Question

The length of rectangle R is 10% more than the side of square S. The width of the rectangle is 10% less than the side of the square. What is the ratio of the areas, R and S?
(a) 99 : 100
(b) 101 : 100
(c) 1 : 1
(d) 199 : 200

Answer 1

Hint: Here, first of all we will assume the length of the side of the square to be x. Then, using the concept of percentage we will find the length and width of the rectangle in terms of x. Area of the rectangle is given as length multiplied by breadth and the area of square is given as the square of its side length.

Complete step-by-step solution -
Let us assume that the side of the square S is x.
Since, it is given that the length of rectangle R is 10% more than the side of square S. So, the increase in length of x is $\dfrac{10}{100}\times x$ .
Therefore, length of rectangle R is = $x+\dfrac{10}{100}\times x=\dfrac{100x+10x}{100}=\dfrac{110x}{100}=\dfrac{11x}{10}.........\left( 1 \right)$
Now, it is also given that the width of the rectangle is 10% less than the side of the square. So, the decrease in the length of the x is $\dfrac{10}{100}\times x$.

Therefore, width of the rectangle R is given as $x-\dfrac{10}{100}\times x=\dfrac{100x-10x}{100}=\dfrac{90x}{100}=\dfrac{9x}{10}.........\left( 2 \right)$
Now, area of rectangle R = length $\times $ breadth
On putting the values of length and breadth from equation (1) and (2), we get:
Area of rectangle R = $\dfrac{11x}{10}\times \dfrac{9x}{10}=\dfrac{99{{x}^{2}}}{100}$
Also, are of square S = ${{\left( side \right)}^{2}}={{x}^{2}}$
So, ratio of areas of R and S is given as:
$\dfrac{\text{area of rectangle}}{\text{area of square}}=\dfrac{\left( \dfrac{99{{x}^{2}}}{100} \right)}{{{x}^{2}}}=\dfrac{99}{100}$
So, the required ratio is 99 : 100.
Hence, option (a) is the correct answer.

Note: Students should note here that the length of the rectangle is 10% more than the side of the square, thus we add $\dfrac{10}{100}\times x$ whereas in case of width we subtracted $\dfrac{10}{100}\times x$ from x. Students should remember the correct formula for the area of rectangle and square.