Question

The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

Hint:

For area division problems, the number of squares can also be found directly by multiplying the simplified division ratios:
$\frac{156}{12} = 13$ squares along the width, and $\frac{216}{12} = 18$ squares along the height.
Total squares $= 13 \times 18 = 234$.

Answer 1

Step 1: Understanding the Question:
We are given a rectangular window of size $156\text{ cm}$ by $216\text{ cm}$.
Arjun wants to cover the window with complete squares of the maximum possible size.
To find the maximum side length of such squares, we must find the Highest Common Factor (H.C.F.) of the two dimensions.
Once the side length is found, the total number of squares is the total window area divided by the area of one square.

Step 2: Key Formula or Approach:
- Side length of the square, $s = \text{H.C.F.}(156, 216)$
- Total number of squares, $N = \frac{\text{Area of the window}}{\text{Area of one square}} = \frac{156 \times 216}{s^2}$

Step 3: Detailed Explanation:

• Find the prime factorization of 156 and 216:
\[ 156 = 2 \times 78 = 2^2 \times 39 = 2^2 \times 3 \times 13 \]
\[ 216 = 2 \times 108 = 2^2 \times 54 = 2^3 \times 27 = 2^3 \times 3^3 \]

• Find the H.C.F. by taking the lowest powers of the common prime factors:
\[ \text{H.C.F.}(156, 216) = 2^2 \times 3^1 = 4 \times 3 = 12 \]
Therefore, the maximum side length of the square is $12\text{ cm}$.

• Calculate the total number of squares formed:
\[ \text{Number of squares } (N) = \frac{156 \times 216}{12 \times 12} \]
\[ N = \left(\frac{156}{12}\right) \times \left(\frac{216}{12}\right) \]
\[ N = 13 \times 18 = 234 \]


Step 4: Final Answer:
The side length of each square is $12\text{ cm}$ and the total number of squares formed is $234$.