The area of a rectangle is $252\text{ cm}^2$ and its length and breadth are in the ratio of 9 : 7 respectively. What is its perimeter? (Bank. P.O., 2009)

### Concept & Logic When dimensions are given as a ratio, introduce a common multiplier (like $x$) to express the exact length and breadth. Set up an equation using the area to solve for $x$. $$ \text{Area} = \text{Length} \times \text{Breadth} $$ ### Step-by-Step Solution * **Given:** Area = $252\text{ cm}^2$, Ratio of Length to Breadth = $9:7$. * **Step 1:** Express dimensions using variable $x$. Let Length ($L$) = $9x$ Let Breadth ($B$) = $7x$ * **Step 2:** Set up the area equation and solve for $x$. $9x \times 7x = 252$ $63x^2 = 252$ $x^2 = \frac{252}{63}$ $x^2 = 4$ $x = 2$ * **Step 3:** Find actual dimensions. $L = 9 \times 2 = 18\text{ cm}$ $B = 7 \times 2 = 14\text{ cm}$ * **Step 4:** Calculate the perimeter. $\text{Perimeter} = 2(L + B)$ $\text{Perimeter} = 2(18 + 14) = 2(32) = 64\text{ cm}$ ### Exam Strategy & Shortcut Notice the ratio sum is $9 + 7 = 16$. The semi-perimeter is $16x$, so the full perimeter is $32x$. This tells you immediately that the perimeter must be a multiple of $32$. Looking at the options ($64, 68, 96, 128$), $64, 96$, and $128$ are multiples of $32$. Testing $x=2$ gives a perimeter of $64$ and an area of $(9\times2)\times(7\times2) = 18 \times 14 = 252$, which matches perfectly. ### Common Pitfall A frequent error is forgetting that multiplying $9x$ and $7x$ yields an $x^2$ term, not just $x$. Solving $63x = 252 \implies x = 4$ will lead to completely incorrect dimensions. ### Final Answer Therefore, the correct answer is **$64\text{ cm}$**.