Question

The length of rectangle R is 10 percent more than the side of Square S. The width of the rectangle is 10 percent less than the side of the square. The ratio of the areas, R: S, is:
A. 99:100
B. 101:100
C. 1:1
D. 199:200
E. 201:200

Answer 1

Hint: Let the side of the square be a unit. Then the length of the rectangle is $ a + 10\% ofa $ units and the breadth (width) of the rectangle is $ a - 10\% ofa $ units. Find the length and breadth of the rectangle in terms of a. Then find the area of the rectangle and area of square and calculate the ratio of areas.
Formulas used:
Area of a rectangle with length l and breadth b is $ l \times b $
Area of a square with side a is $ {a^2} $

Complete step by step solution:
We are given that the length of rectangle R is 10 percent more and the width (breadth) of the rectangle is 10 percent less than the side of the square S.
We have to find the ratio of their areas.
Let the side of the square be a unit.
Then the length of the rectangle is 10 percent more than a,
 $ \Rightarrow a + 10\% ofa = a + \dfrac{{10}}{{100}}a = a + 0.1a = 1.1a $ units.
Breadth of the rectangle is 10 percent less than a, $ a - 10\% ofa = a - \dfrac{{10}}{{100}}a = a - 0.1a = 0.9a $ units.

Area of the square is $ {a^2} sq.units $
Area of the rectangle is $ 1.1a \times 0.9a = 0.99{a^2} sq.units $
Ratio of the area of rectangle to the area of square is
 $ \Rightarrow \dfrac{R}{S} = \dfrac{{0.99{a^2}}}{{{a^2}}} = \dfrac{{0.99}}{1} = \dfrac{{99}}{{100}} = 99:100 $
The ratio of the areas, R:S, is $ 99:100 $ .
So, the correct answer is “Option A”.

Note: Length and breadth of a rectangle are not equal, only opposite sides are equal. When length and breadth (all the sides) are equal, then the rectangle is considered as a square. Rectangles and squares are called equiangular quadrilaterals, which mean they both have equal angles. All squares are rectangles, but all rectangles are not squares.