Question

How many rectangles can make with a $24$ cm long string with integral sides and what are the sides of those rectangles in cm?

Answer 1

Hint: In this particular sum we are given that the length of the string is $24$cm. From this, we can say that the perimeter of the rectangle would be $24$cm. Thus we know that the perimeter of a rectangle is given by the formula $2(a + b) = P$. Where P is the perimeter of the rectangle and $a \& b$ is the sides of the rectangle. In this particular sum, we can see that the perimeter is double the sum of the sides of the rectangle. Since the perimeter is $24$cm, the sum of the sides of a rectangle is $12$cm. Therefore we need to find the number of rectangles possible whose sum of the sides is $12$.

Complete step by step solution:
It is given that the length of the string is $24$cm. Thus we can say that the perimeter of the Rectangle has to be $24$cm.
Formula for the perimeter of a rectangle is given by $2(a + b) = P$.
$\therefore 2(a + b) = 24$
Thus we can say that $(a + b) = 12$
We can start with $a = 6,b = 6$. Though we know that it is a square, it is said that every square is a rectangle as it is a special case.
Similarly increasing the value of $a$ by 1 we find other values. Following are the other sides of the rectangle
$
  a = 7,b = 5 \\
  a = 8,b = 4 \\
  a = 9,b = 3 \\
  a = 10,b = 2 \\
  a = 11,b = 1 \\
$

Thus all we can say $6$ rectangles are possible and following are those rectangles:
$
  1. 6,6,6,6 \\
  2.5,7,5,7 \\
  3.4,8,4,8 \\
  4.3,9,3,9 \\
  5.2,10,2,10 \\
  6.1,11,1,11 \\
$

Note: It is always to find the values of sides by equating it with the perimeter. Students should never use the area formula to calculate the sides. Even when the question is asked related to a circle, the student should use the formula of circumference to find the radius of the circle.