, = \dfrac{{decrease\,\,in\,\,area}}{{original\,\,area}} \times 100 \\
   = \dfrac{{\dfrac{{xy}}{9}}}{{xy}} \times 100 \\
   = \dfrac{{100}}{9} \;
   = 11.11\% ;
  $

Hence, from above we see that there is a decrease of $ 11.11\% $ of area when length is increased by one third and breadth is decreased by one third.
So, the correct answer is “11.11%”.

Note: To find percentage decrease in area. We first calculate the area of the rectangle by taking length and breadth as ‘x’ and ‘y’ and then find the new length and breadth from the given and so new area of the rectangle. Then dividing the difference of areas by the original area of the rectangle and multiplying the result with $ 100 $ to get the required percentage.