Question

The length of one pair of opposite sides of a square is reduced by 10 % and that of the other pair is increased by 10 %. Find the ratio of the area of the new rectangle to the area of the original square.
(a) \[\dfrac{1}{100}\]
(b) \[\dfrac{49}{50}\]
(c) \[\dfrac{99}{100}\]
(d) \[\dfrac{1}{50}\]

Answer 1

Hint: To solve this question, we will assume that the side of the square is x. After changing the side’s length, one side will become 110 % of x and the other side will become 90 % of x. Now, we will calculate the area of this new rectangle by the formula: Area of rectangle = \[l\times b,\] where l and b are the length and breadth of the new rectangle formed. Finally, we will take the ratio of the areas to get the answer.

Complete step-by-step answer:
Now, it is given in the question that originally the square is present. Here, we are going to assume that the length of the side of the square is x. Now, we are going to calculate the area of this square. The area of the square with side ‘s’ is given by:
\[\text{Area }={{\left( \text{side} \right)}^{2}}={{\left( s \right)}^{2}}\]
Thus, we have,
\[\text{Area of square }={{x}^{2}}.....\left( i \right)\]
Now, the question says that one side of the square has increased by 10 %. Let this new side be denoted by \[{{x}_{1}}.\] So, we have,
\[{{x}_{1}}=\] original side + (10 % of original side)
\[\Rightarrow {{x}_{1}}=\text{originalside}+\left( \dfrac{10}{100}\times \text{ original side} \right)\]
\[\Rightarrow {{x}_{1}}=x+\dfrac{10x}{100}\]
\[\Rightarrow {{x}_{1}}=\dfrac{110x}{100}.....\left( ii \right)\]
Another thing given in the question is that the other side of the square is decreased by 10 %. Let this new side be denoted by \[{{x}_{2}}.\] So, we have,
\[{{x}_{2}}=\left( \left( 100-10 \right)\text{Percent of }x \right)\]
\[\Rightarrow {{x}_{2}}=\left( \text{90 Percent of }x \right)\]
\[\Rightarrow {{x}_{2}}=\dfrac{90}{100}\times x\]
\[\Rightarrow {{x}_{2}}=\dfrac{90x}{100}....\left( iii \right)\]
The new figure formed after doing these changes will be a rectangle because one side of the square has been increased and the other side has been decreased. Now, we have to calculate the area of the rectangle. The area of the rectangle with given length and breadth is,
\[\text{Area}=l\times b\]
Thus, the area of the rectangle in our case will be,
\[\text{Area of rectangle}={{x}_{1}}\times {{x}_{2}}.....\left( iv \right)\]
Now, we will put the values of \[{{x}_{1}}\text{ and }{{x}_{2}}\] from (ii) and (iii) into (iv). After doing this, we get,
\[\text{Area of rectangle}=\left( \dfrac{110x}{100} \right)\left( \dfrac{90x}{100} \right)\]
\[\text{Area of rectangle}=\dfrac{99{{x}^{2}}}{100}.....\left( v \right)\]
Now, we have to find the ratio. For this, we will divide (v) by (i). So, we get,
\[\dfrac{\text{Area of rectangle}}{\text{Area of square}}=\dfrac{\left( \dfrac{99{{x}^{2}}}{100} \right)}{{{x}^{2}}}=\dfrac{99{{x}^{2}}}{100{{x}^{2}}}=\dfrac{99}{100}\]
Hence, the option (c) is the right answer.

Note: We can solve this question alternately by the following method: Let us assume that the new rectangle formed has length = y and breadth = z. The side of the square is x. So, the ratio = \[\dfrac{{{y}{z}}}{{{x}^{2}}}=\left( \dfrac{y}{x} \right)\left( \dfrac{z}{x} \right).\] Now, we know that \[\left( \dfrac{y}{x} \right)=\dfrac{110}{100}\text{ and }\left( \dfrac{z}{x} \right)=\dfrac{90}{100}.\] So, we have
\[\text{Ratio }=\left( \dfrac{110}{100} \right)\left( \dfrac{90}{100} \right)=\dfrac{99}{100}\]