Sideways ratio
Weekly Problem 33 - 2014
A rectangle with area $125\text{cm}^2$ has sides in the ratio $4:5$. What is the perimeter of the rectangle?
A rectangle with area $125\text{cm}^2$ has sides in the ratio $4:5$. What is the perimeter of the rectangle?
Age
11 to 14
Challenge level
Problem
A rectangle with area $125\;\mathrm{cm}^2$ has sides in the ratio $4:5$.
What is the perimeter of the rectangle?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Let the sides of the rectangle, in $\mathrm{cm}$, be $4x$ and $5x$.
Then the area of the square is $4x \times 5x\;\mathrm{cm}^2 = 20x^2\;\mathrm{cm}^2$. So $20x^2 = 125$, that is $x^2 = \frac{25}{4}$. Therefore $x = \pm\frac{5}{2}$, but $x$ cannot be negative so $x = \frac{5}{2}$ and so the sides of the rectangle are $10\;\mathrm{cm}$ and $12.5\;\mathrm{cm}$.
Hence the rectangle has perimeter $45\;\mathrm{cm}$.