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: Find the H.C.F. of 156 and 216 using the Euclidean algorithm.
$216 = 1 \times 156 + 60$; $156 = 2 \times 60 + 36$; $60 = 1 \times 36 + 24$; $36 = 1 \times 24 + 12$; $24 = 2 \times 12 + 0$. The last nonzero remainder is $12$, so $\text{H.C.F.}(156,216) = 12$.
Step 2: This H.C.F. is the maximum square side length.
The largest square that can tile both dimensions exactly has side $12\text{ cm}$.
Step 3: Count the squares along each side and multiply.
Along the $156\text{ cm}$ side there are $156 \div 12 = 13$ squares, and along the $216\text{ cm}$ side there are $216 \div 12 = 18$ squares, so the total number of squares is $13 \times 18 = 234$.
\[ \boxed{\text{Side} = 12\text{ cm}, \ \text{Number of squares} = 234} \]